Long ago, someone drew a triangle and three segments across it, each starting at a vertex and stopping at the midpoint of the opposite side. The segments met in a point. The person was impressed and repeated the experiment on a different shape of triangle. Again the segments met in a point. The person drew yet a third triangle, very carefully, with the same result. He told his friends. To their surprise and delight, the coincidence worked for them, too. Word spread, and the magic of the three segments was regarded as the work of a higher power.

Centuries passed, and someone proved that the three medians do indeed concur in a point, now called the centroid. The ancients found and proved other points, too, now called the incenter, circumcenter and orthocenter. More centuries passed, more special points were discovered, and a definition of triangle center emerged. Like the definition of continuous function, this definition is satisfied by infinitely many objects, of which only finitely many will ever be published. The Encyclopedia of Triangle Centers (ETC) extends a list of 400 triangle centers published in the book Triangle Centers and Central Triangles.

Last update: 10/19/2001 14:20:53

NOTATION AND COORDINATES

The reference triangle is ABC, with sidelengths a,b,c. Each triangle center P has homogeneous trilinear coordinates, or simply trilinears, of the form x : y : z. This means there is a nonzero function h of (a,b,c) such that

x = hx', y = hy', z = hz',

where x', y', z' are the directed distances from P to sidelines BC, CA, AB. Likewise, u : v : w are barycentrics if there is a function k of (a,b,c) such that

u = ku', v = kv', w = kw',

where u', v', w' are the signed areas of triangles PBC, PCA, PAB. Both coordinate systems are widely used; if trilinears for a point are x : y : z, then barycentrics are ax : by : cz.

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HOW TO USE ETC

You won't have to scroll down very far to find well known centers. Other named centers can be found using your computer's searcher - for example, search for "Nagel" to find "Nagel point" as X(8).

To determine if a possibly new center is already listed, click SEARCH at the top of this page. If you're unsure of a term, click GLOSSARY. For visual constructions of selected centers, click SKETCHES.

X(1) = INCENTER

Trilinears 1 : 1 : 1Barycentrics a : b : c

The point of concurrence of the interior angle bisectors of ABC; the point inside ABC whose distances from sidelines BC, CA, AB are equal. This equal distance, r, is the radius of the incircle, and is given by

r = 2*area(ABC)/(a + b + c).

Three more points are also equidistant from the sidelines; they are given by these names and trilinears:

A-excenter = -1 : 1 : 1, B-excenter = 1 : -1 : 1, C-excenter = 1 : 1 : -1.

The radii of the excircles are

2*area(ABC)/(-a + b + c), 2*area(ABC)/(a - b + c), 2*area(ABC)/(a + b - c).

Writing the A-exradius as r(a,b,c), the others are r(b,c,a) and r(c,a,b). If these exradii are abbreviated as ra, rb, rc, then 1/r = 1/ra + 1/rb + 1/rc. Moreover,

area(ABC) = sqrt(r*ra*rb*rc) and ra + rb + rc = r + 4R,

where R denotes the radius of the circumcircle.

The incenter is the identity of the group of triangle centers under "trilinear multiplication" defined by

(x : y : z)*(u : v : w) = xu : yv : zw.

A construction for * is easily obtained from the construction for "barycentric multiplication" mentioned in connection with X(2), just below.

The incenter and the other classical centers are discussed in these highly recommended books:

Nathan Altshiller Court, College Geometry, Barnes & Noble, 1952;Roger A. Johnson, Advanced Euclidean Geometry, Dover, 1960.

X(1) lies on these lines:2,8 3,35 4,33 5,11 6,9 7,20 19,28 21,31 24,1061 25,1036 29,92 30,79 32,172 39,291 41,101 49,215 60,110 61,203 62,202 71,579 75,86 76,350 82,560 84,221 87,192 88,100 90,155 99,741 102,108 104,109 142,277 147,150 163,293 164,258 167,174 168,173 181,970 182,983 185,296 188,361 190,537 196,207 201,212 224,377 229,267 256,511 257,385 281,282 289,363 312,1089 320,752 321,964 329,452 335,384 336,811 341,1050 364,365 376,553 378,1063 393,836 512,875 513,764 514,663 528,1086 561,718 564,1048 572,604 573,941 607,949 631,1000 647,1021 659,891 662,897 672,1002 689,719 727,932 731,789 748,756 761,825 765,1052 908,998 1037,1041 1053,1110

X(1) = midpoint between X(I) and X(J) for these (I,J): (7,390), (8,145)

X(1) = reflection of X(I) about X(J) for these (I,J): (8,10), (40,3), (46,56), (80,11), (100,214), (191,21), (267,229), (355,5), (484,36)

X(1) = isogonal conjugate of X(1)X(1) = isotomic conjugate of X(75)X(1) = inverse of X(36) in the circumcircleX(1) = inverse of X(80) in the Fuhrmann circleX(1) = complement of X(8)X(1) = anticomplement of X(10)X(1) = eigencenter of cevian triangle of X(I) for I = 1, 88, 162X(1) = eigencenter of anticevian triangle of X(I) for I = 1, 44, 513

X(1) = X(I)-Ceva conjugate of X(J) for these (I,J):(2,9), (4,46), (6,43), (7,57), (8,40), (9,165), (10,191), (21,3), (29,4), (75,63), (77,223), (80,484), (81,6), (82,31), (85,169), (86,2), (88,44), (92,19), (100,513), (104,36), (105,238), (174,173), (188,164), (220,170), (259,503), (266,361), (280,84), (366,364), (508,362)

X(1) = cevapoint of X(I) and X(J) for these (I,J):(2,192), (6,55), (11,523), (15,202), (16,203), (19,204), (31,48), (34,207), (37,42), (50,215), (56,221), (65,73), (244,513)

X(1) = X(I)-cross conjugate of X(J) for these (I,J):(2,87), (3,90), (6,57), (31,19), (33,282), (37,2), (38,75), (42,6), (44,88), (48,63), (55,9), (56,84), (58,267), (65,4), (73,3), (192,43), (221,40), (244,513), (259,258), (266,505), (354,7), (367,366), (500,35), (513,100), (517,80), (518,291)

X(1) = crosspoint of X(I) and X(J) for these (I,J):(2,7), (8,280), (21,29), (59,110), (75,92), (81,86)

X(1) = X(I)-Hirst inverse of X(J) for these (I,J):(2,239), (4,243), (6,238), (9,518), (19,240), (57,241), (105,294), (291,292).

X(1) = X(6)-line conjugate of X(44)

X(1) = X(I)-aleph conjugate of X(J) for these (I,J):(1,1), (2,63), (4,920), (21,411), (29,412), (88,88), (100,100), (162,162), (174,57),

(188,40), (190,190), (266,978), (365,43), (366,9), (507,173), (508,169), (513,1052), (651, 651), (653,653), (655,655), (658,658), (660,660), (662,662), (673,673), (771,771), (799,799), (823,823), (897,897)

Let X = X(1) and let V be the vector-sum XA + XB + XC; then V = X(8)X(1) = X(1)X(145).

Let W be the vector-sum XA' + XB' + XC', where A'B'C' is the pedal triangle of X;then W = X(65)X(1) = X(8)X(72).

X(2) = CENTROID

Trilinears 1/a : 1/b : 1/c= bc : ca : ab= csc A : csc B : csc C

Barycentrics 1 : 1 : 1

The point of concurrence of the medians of ABC, situated 1/3 of the distance from each vertex to the midpoint of the opposite side. More generally, if L is any line in the plane of ABC, then the distance from X(2) to L is the average of the distances from A, B, C to L. An idealized triangular sheet balances atop a pin head located at X(2) and also balances atop any knife-edge that passes through X(2). The triangles BXC, CXA, AXB have equal areas if and only if X = X(2).

X(2) is the centroid of the set of 3 vertices, the centroid of the triangle including its interior, but not the centroid of the triangle without its interior; that centroid is X(10).

X(2) is the identity of the group of triangle centers under "barycentric multiplication" defined by

(x : y : z)*(u : v : w) = xu : yv : zw.

A simple construction for * (and for square roots of points) is known:

Paul Yiu, "The uses of homogeneous barycentric coordinates in plane euclidean geometry," International Journal of Mathematical Education in Science and Technology, forthcoming.

A preprint can be downloaded from Paul Yiu's website.

X(2) lies on these lines:1,8 3,4 6,69 7,9 11,55 12,56 13,16 14,15 17,62 18,61 19,534 31,171 32,83 33,1040 34,1038 36,535 37,75 38,244 39,76 40,946 44,89 45,88 51,262 54,68 58,540 65,959 66,206 74,113 77,189 80,214 85,241 92,273 94,300 95,97 98,110 99,111 101,116 102,117 103,118 104,119 106,121 107,122 108,123 109,124 112,127 136,925 137,930 165,516 174,236 178,188 187,316 196,653 210,354 216,232 222,651 253,1073 254,847 261,593 271,1034 254,847 261,593 271,1034 272,284 280,318 283,580 290,327 292,334 294,949 308,702 311,570 314,941 319,1100 322,1108 330,1107 351,804 355,944

366,367 371,486 372,485 392,517 476,842 495,956 496,1058 514,1022 561,716 578,1092 647,850 650,693 668,1015 670,1084 689,733 743,789 799,873

X(2) = midpoint between X(I) and X(J) for these (I,J): (3,381), (4,376), (210,354)X(2) = reflection of X(I) about X(J) for these (I,J): (4,381), (20,376), (376,3), (381,5)X(2) = isogonal conjugate of X(6)X(2) = isotomic conjugate of X(2)X(2) = cyclocevian conjugate of X(4)X(2) = inverse of X(23) in the circumcircleX(2) = inverse of X(858) in the nine-point circleX(2) = inverse of X(110) in the Brocard circleX(2) = complement of X(2)X(2) = anticomplement of X(2)

X(2) = X(I)-Ceva conjugate of X(J) for these (I,J):(1,192), (4,193), (6,194), (7,145), (8,144), (69,20), (75,8), (76,69), (83,6), (85,7), (86,1), (87,330), (95,3), (98,385), (99,523), (190,514), (264,4), (274,75), (276, 264), (287,401), (290,511), (308,76), (312,329), (325,147), (333,63), (348,347), (491,487), (492,488), (523,148)

X(2) = cevapoint of X(I) and X(J) for these (I,J):(1,9), (3,6), (5,216), (10,37), (32,206), (39,141), (44,214), (57,223), (114,230), (115,523), (128,231), (132,232), (140,233), (188,236)

X(2) = X(I)-cross conjugate of X(J) for these (I,J):(1,7), (3,69), (4,253), (5,264), (6,4), (9,8), (10,75), (32,66), (37,1), (39,6), (44,80), (57,189), (75,330), (114,325), (140,95), (141,76), (142,85), (178,508), (187,67), (206,315), (214,320), (216,3), (223,329), (226,92), (230,98), (233,5), (281,280), (395,14), (396,13), (440,306), (511,290), (514,190), (523,99)

X(2) = crosspoint of X(I) and X(J) for these (I,J):(1,87), (75,85), (76,264), (83,308), (86,274), (95,276)

X(2) = X(I)-Hirst inverse of X(J) for these (I,J):(1,239), (3,401), (4,297), (6,385), (21,448), (27,447), (69,325), (75,350), (98,287), (115,148), (193,230), (291,335), (298,299), (449,452)

X(2) = X(3)-line conjugate of X(237) = X(316)-line conjugate of X(187)

X(2) = X(I)-aleph conjugate of X(J) for these (I,J):(1,1045), (2,191), (86,2), (174,1046), (333,20), (366,846)

X(2) is the unique point X (as a function of a,b,c) for which the vector-sum XA + XB + XC is the zero vector.

Let W be the vector-sum XA' + XB' + XC', where A'B'C' is the pedal triangle of X; then W = X(51)X(2).

X(3) = CIRCUMCENTER

Trilinears cos A : cos B : cos C= a(b2 + c2 - a2) : b(c2 + a2 - b2) : c(a2 + b2 - c2)

Barycentrics sin 2A : sin 2B : sin 2C

The point of concurrence of the perpendicular bisectors of the sides of ABC. The lengths of segments AX, BX, CX are equal if and only if X = X(3). This common distance is the radius of the circumcircle, which passes through vertices A,B,C. Called the circumradius, it is given by

R = a/(2 sin A) = abc/(4*area(ABC)).

X(3) lies on these lines:1,35 2,4 6,15 7,943 8,100 9,84 10,197 11,499 12,498 13,17 14,18 31,601 37,975 38,976 41,218 42,967 48,71 49,155 54,97 63,72 64,154 66,141 67,542 68,343 69,332 73,212 74,110 76,98 83,262 95,264 101,103 102,109 105,277 113,122 114,127 119,123 125,131 142,516 158,243 169,910 194,385 200,963 223,1035 225,1074 238,978 252,930 256,987 269,939 296,820 298,617 299,616 302,621 303,622 315,325 352,353 388,495 390,1058 395,398 396,397 476,477 485,590 486,615 489,492 490,491 496,497 525,878 595,995 618,635 619,636 623,629 624,630 639,641 640,642 662,1098 667,1083 691,842 847,925 901,953 934,972 960,997 1037,1066 1093,1105

X(3) = midpoint between X(I) and X(J) for these (I,J):(1,40), (2,376), (4,20), (22,378), (74,110), (98,99), (100,104), (101,103), (102,109), (476,477)

X(3) = reflection of X(I) about X(J) for these (I,J):(4,5), (5,140), (6,182), (52,389), (195,54), (265,125), (355,10), (381,2), (382,4), (399,110)

X(3) = isogonal conjugate of X(4)X(3) = isotomic conjugate of X(264)X(3) = inverse of X(5) in the orthocentric circleX(3) = complement of X(4)X(3) = anticomplement of X(5)X(3) = eigencenter of the medial triangleX(3) = eigencenter of the tangential triangle

X(3) = X(I)-Ceva conjugate of X(J) for these (I,J):(2,6), (4,155), (5,195), (21,1), (22,159), (30,399), (63,219), (69,394), (77,222), (95,2), (96,68), (99,525), (100,521), (110,520), (250, 110), (283,255)

X(3) = cevapoint of X(I) and X(J) for these (I,J):(6,154), (48,212), (55,198), (71,228), (185,417), (216,418)

X(3) = X(I)-cross conjugate of X(J) for these (I,J):(48,222), (55,268), (71,63), (73,1), (184,6), (185,4), (212,219), (216,2), (228,48), (520,110)

X(3) = crosspoint of X(I) and X(J) for these (I,J):(1,90), (2,69), (4,254), (21,283), (54,96), (59,100), (63,77), (78,271), (81,272), (95,97), (99,249), (110,250), (485,486)

X(3) = X(I)-Hirst inverse of X(J) for these (I,J): (2, 401), (4,450), (6,511), (21,416), (194, 385)X(3) = X(2)-line conjugate of X(468)X(3) = X(3I)-aleph conjugate of X(J) for these (I,J): (1,1046), (21,3), (188,191), (259,1045)

Let X = X(3) and let V be the vector-sum XA + XB + XC; then V = X(4)X(3) = X(382)X(4) = X(3)X(20) = X(355)X(40) = X(265)X(74) = X(52)X(185) = X(381)X(376) = X(146)X(399).

Let W be the vector-sum XA' + XB' + XC', where A'B'C' is the pedal triangle of X;then W = X(4)X(3) = X(382)X(4) = X(3)X(20) = X(355)X(40) = X(265)X(74) = X(52)X(185) = X(381)X(376) = X(146)X(399). These are the same vectors as in the preceding list; i.e., XA + XB + XC = XA' + XB' + XC'. It is easy to prove that the unique solution X of this equation is X(3).

X(4) = ORTHOCENTER

Trilinears sec A : sec B : sec CBarycentrics tan A : tan B : tan C

The point of concurrence of the altitudes of ABC. The orthocenter and the vertices A,B,C comprise an orthocentric system, defined as a set of four points, one of which is the orthocenter of the triangle of the other three (so that each is the orthocenter of the other three). The incenter and excenters also form an orthocentric system.

Suppose P is not on a sideline of ABC. Let A'B'C' be the cevian triangle of P. Let D,E,F be the circles having diameters AA', BB',CC'. The radical center of D,E,F is X(4). (Floor van Lamoen, Hyacinthos@onelist, Jan. 24, 2000.)

Ross Honsberger, Episodes in Nineteenth and Twentieth Century Euclidean Geometry, Mathematical Association of America, 1995. Chapter 2: The Orthocenter.

X(4) lies on these lines:1,33 2,3 6,53 7,273 8,72 9,10 11,56 12,55 13,61 14,62 15,17 16,18 32,98 35,498 36,499 39,232 46,90 49,156 51,185 52,68 54,184 57,84 65,158 67,338 69,76 74,107 78,908 83,182 93,562 94,143 96,231 99,114 100,119 101,118 102,124 103,116 109,117 110,113 128,930 131,135 137,933 145,149 147,148 150,152 155,254 162,270 171,601 195,399 218,294 238,602 240,256 276,327 371,485 372,486 390,495 487,489 488,490 496,999

512,879 542,576 575,598 616,627 617,628 801,1092 842,935 1036,1065 1037,1067 1038,1076 1039,1096 1040,1074

X(4) = midpoint between X(I) and X(J) for these (I,J):(3,382), (147,148), (149,153), (150,152)

X(4) = reflection of X(I) about X(J) for these (I,J):(2,381), (3,5), (8,355), (20,3), (24,235), (40,10), (74,125), (98,115), (99,114), (100,119), (101,118), (102,124), (103,116), (104,11), (107,133), (109,117), (110,113), (112,132), (185,389), (186,403), (376,2), (378,427)

X(4) = isogonal conjugate of X(3)X(4) = isotomic conjugate of X(69)X(4) = cyclocevian conjugate of X(2)X(4) = inverse of X(186) in the circumcircleX(4) = inverse of X(403) in the nine-point circleX(4) = complement of X(20)X(4) = anticomplement of X(3)X(4) = eigencenter of cevian triangle of X(i) for i = 1, 88, 162X(4) = eigencenter of anticevian triangle of X(i) for i = 1, 44, 513

X(4) = X(I)-Ceva conjugate of X(J) for these (I,J):(7,196), (27,19), (29,1), (92,281), (107,523), (264,2), (273,278), (275,6), (286,92), (393, 459)

X(4) = cevapoint of X(I) and X(J) for these (I,J):(1,46), (2,193), (3,155), (5,52), (6,25), (11,513), (19,33), (30,113), (34,208), (37,209), (39,211), (51,53), (65,225), (114,511), (115,512), (116,514), (117,515), (118,516), (119,517), (120,518), (121,519), (122,520), (123,521), (124,522), (125,523), (126,524), (127,525), (185,235), (371,372), (487,488)

X(4) = X(I)-cross conjugate of X(J) for these (I,J):(3,254), (6,2), (19,278), (25,393), (33,281), (51,6), (52,24), (65,1), (113,403), (125,523), (185,3), (193,459), (225,158), (389,54), (397,17), (398,18), (407,225), (427,264), (512,112), (513,108), (523,107)

X(4) = crosspoint of X(I) and X(J) for these (I,J): (2,253), (7,189), (27,286), (92,273)

X(4) = X(I)-Hirst inverse of X(J) for these (I,J):(1,243), (2,297), (3,350), (19,242), (21,425), (24,421), (25,419), (27,423), (28,422), (29,415), (420,427), (424,451), (459,460), (470,471)

X(4) = X(I)-aleph conjugate of X(J) for these (I,J): (1,1047), (29,4)

Let X = X(4) and let V be the vector-sum XA + XB + XC; then V = X(20)X(4) = X(3)X(382).

Let W be the vector-sum XA' + XB' + XC', where A'B'C' is the pedal triangle of X;then W = X(185)X(4) = X(52)X(382).

X(5) = NINE-POINT CENTER

Trilinears cos(B - C) : cos(C - A) : cos(A - B)= f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = cos A + 2 cos B cos C= g(a,b,c) : g(b,c,a): g(c,a,b), where g(a,b,c) = bc[a2(b2 + c2) - (b2 - c2)2]

Barycentrics a cos(B - C) : b cos(C - A) : c cos(A - B) = h(a,b,c) : h(b,c,a) : h(c,a,b), where h(a,b,c) = a2(b2 + c2) - (b2 - c2)2

The center of the nine-point circle. Euler showed in that this circle passes through the midpoints of the sides of ABC and the feet of the altitudes of ABC, hence six of the nine points. The other three are the midpoints of segments A-to-X(4), B-to-X(4), C-to-X(4). The radius of the nine-point circle is one-half the circumradius.

Dan Pedoe, Circles: A Mathematical View, Mathematical Association of America, 1995.

X(5) lies on these lines:1,11 2,3 6,68 10,517 13,18 14,17 32,230 33,1062 34,1060 39,114 49,54 51,52 53,216 55,498 56,499 72,908 76,262 83,98 113,125 116,118 117,124 122,133 127,132 128,137 129,130 131,136 141,211 142,971 156,184 182,206 183,315 226,912 264,1093 298,634 299,633 302,622 303,621 371,590 372,615 388,999 491,637 492,638 524,576 542,575 601,750 602,748 618,629 619,630 1090,1091

X(5) = midpoint between X(I) and X(J) for these (I,J):(1,355), (2,381), (3,4), (11,119), (20,382), (68,155), (110,265), (113,125), (114,115), (116,118), (117,124), (122,133), (127,132), (128,137), (129,130), (131,136)

X(5) = reflection of X(I) about X(J) for these (I,J): (3,140), (52,143)X(5) = isogonal conjugate of X(54)X(5) = isotomic conjugate of X(95)X(5) = inverse of X(3) in the orthocentroidal circleX(5) = complement of X(3)X(5) = anticomplement of X(140)X(5) = eigencenter of anticevian triangle of X(523)

X(5) = X(I)-Ceva conjugate of X(J) for these (I,J):(2,216), (4,52), (110,523), (264, 324), (265,30), (311,343), (324,53)

X(5) = cevapoint of X(I) and X(J) for these (I,J): (3,195), (51,216)

X(5) = X(I)-cross conjugate of X(J) for these (I,J): (51,53), (216,343), (233,2)X(5) = crosspoint of X(I) and X(J) for these (I,J): (2,264), (311,324)X(5) = X(1)-aleph conjugate of X(1048)

Let X = X(5) and let V be the vector-sum XA + XB + XC; then V = X(5)X(4) = X(3)X(5).

X(6) = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT)

Trilinears a : b : c= sin A : sin B : sin C

Barycentrics a2 : b2 : c2

The point of concurrence of the symmedians (reflections of medians about corresponding angle bisectors); the point (x, y, z), given here in actual trilinear distances, that minimizes x2 + y2 + z2.

Let A'B'C' be the pedal triangle of an arbitrary point X, and let S(X) be the vector sum XA' + XB' + XC'. Then

S(X) = (0 vector) if and only if X = X(6).

The "if" implication is equivalent to the well known fact that X(6) is the centroid of its pedal triangle, and the converse was proved by Barry Wolk on Hyacinthos@onelist.com,

Dec. 23, 1999.

Ross Honsberger, Episodes in Nineteenth and Twentieth Century Euclidean Geometry, Mathematical Association of America, 1995. Chapter 7: The Symmedian Point.

X(6) lies on these lines:1,9 2,69 3,15 4,53 5,68 7,294 8,594 13,14 17,18 19,34 21,941 22,251 23,353 24,54 25,51 26,143 31,42 33,204 36,609 40,380 41,48 43,87 57,222 64,185 66,427 67,125 74,112 75,239 76,83 77,241 88,89 98,262 99,729 100,739 101,106 105,1002 110,111 145,346 157,248 160,237 169,942 181,197 190,192 194,384 210,612 264,287 291,985 292,869 297,317 314,981 354,374 513,1024 517,998 519,996 523,879 561,720 598,671 603,1035 662,757 688,882 689,703 691,843 694,1084 717,789 750,899 753,825 755,827 840,919 846,1051 959,961 971,990 986,1046

X(6) = midpoint between X(69) and X(193)X(6) = reflection of X(I) about X(J) for these (I,J): (3,182), (67,125), (69,141), (159,206)X(6) = isogonal conjugate of X(2)X(6) = isotomic conjugate of X(76)X(6) = cyclocevian conjugate of X(1031)X(6) = inverse of X(187) in the circumcircleX(6) = inverse of X(115) in the orthocentroidal circleX(6) = complement of X(69)X(6) = anticomplement of X(141)

X(6) = X(I)-Ceva conjugate of X(J) for these (I,J):(1,55), (2,3), (3,154), (4,25), (8,197), (9,198), (10,199), (54,184), (57, 56), (58,31), (68,161), (69,159), (76,22), (81,1), (83,2), (88,36), (95,160), (98,237), (110,512), (111,187), (222,221), (249,110), (251,32), (264,157), (275,4), (284,48), (288,54), (323,399)

X(6) = cevapoint of X(I) and X(J) for these (I,J): (1,43), (2,194), (31,41), (32,184), (42,213), (51,217)

X(6) = X(I)-cross conjugate of X(J) for these (I,J):(25,64), (31,56), (32,25), (39,2), (41,55), (42,1), (51,4), (184,3), (187,111), (213,31), (217,184), (237,98), (512,110)

X(6) = crosspoint of X(I) and X(J) for these (I,J):(1,57), (2,4), (9,282), (54,275), (58,81), (83, 251), (110,249), (266,289)

X(6) = X(I)-Hirst inverse of X(J) for these (I,J): (1,238), (2,385), (3,511), (15,16), (25,232)X(6) = X(I)-line conjugate of X(J) for these (I,J): (1,518), (2,524), (3,511)X(6) = X(I)-aleph conjugate of X(J) for these (I,J): (1,846), (81,6), (365,1045), (366,191), (509,1046)

Let X = X(6) and let V be the vector-sum XA + XB + XC; then V = X(6)X(193) = X(69)X(6).

X(7) = GERGONNE POINT

Trilinears bc/(b + c - a) : ca/(c + a - b) : ab/(a + b - c)= sec2(A/2) : sec2(B/2) : sec2(C/2)

Barycentrics 1/(b + c - a) : 1/(c + a - b) : 1/(a + b - c)

Let A', B', C' denote the points in which the incircle meets the sides BC, CA, AB, respectively. The lines A'A', BB', CC' concur in X(7).

X(7) lies on these lines:1,20 2,9 3,943 4,273 6,294 8,65 11,658 21,56 27,81 37,241 33,1041 34,1039 58,272 72,443 80,150 92,189 100,1004 104,934 108,1013 109,675 171,983 174,234 177,555 190,344 192,335 193,239 218,277 225,969 253,280 256,982 274,959 281,653 286,331 310,314 354,479 513,885 517,1000 528,664 554,1082 594,599 840,927 987,1106

X(7) = reflection of X(I) about X(J) for these (I,J): (9,142), (144,9), (390,1)X(7) = isogonal conjugate of X(55)X(7) = isotomic conjugate of X(8)X(7) = cyclocevian conjugate of X(7)X(7) = complement of X(144)X(7) = anticomplement of X(9)X(7) = X(I)-Ceva conjugate of X(J) for these (I,J): (75,347), (85,2), (86,77), (286,273), (331,278)

X(7) = cevapoint of X(I) and X(J) for these (I,J):(1,57), (2,145), (4,196), (11,514), (55,218), (56,222), (63,224), (65,226), (81,229), (177,234)

X(7) = X(I)-cross conjugate of X(J) for these (I,J):(1,2), (4,189), (11,514), (55,277), (56,278), (57,279), (65,57), (177,174), (222,348), (226,85), (354,1), (497,8), (517,88)

X(7) = crosspoint of X(I) and X(J) for these (I,J): (75,309), (86,286)

X(8) = NAGEL POINT

Trilinears (b + c - a)/a : (c + a - b)/b : (a + b - c)/c= csc2(A/2) : csc2(B/2) : csc2(C/2)

Barycentrics b + c - a : c + a - b : a + b - c

Let A'B'C' be the points in which the A'-excircle meets BC, the B-excircle meets CA, and the C-excircle meets AB, respectively. The lines A'A', BB', CC' concur in X(8). Another construction of A' is to start at A and trace around ABC half its perimeter, and similarly for B' and C'. Also, X(8) is the incenter of the anticomplementary triangle.

X(8) lies on these lines:1,2 3,100 4,72 6,594 7,65 9,346 20,40 21,55 29,219 31,987 33,1039 34,1041 35,993 37,941 38,986 56,404 58,996 76,668 79,758 80,149 81,1010 144,516 177,556 178,236 181,959 190,528 192,256 193,894 194,730 210,312 213,981 220,294 221,651 224,914 238,983 253,307 274,1002 291,330 315,760 344,480 348,664 392,1000 405,943 406,1061 442,495 443,942 474,999 475,1063 599,1086 643,1098 860,1068 908,946 1016,1083

X(8) = reflection of X(I) about X(J) for these (I,J): (1,10), (4,355), (20,40), (145,1), (149,80), (390,9)X(8) = isogonal conjugate of X(56)X(8) = isotomic conjugate of X(7)X(8) = cyclocevian conjugate of X(189)X(8) = complement of X(145)X(8) = anticomplement of X(1)X(8) = X(I)-Ceva conjugate of X(J) for these (I,J): (69,329), (72,2), (312,346), (314,312), (333,9)

X(8) = X(I)-cross conjugate of X(J) for these (I,J):(1,280), (9,2), (10,318), (11,522), (55,281), (72,78), (200,346), (210,9), (219,345), (497,7), (521,100)

X(8) = cevapoint of X(I) and X(J) for these (I,J):(1,40), (2,144), (6,197), (9,200), (10,72), (11,522), (55,219), (175,176)

X(8) = crosspoint of X(I) and X(J) for these (I,J): (75,312), (314,333)X(8) = X(1)-alpeh conjugate of X(1050)

X(9) = MITTENPUNKT

Trilinears b + c - a : c + a - b : a + b - c= cot(A/2) : cot(B/2) : cot(C/2)

Barycentrics a(b + c - a) : b(c + a - b) : c(a + b - c)

The symmedian point of the excentral triangle.

X(9) lies on these lines:1,6 2,7 3,84 4,10 8,346 21,41 31,612 32,987 33,212 34,201 35,90 38,614 39,978 42,941 43,256 46,79 48,101 55,200 58,975 100,1005 164,168 165,910 173,177 192,239 223,1073 228,1011 241,269 261,645 312,314 342,653 348,738 364,366 374,517 478,1038 498,920 522,657 607,1039 608,1041 750,896

X(9) = midpoint between X(I) and X(J) for these (I,J): (7,144), (8,390)X(9) = reflection of X(7) about X(142)X(9) = isogonal conjugate of X(57)X(9) = isotomic conjugate of X(85)X(9) = complement of X(7)X(9) = anticomplement of X(142)X(9) = X(I)-Ceva conjugate of X(J) for these (I,J):(2,1), (8,200), (21,55), (63,40), (190,522), (312,78), (318,33), (333,8)

X(9) = cevapoint of X(I) and X(J) for these (I,J): (1,165), (6,198), (31,205), (37,71), (41,212), (55,220)

X(9) = X(I)-cross conjugate of X(J) for these (I,J):(6,282), (37,281), (41,33), (55,1), (71,219), (210,8), (212,78), (220,200)

X(9) = crosspoint of X(I) and X(J) for these (I,J): (2,8), (21,333), (63,271), (312,318)X(9) = X(I)-Hirst inverse of X(J) for these (I,J): (1, 518), (192,239)

X(9) = X(I)-aleph conjugate of X(J) for these (I,J):(1,43), (2,9), (9,170), (188,165), (190,1018), (366,1), (507,361), (508,57), (509,978)

X(10) = SPIEKER CENTER

Trilinears bc(b + c) : ca(c + a) : ab(a + b)Barycentrics b + c : c + a : a + b

The Spieker circle is the incircle of the medial triangle; its center, X(10), is the centroid of the perimeter of ABC.

X(10) lies on these lines:1,2 3,197 4,9 5,517 11,121 12,65 20,165 21,35 31,964 33,406 34,475 36,404 37,594 38,596 39,730 44,752 46,63 55,405 56,474 57,388 58,171 69,969 75,76 82,83 86,319 87,979 98,101 116,120 117,123 119,124 140,214 141,142 158,318 190,671 191,267

201,225 219,965 274,291 321,756 480,954 514,764 537,1086 626,760 631,944 775,801 894,1046 908,994

X(10) = midpoint between X(I) and X(J) for these (I,J): (1,8), (3,355), (4,40), (65,72), (80,100)X(10) = isogonal conjugate of X(58)X(10) = isotomic conjugate of X(86)X(10) = complement of X(1)

X(10) = X(I)-Ceva conjugate of X(J) for these (I,J):(2,37), (8,72), (75,321), (80,519), (100,522), (313,306)

X(10) = cevapoint of X(I) and X(J) for these (I,J):(1,191), (6,199), (12,201), (37,210), (42,71), (65,227)

X(10) = X(I)-cross conjugate of X(J) for these (I,J): (37,226), (71,306), (191,502), (201,72)X(10) = crosspoint of X(I) and X(J) for these (I,J): (2,75), (8,318)

X(11) = FEUERBACH POINT

Trilinears 1 - cos(B - C) : 1 - cos(C - A) : 1 - cos(A - B)= f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = sin2(B/2 - C/2)= g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = bc(b + c - a)(b - c)2

Barycentrics a(1 - cos(B - C)) : b(1 - cos(C - A)) : c(1 - cos(A - B))= h(a,b,c) : h(b,c,a) : h(c,a,b), where h(A,B,C) = (b + c - a)(b - c)2

The point of tangency of the nine-point circle and the incircle. The nine-point circle is the circumcenter of the medial triangle, as well as the orthic triangle. Feuerbach's famous theorem states that the nine-point circle is tangent to the incircle and the three excircles.

X(11) lies on these lines:1,5 2,55 3,499 4,56 7,658 10,121 13,202 14,203 30,36 33,427 34,235 35,140 65,117 68,1069 110,215 113,942 115,1015 118,226 153,388 212,748 214,442 244,867 325,350 381,999 429,1104 518,908 523,1090

X(11) = midpoint between X(I) and X(J) for these (I,J): (1,80), (4,104), (100,149)X(11) = reflection of X(119) about X(5)X(11) = isogonal conjugate of X(59)X(11) = X(I)-Ceva conjugate of X(J) for these (I,J): (1,523), (4,513), (7,514), (8,522)X(11) = crosspoint of X(I) and X(J) for these (I,J): (7,514), (8,522)

Let X = X(11) and let V be the vector-sum XA + XB + XC; then V = X(100)X(11) = X(11)X(149).

X(12) = HARMONIC CONJUGATE OF X(11) WRT X(1) AND X(5)

Trilinears 1 + cos(B - C) : 1 + cos(C - A) : 1 + cos(A - B)= f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = cos2(B/2 - C/2)= g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = bc(b + c)2/(b + c - a)

Barycentrics a(1 + cos(B - C)) : b(1 + cos(C - A)) : c(1 + cos(A - B))= h(a,b,c) : h(b,c,a) : h(c,a,b), where h(a,b,c) = (b + c)2/(b + c - a)

Let A'B'C' be the touch points of the nine-point circle with the A-, B-, C- excircles, respectively. The lines AA', BB', CC' concur in X(12).

X(12) lies on these lines:1,5 2,56 3,498 4,55 10,65 17,203 18,202 30,35 33,235 34,427 36,140 37,225 54,215 79,484 85,120 108,451 172,230 201,756 228,407 313,349 499,999 603,750 908,960 1091,1109

X(12) = isogonal conjugate of X(60)X(12) = isotomic conjugate of X(261)X(12) = X(10)-Ceva conjugate of X(201)

X(13) = 1st ISOGONIC CENTER (FERMAT POINT, TORRICELLI POINT)

Trilinears csc(A + π/3) : csc(B + π/3) : csc(C + π/3)= sec(A - π/6) : sec(B - π/6) : sec(C - π/6)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), wheref(a,b,c) = a4 - 2(b2 - c2)2 + a2(b2 + c2 + 4*SQR(3)*Area(ABC))

Construct the equilateral triangle BA'C having base BC and vertex A' on the negative side of BC; similarly construct equilateral triangles CB'A and AC'B based on the other two sides. The lines AA',BB',CC' concur in X(13). If each of the angles A, B, C is < 2*π/3, then X(13) is the only center X for which the angles BXC, CXA, AXB are equal, and X(13) minimizes the sum |AX| + |BX| + |CX|. The antipedal triangle of X(13) is equilateral.

X(13) lies on these lines:2,16 3,17 4,61 5,18 6,14 11,202 15,30 76,299 98,1080 99,303 148,617 226,1082 262,383 275,472 298,532 531,671 533,621 634,635

X(13) = reflection of X(I) about X(J) for these (I,J): (14,115), (15,396)X(13) = isogonal conjugate of X(15)X(13) = isotomic conjugate of X(298)X(13) = inverse of X(14) in the orthocentroidal circleX(13) = complement of X(616)X(13) = anticomplement of X(618)

X(13) = cevapoint of X(15) and X(62)X(13) = X(I)-cross conjugate of X(J) for these (I,J): (15,18), (30,14), (396,2)

X(14) = 2nd ISOGONIC CENTER

Trilinears csc(A - π/3) : csc(B - π/3) : csc(C - π/3)= sec(A + π/6) : sec(B + π/6) : sec(C + π/6)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), wheref(a,b,c) = a4 - 2(b2 - c2)2 + a2(b2 + c2 - 4*SQR(3)*Area(ABC))

Construct the equilateral triangle BA'C having base BC and vertex A' on the positive side of BC; similarly construct equilateral triangles CB'A and AC'B based on the other two sides. The lines AA',BB',CC' concur in X(14). The antipedal triangle of X(14) is equilateral.

X(14) lies on these lines:2,15 3,18 4,62 5,17 6,13 11,203 16,30 76,298 98,383 99,302 148,616 226,554 262,1080 275,473 299,533 397,546 530,671 532,622 633,636

X(14) = reflection of X(I) about X(J) for these (I,J): (13,115), (16,395)X(14) = isogonal conjugate of X(16)X(14) = isotomic conjugate of X(299)X(14) = inverse of X(13) in the orthocentroidal circleX(14) = complement of X(617)X(14) = anticomplement of X(619)X(14) = cevapoint of X(16) and X(61)X(14) = X(I)-cross conjugate of X(J) for these (I,J): (16,17), (30,13), (395,2)

X(15) = 1st ISODYNAMIC POINT

Trilinears sin(A + π/3) : sin(B + π/3) : sin(C + π/3)= cos(A - π/6) : cos(B - π/6) : cos(C - π/6)

Barycentrics a sin(A + π/3) : b sin(B + π/3) : c sin(C + π/3)

Let U and V be the points on sideline BC met by the interior and exterior bisectors of angle A. The circle having diameter UV is the A-Apollonian circle. The B- and C- Apollonian circles are similarly constructed. Each circle passes through one vertex and both isodynamic points. The pedal triangle of X(15) is equilateral.

X(15) lies on these lines:2,14 3,6 4,17 13,30 18,140 36,202 55,203 298,533 303,316 395,549 397,550 532,616 628,636

X(15) = reflection of X(I) about X(J) for these (I,J): (13,396), (16,187)X(15) = isogonal conjugate of X(13)

X(15) = isotomic conjugate of X(300)X(15) = inverse of X(16) in the circumcircleX(15) = inverse of X(16) in Brocard circleX(15) = complement of X(621)X(15) = anticomplement of X(623)X(15) = X(I)-Ceva conjugate of X(J) for these (I,J): (1,202), (13,62), (74,16)X(15) = crosspoint of X(I) and X(J) for these (I,J): (13,18), (298,470)X(15) = X(6)-Hirst inverse of X(16)

X(16) = 2nd ISODYNAMIC POINT

Trilinears sin(A - π/3) : sin(B - π/3) : sin(C - π/3)= cos(A + π/6) : cos(B + π/6) : cos(C + π/6)

Barycentrics a sin(A - π/3) : b sin(B - π/3) : c sin(C - π/3)

Let U and V be the points on sideline BC met by the interior and exterior bisectors of angle A. The circle having diameter UV is the A-Apollonian circle. The B- and C- Apollonian circles are similarly constructed. Each circle passes through a vertex and both isodynamic points. The pedal triangle of X(16) is equilateral.

X(16) lies on these lines:2,13 3,6 4,18 14,30 17,140 36,203 55,202 299,532 302,316 396,549 398,550 533,617 627,635

X(16) = reflection of X(I) about X(J) for these (I,J): (14,395), (15,187)X(16) = isogonal conjugate of X(14)X(16) = isotomic conjugate of X(301)X(16) = inverse of X(15) in the circumcircleX(16) = inverse of X(15) in the BrocardX(16) = complement of X(622)X(16) = anticomplement of X(624)X(16) = X(I)-Ceva conjugate of X(J) for these (I,J): (1,203), (14,61), (74,15)X(16) = crosspoint of X(I) and X(J) for these (I,J): (14,17), (299,471)X(16) = X(6)-Hirst inverse of X(15)

X(17) = 1st NAPOLEON POINT

Trilinears csc(A + π/6) : csc(B + π/6) : csc(C + π/6)= sec(A - π/3) : sec(B - π/3) : sec(C - π/3)

Barycentrics a csc(A + π/6) : b csc(B + π/6) : c csc(C + π/6)

Let X,Y,Z be the centers of the equilateral triangles in the construction of X(13). The lines AX, BY, CZ concur in X(17).

John Rigby, "Napoleon revisited," Journal of Geometry, 33 (1988) 126-146.

X(17) lies on these lines:2,62 3,13 4,15 5,14 6,18 12,203 16,140 76,303 83,624 202,499 275,471 299,635 623,633

X(17) = isogonal conjugate of X(61)X(17) = isotomic conjugate of X(302)X(17) = complement of X(627)X(17) = anticomplement of X(629)X(17) = X(I)-cross conjugate of X(J) for these (I,J): (16,14), (140,18), (397,4)

X(18) = 2nd NAPOLEON POINT

Trilinears csc(A - π/6) : csc(B - π/6) : csc(C - π/6)= sec(A + π/3) : sec(B + π/3) : sec(C + π/3)

Barycentrics a csc(A - π/6) : b csc(B - π/6) : c csc(C - π/6)

Let X,Y,Z be the centers of the equilateral triangles in the construction of X(14). The lines AX, BY, CZ concur in X(18).

X(18) lies on these lines:2,61 3,14 4,16 5,13 6,17 12,202 15,140 76,302 83,623 203,499 275,470 298,636 624,634

X(18) = isogonal conjugate of X(62)X(18) = isotomic conjugate of X(303)X(18) = complement of X(628)X(18) = anticomplement of X(630)X(18) = X(I)-cross conjugate of X(J) for these (I,J): (15,13), (140,17), (398,4)

X(19) = CLAWSON POINT

Trilinears tan A : tan B : tan C= f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = sin 2B + sin 2C - sin 2A= g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = 1/(b2 + c2 - a2)

Barycentrics a tan A : b tan B : c tan C

The homothetic center of the orthic and extangents triangles. Further information is available from Paul Yiu's Website.

X(19) lies on these lines:1,28 2,534 4,9 6,34 25,33 27,63 31,204 46,579 47,921 56,207 57,196 81,969 91,920 101,913 102,282 112,759 162,897 163,563 208,225 219,517 232,444 273,653 294,1041 604,609 960,965

X(19) = isogonal conjugate of X(63)X(19) = isotomic conjugate of X(304)

X(19) = X(I)-Ceva conjugate of X(J) for these (I,J):(1,204), (4,33), (27,4), (28,25), (57,208), (92,1), (196,207), (278,34)

X(19) = X(I)-cross conjugate of X(J) for these (I,J): (25,34), (31,1)X(19) = crosspoint of X(I) and X(J) for these (I,J): (4,278), (27,28), (57,84), (92,158)X(19) = X(I)-Hirst inverse of X(J) for these (I,J): (1,240), (4,242)X(19) = X(I)-aleph conjugate of X(J) for these (I,J): (2,610), (92,19), (508,223), (648,163)

Centers 20- 30, 2- 5, 140, 186, 199, 235, 237, 297, 376- 379, 381- 384,

401- 475, 546- 550, 631, 632 lie on the Euler line.

X(20) = DE LONGCHAMPS POINT

Trilinears cos A - cos B cos C : cos B - cos C cos A : cos C - cos A cos B

Barycentrics tan B + tan C - tan A : tan C + tan A - tan B: tan A + tan B - tan C= f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = [-3a4 + 2a2(b2 + c2) + (b2 - c2)2]

The reflection of X(4) about X(3); also, the orthocenter of the anticomplementary triangle.

X(20) lies on these lines:1,7 2,3 8,40 10,165 33,1038 34,1040 55,388 56,497 57,938 58,387 64,69 68,74 72,144 78,329 98,148 99,147 100,153 101,152 103,150 104,149 109,151 110,146 145,517 155,323 185,193 391,573 393,577 394,1032 487,638 488,637 616,633 617,635 621,627 622,628 999,1058

X(20) = reflection of X(I) about X(J) for these (I,J): (2,376), (4,3), (8,40), (146,110), (147,99), (148,98), (149,104), (150,103), (151,109), (152,101), (153,100), (382,5)

X(20) = isogonal conjugate of X(64)X(20) = isotomic conjugate of X(253)X(20) = cyclocevian conjugate of X(1032)X(20) = anticomplement of X(4)X(20) = X(I)-Ceva conjugate of X(J) for these (I,J): (69,2), (489,487), (490,488)X(20) = X(I)-aleph conjugate of X(J) for these (I,J): (8,191), (9,1045), (188,1046), (333,2), (1043,20)

X(21) = SCHIFFLER POINT

Trilinears 1/(cos B + cos C) : 1/(cos C + cos A) : 1/(cos A + cos B)= f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c - a)/(b + c)

Barycentrics a/(cos B + cos C) : b/(cos C + cos A) : c/(cos A + cos B)

Write I for the incenter; the Euler lines of the four triangles IBC, ICA, IAB, and ABC concur in X(21).

X(21) lies on these lines:1,31 2,3 6,941 7,56 8,55 9,41 10,35 32,981 36,79 37,172 51,970 60,960 72,943 75,272 84,285 90,224 99,105 104,110 144,954 145,956 238,256 261,314 268,280 332,1036 612,989 614,988 741,932 748,978 884,885 915,925 976,983 1038,1041 1039,1040 1060,1063 1061,1062

X(21) = midpoint between X(1) and X(191)X(21) = isogonal conjugate of X(65)X(21) = anticomplement of X(422)X(21) = X(I)-Ceva conjugate of X(J) for these (I,J): (86,81), (261,333)X(21) = cevapoint of X(I) and X(J) for these (I,J): (1,3), (9,55)

X(21) = X(I)-cross conjugate of X(J) for these (I,J):(1,29), (3,283), (9,333), (55,284), (58,285), (284,81), (522,100)

X(21) = crosspoint of X(86) and X(333)X(21) = X(I)-Hirst inverse of X(J) for these (I,J): (2,448), (3,416), (4,425)Let X = X(21) and let V be the vector-sum XA + XB + XC; then V = X(79)X(1).

X(22) = EXETER POINT

Trilinears a(b4 + c4 - a4) : b(c4 + a4 - a4) : c(a4 + b4 - c4)Barycentrics a2(b4 + c4 - a4) : b2(c4 + a4 - a4) : c2(a4 + b4 - c4)

The perspector of the circummedial triangle and the tangential triangle; also X(22) = X(55)-of-the-tangential triangle if ABC is acute.

X(22) lies on these lines:2,3 6,251 35,612 36,614 51,182 56,977 69,159 98,925 99,305 100,197 110,154 157,183 160,325 161,343 184,511 232,577

X(22) = reflection of X(378) about X(3)X(22) = isogonal conjugate of X(66)X(22) = inverse of X(858) in the circumcircleX(22) = anticomplement of X(427)X(22) = X(76)-Ceva conjugate of X(6)X(22) = cevapoint of X(3) and X(159)X(22) = crosspoint of X(99) and X(250)

X(23) = FAR-OUT POINT

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a[b4 + c4 - a4 - b2c2] Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)

The inverse of the centroid in the circumcircle.

X(23) lies on these lines:2,3 6,353 51,575 94,98 110,323 111,187 159,193 184,576 232,250 385,523

X(23) = reflection of X(323) about X(110)X(23) = isogonal conjugate of X(67)X(23) = inverse of X(2) in the circumcircleX(23) = anticomplement of X(427)X(23) = crosspoint of X(111) and X(251)

X(24) = PERSPECTOR OF ABC AND ORTHIC-OF-ORTHIC TRIANGLE

Trilinears sec A cos 2A : sec B cos 2B : sec C cos 2C= sec A - 2 cos A : sec B - 2 cos B : sec C - 2 cos C

Barycentrics tan A cos 2A : tan B cos 2B : tan C cos 2C= tan A - sin 2A : tan A - sin 2B : tan C - sin 2C

Constructed as indicated by the name; also X(24) = X(56)-of-the-tangential triangle if ABC is acute.

X(24) lies on these lines:2,3 6,54 32,232 33,35 34,36 49,568 51,578 64,74 96,847 107,1093 108,915 110,155 184,389 254,393 511,1092

X(24) = reflection of X(4) about X(235)X(24) = isogonal conjugate of X(68)X(24) = inverse of X(403) in the circumcircleX(24) = X(249)-Ceva conjugate of X(112)X(24) = X(52)-cross conjugate of X(4)X(24) = crosspoint of X(107) and X(250)X(24) = X(4)-Hirst inverse of X(421)

X(25) = HOMOTHETIC CENTER OF ORTHIC AND TANGENTIAL TRIANGLES

Trilinears sin A tan A : sin B tan B : sin C tan C = cos A - sec A : cos B - sec B cos C - sec C= f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a/(b2 + c2 - a2)

Barycentrics sin 2A - 2 tan A : sin 2B - 2 tan B : sin 2C - 2 tan C= g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a2/(b2 + c2 - a2)

Constructed as indicated by the name; also X(25) is the pole of the orthic axis (the line having trilinear coefficients cos A : cos B : cos C) with respect to the circumcircle. Also, X(25) is X(57)-of-the-tangential triangle.

X(25) lies on these lines:1,1036 2,3 6,51 19,33 31,608 34,56 41,42 52,155 53,157 58,967 92,242 98,107 105,108 111,112 114,135 132,136 143,156 183,264 185,1078 262,275 317,325 371,493 372,494 393,1033 394,511 669,878 692,913

X(25) = isogonal conjugate of X(69)X(25) = isotomic conjugate of X(305)X(25) = inverse of X(468) in the circumcircleX(25) = inverse of X(427) in the orthocentroidal circleX(25) = X(I)-Ceva conjugate of X(J) for these (I,J): (4,6), (28,19), (250,112)X(25) = X(32)-cross conjugate of X(6)X(25) = crosspoint of X(I) and X(J) for these (I,J): (4,393), (6,64), (19,34), (112,250)X(25) = X(I)-Hirst inverse of X(J) for these (I,J): (4,419), (6,232)

X(26) = CIRCUMCENTER OF THE TANGENTIAL TRIANGLE

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a[b2cos 2B + c2cos 2C - a2cos 2A]Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a2(b2cos 2B + c2cos 2C - a2cos 2A)

Theorems involving X(26), published in 1889 by A. Gob, are discussed inRoger A. Johnson, Advanced Euclidean Geometry, Dover, 1960, 259-260.

X(26) lies on these lines: 2,3 6,143 52,184 68,161 154,155 206,511

X(26) = reflection of X(155) about X(156)X(26) = isogonal conjugate of X(70)

X(27) CEVAPOINT OF ORTHOCENTER AND CLAWSON CENTER

Trilinears (sec A)/(b + c) : (sec B)/(c + a) : (sec C)/(a + b)Barycentrics (tan A)/(b + c) : (tan B)/(c + a) : (tan C)/(a + b)

X(27) lies on these lines:2,3 7,81 19,63 57,273 58,270 103,107 110,917 226,284 295,335 306,1043 393,967 648,903 662,913

X(27) = isogonal conjugate of X(71)X(27) = isotomic conjugate of X(306)X(27) = inverse of X(469) in the orthocentroidal circle

X(27) = anticomplement of X(440)X(27) = X(286)-Ceva conjugate of X(29)X(27) = cevapoint of X(I) and X(J) for these (I,J): (4,19), (57,278)X(27) = X(I)-cross conjugate of X(J) for these (I,J): (4,286), (19,28), (57,81), (58,86)X(27) = X(I)-Hirst inverse of X(J) for these (I,J): (2,447), (4,423)

X(28)

Trilinears (tan A)/(b + c) : (tan B)/(c + a) : (tan C)/(a + b)Barycentrics (sin A tan A)/(b + c) : (sin B tan B)/(c + a) : (sin C tan C)/(a + b)

X(28) lies on these lines:1,19 2,3 33,975 34,57 56,278 60,81 88,162 104,107 105,112 108,225 110,915 228,943 242,261 272,273 279,1014 281,958 607,1002 608,959

X(28) = isogonal conjugate of X(72)X(28) = X(I)-Ceva conjugate of X(J) for these (I,J): (270,58), (286,81)X(28) = cevapoint of X(I) and X(J) for these (I,J): (19,25), (34,56)X(28) = X(I)-cross conjugate of X(J) for these (I,J): (19,27), (58,58)X(28) = X(4)-Hirst inverse of X(422)

X(29) CEVAPOINT OF INCENTER AND ORTHOCENTER

Trilinears (sec A)/(cos B + cos C) : (sec B)/(cos C + cos A) : (sec C)/(cos A + cos B)Barycentrics (tan A)/(cos B + cos C) : (tan B)/(cos C + cos A) : (tan C)/(cos A + cos B)

X(29) lies on these lines:1,92 2,3 8,219 33,78 34,77 58,162 65,296 81,189 102,107 226,951 242,257 270,283 284,950 314,1039 388,1037 497,1036 515,947 1056,1059 1057,1058

X(29) = isogonal conjugate of X(73)X(29) = isotomic conjugate of X(307)X(29) = X(286)-Ceva conjugate of X(27)X(29) = cevapoint of X(I) and X(J) for these (I,J): (1,4), (33,281)X(29) = X(I)-cross conjugate of X(J) for these (I,J): (1,21), (284,333), (497,314)X(29) = X(4)-Hirst inverse of X(415)

X(30) = EULER INFINITY POINT

Trilinears cos A - 2 cos B cos C : cos B - 2 cos C cos A : cos C - 2 cos A cos B= f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc[2a4 - (b2 - c2)2 - a2(b2 + c2)]

Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = 2a4 - (b2 - c2)2 - a2(b2 + c2)

The point of intersection of the Euler line and the line at infinity. Thus, each of the 22 lines listed below is parallel to the Euler line.

X(30) lies on these lines:1,79 2,3 11,36 12,35 13,15 14,16 33,1060 34,1062 40,191 52,185 53,577 55,495 56,496 61,397 62,398 64,68 74,265 80,484 98,671 99,316 110,477 115,187 143,389 146,323 148,385 155,1078 182,597 262,598 298,616 299,617 390,1056 489,638 490,637 497,999 511,512 551,946 553,942 618,623 619,624 620,625 944,962

X(30) = isogonal conjugate of X(74)X(30) = X(I)-Ceva conjugate of X(J) for these (I,J): (4,113), (265,5), (476,523)X(30) = cevapoint of X(3) and X(399)X(30) = crosspoint of X(I) and X(J) for these (I,J): (13,14), (94,264)

X(31) = 2nd POWER POINT

Trilinears a2 : b2 : c2

Barycentrics a3 : b3 : c3

X(31) lies on these lines:1,21 2,171 3,601 6,42 8,987 9,612 10,964 19,204 25,608 32,41 35,386 36,995 40,580 43,100 44,210 48,560 51,181 56,154 57,105 65,1104 72,976 75,82 76,734 91,1087 92,162 101,609 110,593 163,923 184,604 etc.

X(31) = isogonal conjugate of X(75)X(31) = isotomic conjugate of X(561)X(31) = X(I)-Ceva conjugate of X(J) for these (I,J): (1,48), (6,41), (9,205), (58,6), (82,1)X(31) = X(213)-cross conjugate of X(6)X(31) = crosspoint of X(I) and X(J) for these (I,J): (1,19), (6,56)X(31) = X(I)-aleph conjugate of X(J) for these (I,J): (82,31), (83,75)

X(32) = 3rd POWER POINT

Trilinears a3 : b3 : c3

= sin(A - ω) : sin(B - ω) : sin(C - ω)

Barycentrics a4 : b4 : c4

X(32) lies on these lines:1,172 2,83 3,6 4,98 5,230 9,987 21,981 24,232 31,41 56,1015 75,746 76,384 81,980 99,194 100,713 101,595 110,729 163,849 184,211 218,906 512,878 538,1003 561,724 590,640 604,1106 615,639 731,825 733,827 910,1104 993,1107

X(32) = midpoint between X(371) and X(372)X(32) = isogonal conjugate of X(76)X(32) = inverse of X(39) in the Brocard circleX(32) = complement of X(315)X(32) = anticomplement of X(626)X(32) = X(I)-Ceva conjugate of X(J) for these (I,J): (2,206), (6,184), (112,512), (251,6)

X(32) = crosspoint of X(I) and X(J) for these (I,J): (2,66), (6,25)X(32) = X(184)-Hirst inverse of X(237)

X(33) = PERSPECTOR OF THE ORTHIC AND INTANGENTS TRIANGLES

Trilinears 1 + sec A : 1 + sec B : 1 + sec C = tan A cot(A/2) : tan B cot(B/2) : tan C cot(C/2) = f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c - a)/(b2 + c2 - a2) = g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = sec A cos2(A/2)

Barycentrics sin A + tan A : sin B + tan B : sin C + tan C= h(A,B,C) : h(B,C,A) : h(C,A,B), where h(A,B,C) = tan A cos2(A/2)

X(33) lies on these lines:1,4 2,1040 5,1062 6,204 7,1041 8,1039 9,212 10,406 11,427 12,235 19,25 20,1038 24,35 28,975 29,78 30,1060 36,378 40,201 42,393 47,90 56,963 57,103 63,1013 64,65 79,1063 80,1061 84,603 112,609 200,281 210,220 222,971 264,350

X(33) = isogonal conjugate of X(77)X(33) = X(I)-Ceva conjugate of X(J) for these (I,J): (4,19), (29,281), (318,9)X(33) = X(I)-cross conjugate of X(J) for these (I,J): (41,9), (42,55)X(33) = crosspoint of X(I) and X(J) for these (I,J): (1,282), (4,281)

X(34)

Trilinears 1 - sec A : 1 - sec B : 1 - sec C= tan A tan(A/2) : tan B tan(B/2) : tan C tan(C/2) = f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[(b + c - a)(b2 + c2 - a2)] = g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = sec A sin2(A/2)

Barycentrics sin A - tan A : sin B - tan B : sin C - tan C= h(A,B,C) : h(B,C,A) : h(C,A,B), where h(A,B,C) = tan A sin2(A/2)

The center of perspective of the orthic triangle and the reflection about the incenter of the intangents triangle.

X(34) lies on these lines:1,4 2,1038 5,1060 6,19 7,1039 8,1041 9,201 10,475 11,235 12,427 20,1040 24,36 25,56 28,57 29,77 30,1062 35,378 40,212 46,47 55,227 79,1061 80,1063 87,242 106,108 196,937 207,1042 222,942 244,1106 331,870 347,452 860,997

X(34) = isogonal conjugate of X(78)X(34) = X(I)-Ceva conjugate of X(J) for these (I,J): (1,207), (4,208), (28,56), (273,57), (278,19)X(34) = X(25)-cross conjugate of X(19)

X(35)

Trilinears 1 + 2 cos A : 1 + 2 cos B : 1 + 2 cos C= f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b2 + c2 - a2 + bc)

Barycentrics sin A + sin 2A : sin B + sin 2B: sin C + sin 2C= g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a2(b2 + c2 - a2 + bc)

X(35) lies on these lines:1,3 4,498 8,993 9,90 10,21 11,140 12,30 22,612 24,33 31,386 34,378 37,267 42,58 43,1011 47,212 71,284 72,191 73,74 79,226 172,187 etc.

X(35) = isogonal conjugate of X(79)X(35) = inverse of X(484) in the circumcircleX(35) = X(500)-cross conjugate of X(1)X(35) = X(943)-aleph conjugate of X(35)

X(36) = INVERSE OF THE INCENTER IN THE CIRCUMCIRCLE

Trilinears 1 - 2 cos A : 1 - 2 cos B : 1 - 2 cos C= f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b2 + c2 - a2 - bc) = g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = sec A cos2(A/2)

Barycentrics sin A - sin 2A : sin B - sin 2B: sin C - sin 2C= g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a2(b2 + c2 - a2 - bc)

X(36) lies on these lines:1,3 2,535 4,499 6,609 10,404 11,30 12,140 15,202 16,203 21,79 22,614 24,34 31,995 33,378 39,172 47,602 48,579 54,73 58,60 59,1110 63,997 80,104 84,90 99,350 100,519 101,672 106,901 109,953 187,1015 191,960 214,758 226,1006 238,513 255,1106 376,497 388,498 474,958 495,549 496,550 573,604 1030,1100

X(36) = midpoint between X(1) and X(484)X(36) = isogonal conjugate of X(80)X(36) = inverse of X(1) in the circumcircleX(36) = inverse of X(942) in the incircle>BR> X(36) = X(I)-Ceva conjugate of X(J) for these (I,J): (88,6), (104,1)X(36) = crosspoint of X(58) and X(106)X(36) = X(104)-aleph conjugate of X(36)

X(37) = CROSSPOINT OF INCENTER AND CENTROID

Trilinears b + c : c + a : a + bBarycentrics a(b + c) : b(c + a) : c(a + b)

X(37) lies on these lines:1,6 2,75 3,975 7,241 8,941 10,594 12,225 19,25 21,172 35,267 38,354 39,596 12,225

41,584 48,205 63,940 65,71 73,836 78,965 82,251 86,190 91,498 100,111 101,284 141,742 142,1086 145,391 158,281 171,846 226,440 256,694 347,948 513,876 517,573 537,551 579,942 626,746 665,900 971,991

X(37) = midpoint between X(I) and X(J) for these (I,J): (75,192), (190,335)X(37) = isogonal conjugate of X(81)X(37) = isotomic conjugate of X(274)X(37) = complement of X(75)

X(37) = X(I)-Ceva conjugate of X(J) for these (I,J):(1,42), (2,10), (4,209), (9,71), (10,210), (190,513), (226,65), (321,72), (335,518)

X(37) = cevapoint of X(213) and X(228)X(37) = X(I)-cross conjugate of X(J) for these (I,J): (42,65), (228,72) X(37) = crosspoint of X(I) and X(J) for these (I,J): (1,2), (9,281), (10,226)X(37) = X(1)-line conjugate of X(238)X(37) = X(1)-aleph conjugate of X(1051)

Let X = X(37) and let V be the vector-sum XA + XB + XC; then V = X(75)X(37) = X(37)X(192).

X(38)

Trilinears b2 + c2 : c2 + a2 : a2 + b2

=csc A sin(A + ω) : csc B sin(B + ω) : csc C sin(C + ω)

Barycentrics a(b2 + c2) : b(c2 + a2) : c(a2 + b2)= sin(A + ω) : sin(B + ω) : sin(C + ω)

X(38) lies on these lines:1,21 2,244 3,976 8,986 9,614 10,596 37,354 42,518 56,201 57,612 75,310 78,988 92,240 99,745 210,899 321,726 869,980 912,1064 1038,1106

X(38) = isogonal conjugate of X(82)X(38) = crosspoint of X(1) and X(75)

X(39) = BROCARD MIDPOINT

Trilinears a(b2 + c2) : b(c2 + a2) : c(a2 + b2)= sin(A + ω) : sin(B + ω) : sin(C + ω)

Barycentrics a2(b2 + c2) : b2(c2 + a2) : c2(a2 + b2)

The midpoint between the 1st and 2nd Brocard points, given by trilinears c/b : a/c : b/a and b/c : c/a : a/b .

X(39) lies on these lines:1,291 2,76 3,6 4,232 5,114 9,978 10,730 36,172 37,596 51,237 54,248 83,99 110,755

140,230 141,732 185,217 213,672 325,626 395,618 493,494 512,881 588,589 590,642 597,1084 615,641

Ross Honsberger, Episodes in Nineteenth and Twentieth Century Euclidean Geometry, Mathematical Association of America, 1995. Chapter 10: The Brocard Points.

X(39) = midpoint between X(76) and X(194)X(39) = isogonal conjugate of X(83)X(39) = isotomic conjugate of X(308)X(39) = inverse of X(32) in the Brocard circleX(39) = complement of X(76)X(39) = eigencenter of anticevian triangle of X(512)X(39) = X(I)-Ceva conjugate of X(J) for these (I,J): (2,141), (4,211), (99,512)X(39) = crosspoint of X(I) and X(J) for these (I,J): (2,6), (141,427)

Let X = X(39) and let V be the vector-sum XA + XB + XC; then V = X(76)X(39) = X(39)X(194).

X(40) = REFLECTION OF THE INCENTER IN CIRCUMCENTER

Trilinears cos B + cos C - cos A - 1 : cos C + cos A - cos B - 1 : cos A + cos B - cos C - 1= f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = b/(c + a - b) + c/(a + b - c) - a/(b + c - a)= g(A,B,C) : g(B,C,A) : g(C,A,B), whereg(A,B,C) = sin2(B/2) + sin2(C/2) - sin2(A/2)

Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)

The point of concurrence of the perpendiculars from the excenters to the respective sides; also, the circumcenter of the excentral triangle.

X(40) lies on these lines:1,3 2,926 4,9 6,380 8,20 30,191 31,580 33,201 34,212 42,581 43,970 58,601 64,72 77,947 78,100 80,90 92,412 101,972 108,207 109,255 164,188 190,341 196,208 219,610 220,910 221,223 256,989 376,519 386,1064 387,579 390,938 392,474 511,1045 550,952 595,602 728,1018 936,960 958,1012 978,1050

X(40) = midpoint between X(8) and X(20)X(40) = reflection of X(I) about X(J) for these (I,J): (1,3), (4,10)X(40) = isogonal conjugate of X(84)X(40) = isotomic conjugate of X(309)X(40) = complement of X(962)X(40) = anticomplement of X(946)X(40) = X(I)-Ceva conjugate of X(J) for these (I,J): (8,1), (63,9), (347,223)X(40) = X(I)-cross conjugate of X(J) for these (I,J): (198,223), (221,1)X(40) = crosspoint of X(I) and X(J) for these (I,J): (329,347)X(40) = X(I)-aleph conjugate of X(J) for these (I,J): (1,978), (2,57), (8,40), (188,1), (556,63)

X(41)

Trilinears a2(b + c - a) : b2(c + a - b) : c2(a + b - c)= a2cot(A/2) : b2cot(B/2) : c2cot(C/2)

Barycentrics a3(b + c - a) : b3(c + a - b) : c3(a + b - c)

X(41) lies on these lines: 1,101 3,218 6,48 9,21 25,42 31,32 37,584 55,220 58,609 65,910 219,1036 226,379 560,872 601,906 603,911 663,884

X(41) = isogonal conjugate of X(85)X(41) = X(I)-Ceva conjugate of X(J) for these (I,J): (6,31), (9,212), (284,55)X(41) = crosspoint of X(I) and X(J) for these (I,J): (6,55), (9,33)

X(42) CROSSPOINT OF INCENTER AND SYMMEDIAN POINT

Trilinears a(b + c) : b(c + a) : c(a + b)= (1 + cos A)(cos B + cos C) : (1 + cos B)(cos C + cos A) : (1 + cos C)(cos A + cos B)Barycentrics a2(b + c) : b2(c + a) : c2(a + b)

X(42) lies on these lines:1,2 3,967 6,31 9,941 25,41 33,393 35,58 37,210 38,518 40,581 48,197 57,1001 65,73 81,100 101,111 165,991 172,199 181,228 244,354 308,313 321,740 517,1064 560,584 649,788 694,893 748,1001 750,940 894,1045 942,1066

X(42) = isogonal conjugate of X(86)X(42) = isotomic conjugate of X(310)X(42) = X(I)-Ceva conjugate of X(J) for these (I,J): (1,37), (6,213), (10,71), (55,228)X(42) = crosspoint of X(I) and X(J) for these (I,J): (1,6), (33,55), (37,65)X(42) = X(1)-line conjugate of X(239)

X(43) X(6)-CEVA CONJUGATE OF X(1)

Trilinears ab + ac - bc : bc + ba - ca : ca + cb - ab= csc B + csc C - csc A : csc C + csc A - csc B : csc A + csc B - csc

Barycentrics a(ab + ac - bc) : b(bc + ba - ca) : c(ca + cb - ab)

X(43) lies on these lines:1,2 6,87 9,256 31,100 35,1011 40,970 46,851 55,238 57,181 58,979 72,986 75,872 81,750 165,573 170,218 210,984 312,740 518,982

X(43) = isogonal conjugate of X(87)X(43) = X(6)-Ceva conjugate of X(1)X(43) = X(192)-cross conjugate of X(1)X(43) = X(55)-Hirst inverse of X(238)

X(43) = X(I)-aleph conjugate of X(J) for these (I,J):(1,9), (6,43), (55,170), (100,1018), (174,169), (259,165), (365,1), (366,63), (507,362), (509,57)

X(44) X(6)-LINE CONJUGATE OF X(1)

Trilinears b + c - 2a : c + a - 2b : a + b - 2cBarycentrics a(b + c - 2a) : b(c + a - 2b) : c(a + b - 2c)

X(44) lies on these lines: 1,6 2,89 10,752 31,210 51,209 65,374 88,679 181,375 190,239 193,344 214,1017 241,651 292,660 354,748 513,649 527,1086 583,992 678,902

X(44) = midpoint between X(190) and X(239)X(44) = isogonal conjugate of X(88)X(44) = complement of X(320)X(44) = X(I)-Ceva conjugate of X(J) for these (I,J): (2,214), (88,1), (104,55)X(44) = crosspoint of X(I) and X(J) for these (I,J): (1,88), (2,80)X(44) = X(6)-line conjugate of X(1)X(44) = X(88)-cross conjugate of X(44)

X(45)

Trilinears 2b + 2c - a : 2c + 2a - b : 2a + 2b - cBarycentrics a(2b + 2c - a) : b(2c + 2a - b) : c(2a + 2b - c)

X(45) lies on these lines: 1,6 2,88 53,281 55,678 141,344 198,1030 210,968 346,594

X(45) = isogonal conjugate of X(89)

X(46) X(4)-CEVA CONJUGATE OF X(1)

Trilinears cos B + cos C - cos A : cos C + cos A - cos B : cos A + cos B - cos CBarycentrics a(cos B + cos C - cos A) : b(cos C + cos A - cos B) : c(cos A + cos B - cos C)

X(46) lies on these lines:1,3 4,90 9,79 10,63 19,579 34,47 43,851 58,998 78,758 80,84 100,224 158,412 169,672 200,1004 218,910 222,227 225,254 226,498 269,1103 404,997 474,960 499,946 595,614 750,975 978,1054

X(46) = reflection of X(1) about X(56)X(46) = isogonal conjugate of X(90)X(46) = X(4)-Ceva conjugate of X(1)

X(46) = X(I)-aleph conjugate of X(J) for these (I,J):(4,46), (174,223), (188,1079), (366,610), (653, 1020)

X(47)

Trilinears cos 2A : cos 2B : cos 2C = f(a,b,c) : f(b,c,a) : f(c,a,b), wheref(a,b,c) = a2[a4 + b4 + c4 - 2a2b2 - 2a2c2]

Barycentrics a cos 2A : b cos 2B : c cos 2C

X(47) lies on these lines:1,21 19,921 33,90 34,46 35,212 36,602 91,92 158,162 171,498 238,499

X(47) = isogonal conjugate of X(91)X(47) = eigencenter of cevian triangle of X(92)X(47) = eigencenter of anticevian triangle of X(48)X(47) = X(92)-Ceva conjugate of X(48)X(47) = X(275)-aleph conjugate of X(92)

X(48)

Trilinears sin 2A : sin 2B : sin 2C= f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = tan B + tan C= g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a2(b2 + c2 - a2)

Barycentrics a sin 2A : b sin 2B : c sin 2C

X(48) lies on these lines:1,19 3,71 6,41 9,101 31,560 36,579 37,205 42,197 55,154 63,326 75,336 163,1094 184,212 220,963 255,563 281,944 282,947 354,584 577,603 692,911 949,1037 958,965

X(48) = isogonal conjugate of X(92)X(48) = X(I)-Ceva conjugate of X(J) for these (I,J): (1,31), (3,212), (63,255), (92,47), (284, 6)X(48) = X(228)-cross conjugate of X(3)X(48) = crosspoint of X(I) and X(J) for these (I,J): (1,63), (3,222), (91,92), (219,268)X(48) = X(1)-line conjugate of X(240)

X(49) = CENTER OF SINE-TRIPLE-ANGLE CIRCLE

Trilinears cos 3A : cos 3B : cos 3CBarycentrics sin A cos 3A : sin B cos 3B : sin C cos 3C

V. Thebault, "Sine-triple-angle-circle," Mathesis 65 (1956) 282-284.

X(49) lies on these lines: 1,215 3,155 4,156 5,54 24,568 52,195 93,94 381,578

X(49) = isogonal conjugate of X(93)X(49) = eigencenter of cevian triangle of X(94)

X(49) = eigencenter of anticevian triangle of X(50)X(49) = X(94)-Ceva conjugate of X(50)

X(50)

Trilinears sin 3A : sin 3B : sin 3CBarycentrics sin A sin 3A : sin B sin 3B : sin C sin 3C

X(50) lies on these lines: 3,6 67,248 112,477 115,231 230,858 338,401 647,654

X(50) = isogonal conjugate of X(94)X(50) = inverse of X(566) in the Brocard circleX(50) = X(I)-Ceva conjugate of X(J) for these (I,J): (1,215), (74,184), (94,49)X(50) = crosspoint of X(I) and X(J) for these (I,J): (93,94), (186,323)

X(51) = CENTROID OF THE ORTHIC TRIANGLE

Trilinears a2cos(B - C) : b2cos(C - A) : c2cos(A - B)= f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a[a2(b2 + c2) - (b2 - c2)2]

Barycentrics a3cos(B - C) : b3cos(C - A) : c3cos(A - B)

X(51) lies on these lines:2,262 4,185 5,52 6,25 21,970 22,182 23,575 24,578 26,569 31,181 39,237 44,209 54,288 107,275 125,132 129,137 130,138 199,572 210,374 216,418 381,568 397,462 398,463 573,1011

X(51) = reflection of X(210) about X(375)X(51) = isogonal conjugate of X(95)X(51) = X(I)-Ceva conjugate of X(J) for these (I,J): (4,53), (5,216), (6,217)X(51) = X(217)-cross conjugate of X(216)X(51) = crosspoint of X(I) and X(J) for these (I,J): (4,6), (5,53)

Let X = X(51) and let V be the vector-sum XA + XB + XC; then V = X(3)X(52) = X(20)X(185)

Let W be the vector-sum XA' + XB' + XC', where A'B'C' is the pedal triangle of X; then W = X(428)X(51).

X(52) = ORTHOCENTER OF THE ORTHIC TRIANGLE

Trilinears cos 2A cos(B - C) : cos 2B cos(C - A) : cos 2C cos(A - B)= f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = sec A (sec 2B + sec 2C)

Barycentrics tan A (sec 2B + sec 2C ) : tan B (sec 2C + sec 2A) : tan C (sec 2A + sec 2B)

X(52) lies on these lines:3,6 4,68 5,51 25,155 26,184 30,185 49,195 113,135 114,211 128,134 129,139

X(52) = reflection of X(I) about X(J) for these (I,J): (3,389), (5,143)X(52) = isogonal conjugate of X(96)X(52) = inverse of X(569) in the Brocard circleX(52) = X(I)-Ceva conjugate of X(J) for these (I,J): (4,5), (317,467), (324,216)X(52) = crosspoint of X(4) and X(24)

X(53) = SYMMEDIAN POINT OF THE ORTHIC TRIANGLE

Trilinears tan A cos(B - C) : tan B cos(C - A) : tan C cos(A - B)Barycentrics a tan A cos(B - C) : b tan B cos(C - A) : c tan C cos(A - B)

X(53) lies on these lines:4,6 5,216 25,157 30,577 45,281 115,133 128,139 137,138 141,264 232,427 273,1086 275,288 311,324 317,524 318,594 395,472 396,473

X(53) = isogonal conjugate of X(97)X(53) = X(I)-Ceva conjugate of X(J) for these (I,J): (4,51), (324,5)X(53) = X(51)-cross conjugate of X(5)

X(54) = KOSNITA POINT

Trilinears sec(B - C) : sec(C - A) : sec(A - B)Barycentrics tan(B - C) : tan(C - A) : tan(A - B)

John Rigby, "Brief notes on some forgotten geometrical theorems," Mathematics and Informatics Quarterly 7 (1997) 156-158.

X(54) lies on these lines:2,68 3,97 4,184 5,49 6,24 12,215 36,73 39,248 51,288 64,378 69,95 71,572 72,1006 74,185 112,217 140,252 156,381 186,389 276,290 575,895 826,879

X(54) = midpoint between X(3) and X(195)X(54) = isogonal conjugate of X(5)X(54) = isotomic conjugate of X(311)X(54) = X(I)-Ceva conjugate of X(J) for these (I,J): (95,97), (288,6)X(54) = cevapoint of X(6) and X(184)X(54) = X(I)-cross conjugate of X(J) for these (I,J): (3,96), (6,275), (186,74), (389,4), (523,110)X(54) = crosspoint of X(95) and X(275)

X(55) = INTERNAL CENTER OF SIMILITUDE OF CIRCUMCIRCLE AND INCIRCLE

Trilinears a(b + c - a) : b(c + a - b) : c(a + b - c)= 1 + cos A : 1 + cos B : 1 + cos C= cos2(A/2) : cos2(B/2) : cos2(B/2)= tan(B/2) + tan(C/2) : tan(C/2) + tan(A/2) : tan(A/2) + tan(B/2)

Barycentrics a2(b + c - a) : b2(c + a - b) : c2(a + b - c)

The center of homothety of three triangles: tangential, intangents, and extangents.

X(55) lies on these lines:1,3 2,11 4,12 5,498 6,31 8,21 9,200 10,405 15,203 16,202 19,25 20,388 30,495 34,227 41,220 43,238 45,678 48,154 63,518 64,73 77,1037 78,960 81,1002 92,243 103,109 104,1000 108,196 140,496 181,573 182,613 183,350 184,215 192,385 199,1030 201,774 204,1033 219,284 226,516 255,601 256,983 329,1005 376,1056 386,595 392,997 411,962 511,611 515,1012 519,956 574,1015 603,963 631,1058 650,884 654,926 748,899 840,901 846,984 869,893 1026,1083 1070,1076 1072,1074

X(55) = isogonal conjugate of X(7)

X(55) = X(I)-Ceva conjugate of X(J) for these (I,J):(1,6), (3,198), (7,218), (8,219), (9,220), (21,9), (59,101), (104,44), (260,259), (284,41)

X(55) = cevapoint of X(42) and X(228) for these (I,J)X(55) = X(I)-cross conjugate of X(J) for these (I,J): (41,6), (42,33), (228,212)X(55) = crosspoint of X(I) and X(J) for these (I,J): (1,9), (3,268), (7,277), (8,281), (21,284), (59,101)X(55) = X(43)-Hirst inverse of X(238)X(55) = X(1)-line conjugate of X(241)

X(56) = EXTERNAL CENTER OF SIMILITUDE OF CIRCUMCIRCLE AND INCIRCLE

Trilinears a/(b + c - a) : b/(c + a - b) : c/(a + b - c)= 1 - cos A : 1 - cos B : 1 - cos C= sin2(A/2) : sin2(B/2) : sin2(C/2)

Barycentrics a2/(b + c - a) : b2/(c + a - b) : c2/(a + b - c)

The perspector of the tangential triangle and the reflection of the intangents triangle about X(1).

X(56) lies on these lines:1,3 2,12 4,11 5,499 6,41 7,21 8,404 10,474 19,207 20,497 22,977 25,34 28,278 30,496 31,154 32,1015 33,963 38,201 58,222 61,202 62,203 63,960 72,997 77,1036 78,480 81,959 85,870 87,238 100,145 101,218 105,279 106,109 140,495 181,386 182,611

197,227 212,939 219,579 220,672 223,937 226,405 255,602 266,289 269,738 330,385 376,1058 411,938 511,613 551,553 607,911 631,1056 667,764 946,1012 978,979 1025,1083 1070,1074 1072,1076

X(56) = midpoint between X(1) and X(46)X(56) = isogonal conjugate of X(8)X(56) = X(I)-Ceva conjugate of X(J) for these (I,J): (1,221), (7,222), (28,34), (57,6), (59,109), (108,513)X(56) = X(31)-cross conjugate of X(6)X(56) = crosspoint of X(I) and X(J) for these (I,J): (1,84), (7,278), (28,58), (57,269), (59,109)X(56) = X(266)-aleph conjugate of X(1050)

X(57) ISOGONAL CONJUGATE OF X(9)

Trilinears 1/(b + c - a) : 1/(c + a - b) : 1/(a + b - c)= tan(A/2) : tan(B/2) : tan(C/2)

Barycentrics a/(b + c - a) : b/(c + a - b) : c/(a + b - c)

X(57) lies on these lines:1,3 2,7 4,84 6,222 10,388 19,196 20,938 27,273 28,34 31,105 33,103 38,612 42,1002 43,181 72,474 73,386 77,81 78,404 79,90 85,274 88,651 92,653 164,177 169,277 173,174 200,518 201,975 234,362 239,330 255,580 279,479 345,728 497,516 499,920 649,1024 658,673 748,896 758,997 955,991 957,995 959,1042 961,1106 978,1046 1020,1086

X(57) = isogonal conjugate of X(9)X(57) = isotomic conjugate of X(312)X(57) = complement of X(329)

X(57) = X(I)-Ceva conjugate of X(J) for these (I,J):(2,223), (7,1), (27,278), (81,222), (85,77), (273,34), (279,269)

X(57) = cevapoint of X(I) and X(J) for these (I,J): (6,56), (19,208)X(57) = X(I)-cross conjugate of X(J) for these (I,J): (6,1), (19,84), (56,269), (65,7)X(57) = crosspoint of X(I) and X(J) for these (I,J): (2,189), (7,279), (27,81), (85,273)X(57) = X(1)-Hirst inverse of X(241)

X(57) = X(I)-aleph conjugate of X(J) for these (I,J): (2,40), (7,57), (57,978), (174,1), (366,165), (507,503), (508,9), (509,43)

X(58) ISOGONAL CONJUGATE OF X(10)

Trilinears a/(b + c) : b/(c + a) : c/(a + b)Barycentrics a2/(b + c) : b2/(c + a) : c2/(a + b)

X(58) lies on these lines:1,21 2,540 3,6 7,272 8,996 9,975 10,171 20,387 25,967 27,270 28,34 29,162 35,42 36,60 40,601 41,609 43,979 46,998 56,222 65,109 82,596 84,990 86,238 87,978 99,727 101,172 103,112 106,110 229,244 269,1014 274,870 314,987 405,940 519,1043 942,1104 977,982 1019,1027

X(58) = isogonal conjugate of X(10)X(58) = isotomic conjugate of X(313)X(58) = inverse of X(386) in the Brocard circleX(58) = X(I)-Ceva conjugate of X(J) for these (I,J): (81,284), (267,501), (270,28)X(58) = cevapoint of X(6) and X(31)X(58) = X(I)-cross conjugate of X(J) for these (I,J): (6,81), (36,106), (56,28), (513,109)X(58) = crosspoint of X(I) and X(J) for these (I,J): (1,267), (21,285), (27,86), (60,270)

X(59) ISOGONAL CONJUGATE OF X(11)

Trilinears 1/[1 - cos(B - C)] : 1/[1 - cos(C - A) : 1/[1 - cos(A - B)]Barycentrics a/[1 - cos(B - C)] : b/[1 - cos(C - A) : c/[1 - cos(A - B)]

X(59) lies on these lines: 36,1110 60,1101 100,521 101,657 109,901 513,651 518,765 523,655

X(59) = isogonal conjugate of X(11)X(59) = cevapoint of X(I) and X(J) for these (I,J): (55,101), (56,109)X(59) = X(I)-cross conjugate of X(J) for these (I,J): (1,110), (3,100), (55,101), (56,109)

X(60)

Trilinears 1/[1 + cos(B - C)] : 1/[1 + cos(C - A) : 1/[1 + cos(A - B)]Barycentrics a/[1 + cos(B - C)] : b/[1 + cos(C - A) : c/[1 + cos(A - B)]

X(60) lies on these lines: 1,110 21,960 28,81 36,58 59,1101 86,272 283,284 404,662 757,1014

X(60) = isogonal conjugate of X(12)X(60) = X(58)-cross conjugate of X(270)

X(61)

Trilinears sin(A + π/6) : sin(B + π/6) : sin(C + π/6)= cos(A - π/3) : cos(B - π/3) : cos(C - π/3)

Barycentrics sin A sin(A + π/6) : sin B sin(B + π/6) : sin C sin(C + π/6)

X(61) lies on these lines:1,203 2,18 3,6 4,13 5,14 30,397 56,202 140,395 299,636 302,629 618,627

X(61) = isogonal conjugate of X(17)X(61) = inverse of X(62) in the Brocard circleX(61) = complement of X(633)X(61) = anticomplement of X(635)X(61) = eigencenter of cevian triangle of X(14)X(61) = eigencenter of anticevian triangle of X(16)X(61) = X(14)-Ceva conjugate of X(16)X(61) = crosspoint of X(302) and X(473)

Let W be the vector-sum XA' + XB' + XC', where A'B'C' is the pedal triangle of X = X(61); then W = X(397)X(61).

X(62)

Trilinears sin(A - π/6) : sin(B - π/6) : sin(C - π/6)= cos(A + π/3) : cos(B + π/3) : cos(C + π/3)

Barycentrics sin A sin(A - π/6) : sin B sin(B - π/6) : sin C sin(C - π/6)

X(62) lies on these lines:1,202 2,17 3,6 4,14 5,13 30,398 56,203 140,396 298,635 303,630 619,628

X(62) = isogonal conjugate of X(18)X(62) = inverse of X(61) in the Brocard circleX(62) = complement of X(634)X(62) = anticomplement of X(636)X(62) = eigencenter of cevian triangle of X(13)X(62) = eigencenter of anticevian triangle of X(15)X(62) = X(13)-Ceva conjugate of X(15)X(62) = crosspoint of X(303) and X(472)

Let W be the vector-sum XA' + XB' + XC', where A'B'C' is the pedal triangle of X = X(62); then W = X(398)X(62).

X(63)

Trilinears cot A : cot B : cot C= b2 + c2 - a2 : c2 + a2 - b2 : c2 + b2 - c2

Barycentrics cos A : cos B : cos C

X(63) lies on these lines:1,21 2,7 3,72 8,20 10,46 19,27 33,1013 36,997 37,940 48,326 55,518 56,960 65,958 69,71 77,219 91,921 100,103 162,204 169,379 171,612 190,312 194,239 201,603 210,1004 212,1040 213,980 220,241 223,651 238,614 240,1096 244,748 304,1102 318,412 354,1001 392,999 404,936 405,942 452,938 484,535 517,956 544,1018 561,799 654,918 750,756

X(63) = isogonal conjugate of X(19)X(63) = isotomic conjugate of X(92)X(63) = anticomplement of X(226)X(63) = X(I)-Ceva conjugate of X(J) for these (I,J): (7,224), (69,78), (75,1), (304,326), (333,2), (348,77)X(63) = cevapoint of X(I) and X(J) for these (I,J): (3,219), (9,40), (48,255), (71,72)X(63) = X(I)-cross conjugate of X(J) for these (I,J): (3,77), (9,271), (48,1), (71,3), (72,69), (219,78), (255,326)X(63) = crosspoint of X(I) and X(J) for these (I,J): (69,348), (75,304)

X(63) = X(I)-aleph conjugate of X(J) for these (I,J): (2,1), (75,63), (92,920), (99,662), (174,978), (190,100), (333,411), (366,43), (514,1052), (556,40), (648,162), (664,651), (668,190), (670,799), (671,897), (903,88)

X(64)

Trilinears 1/(cos A - cos B cos C) : 1/(cos B - cos C cos A) : 1/(cos C - cos A cos B)Barycentrics a/(cos A - cos B cos C) : b/(cos B - cos C cos A) : c/(cos C - cos A cos B)

X(64) lies on these lines:3,154 6,185 20,69 24,74 30,68 33,65 40,72 54,378 55,73 71,198 265,382

X(64) = isogonal conjugate of X(20)X(64) = X(25)-cross conjugate of X(6)

X(65) = ORTHOCENTER OF THE INTOUCH TRIANGLE

Trilinears cos B + cos C : cos C + cos A : cos A + cos B= (b + c)/(b + c - a) : (c + a)/(c + a - b) : (a + b)/(a + b - c)= sin(A/2) cos(B/2 - C/2) : sin(B/2) cos(C/2 - A/2) : sin(C/2) cos(A/2 - B/2)

Barycentrics a(b + c)/(b + c - a) : b(c + a)/(c + a - b) : c(a + b)/(a + b - c)

The perspector of ABC and the Yff central triangle.

X(65) lies on these lines:1,3 2,959 4,158 6,19 7,8 10,12 11,117 29,296 31,1104 33,64 37,71 41,910 42,73 44,374 58,109 63,958 68,91 74,108 77,969 79,80 81,961 110,229 169,218 172,248 224,1004 225,407 243,412 257,894 278,387 279,1002 386,994 409,1098 474,997 497,938 516,950 519,553 604,1100 651,895 1039,1041 1061,1063

X(65) = reflection of X(72) about X(10)X(65) = isogonal conjugate of X(21)X(65) = isotomic conjugate of X(314)X(65) = anticomplement of X(960)X(65) = X(I)-Ceva conjugate of X(J) for these (I,J): (1,73), (4,225), (7,226), (10,227), (109,513), (226,37)

X(65) = X(42)-cross conjugate of X(37)X(65) = crosspoint of X(I) and X(J) for these (I,J): (1,4), (7,57)

X(66)

Trilinears bc/(b4 + c4 - a4) : ca/(c4 + a4 - b4) : ab/(a4 + b4 - c4)Barycentrics 1/(b4 + c4 - a4) : 1/(c4 + a4 - b4) : 1/(a4 + b4 - c4)

X(66) lies on these lines:2,206 3,141 6,427 68,511 73,976 193,895 248,571 290,317 879,924

X(66) = reflection of X(159) about X(141)X(66) = isogonal conjugate of X(22)X(66) = isotomic conjugate of X(315)X(66) = anticomplement of X(206)X(66) = cevapoint of X(125) and X(512)X(66) = X(32)-cross conjugate of X(2)

Let W be the vector-sum XA' + XB' + XC', where A'B'C' is the pedal triangle of X = X(66); then W = X(185)X(64).

X(67)

Trilinears bc/(b4 + c4 - a4 - b2c2) : ca/(c4 + a4 - b4 - c2a2) : ab/(a4 + b4 - c4 - a2b2)Barycentrics 1/(b4 + c4 - a4 - b2c2) : 1/(c4 + a4 - b4 - c2a2) : 1/(a4 + b4 - c4 - a2b2)

X(67) lies on these lines:3,542 4,338 6,125 50,248 74,935 110,141 265,511 290,340 524,858 526,879

X(67) = reflection of X(I) about X(J) for these (I,J): (6,125), (110,141)X(67) = isogonal conjugate of X(23)X(67) = isotomic conjugate of X(316)X(67) = cevapoint of X(141) and X(524)X(67) = X(187)-cross conjugate of X(2)

X(68)

Trilinears cos A sec 2A : cos B sec 2B : cos C sec 2CBarycentrics tan 2A : tan 2B : tan 2C

X(68) lies on these lines:2,54 3,343 4,52 5,6 11,1069 20,74 26,161 30,64 65,91 66,511 73,1060 136,254 290,315 568,973

X(68) = reflection of X(155) about X(5)X(68) = isogonal conjugate of X(24)X(68) = isotomic conjugate of X(317)

X(68) = X(96)-Ceva conjugate of X(3)X(68) = cevapoint of X(I) and X(J) for these (I,J): (6,161), (125,520)X(68) = X(115)-cross conjugate of X(525)

Let W be the vector-sum XA' + XB' + XC', where A'B'C' is the pedal triangle of X = X(68); then W = X(52)X(68).

X(69) = SYMMEDIAN POINT OF THE ANTICOMPLEMENTARY TRIANGLE

Trilinears (cos A)/a2 : (cos B)/b2 : (cos C)/c2

= bc(b2 + c2 - a2) : ca(c2 + a2 - b2) : ab(a2 + b2 - c2)

Barycentrics cot A : cot B : cot C= b2 + c2 - a2 : c2 + a2 - b2 : a2 + b2 - c2

X(69) lies on these lines:2,6 3,332 4,76 7,8 9,344 10,969 20,64 22,159 54,95 63,71 72,304 73,77 74,99 110,206 125,895 144,190 150,668 189,309 192,742 194,695 200,269 248,287 263,308 265,328 274,443 290,670 297,393 347,664 350,497 404,1014 478,651 485,639 486,640 520,879

X(69) = reflection of X(I) about X(J) for these (I,J): (6,141), (193,6)X(69) = isogonal conjugate of X(25)X(69) = isotomic conjugate of X(4)X(69) = cyclocevian conjugate of X(253)X(69) = complement of X(193) = anticomplement of X(6) X(69) = X(I)-Ceva conjugate of X(J) for these (I,J): (76,2), (304,345), (314,75), (332,326)X(69) = cevapoint of X(I) and X(J) for these (I,J): (2,20), (3,394), (6,159), (8,329), (63,78), (72,306), (125,525)

X(69) = X(I)-cross conjugate of X(J) for these (I,J):(3,2), (63,348), (72,63), (78,345), (125,525), (306,304), (307,75), (343,76)

X(69) = crosspoint of X(I) and X(J) for these (I,J): (76,305), (314,332)X(69) = X(2)-Hirst inverse of X(325)

Let W be the vector-sum XA' + XB' + XC', where A'B'C' is the pedal triangle of X = X(69); then W = X(185)X(20).

X(70)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc/[b2 cos 2B + c2 cos 2C - a2 cos 2A]Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = 1/[b2 cos 2B + c2 cos 2C - a2 cos 2A]

X(70) = isogonal conjugate of X(26)

X(71)

Trilinears (b + c) cos A : (c + a ) cos B : (a + b) cos CBarycentrics (b + c) sin 2A : (c + a ) sin 2B : (a + b) sin 2C

X(71) lies on these lines:1,579 3,48 4,9 6,31 35,284 37,65 54,572 63,69 64,198 74,101 165,610 190,290 583,1100

X(71) = isogonal conjugate of X(27)X(71) = X(I)-Ceva conjugate of X(J) for these (I,J): (3,228), (9, 37), (10,42), (63,72)X(71) = X(228)-cross conjugate of X(73)X(71) = crosspoint of X(I) and X(J) for these (I,J): (3,63), (9,219), (10,306)X(71) = X(4)-line conjugate of X(242)

X(72)

Trilinears (b + c) cot A : (c + a) cot B : (a + b) cot C= (b + c)(b2 + c2 - a2) : (c + a)(c2 + a2 - b2) : (a + b)(a2 + b2 - c2)

Barycentrics (b + c) cos A : (c + a) cos B : (a + b) cos C

X(72) lies on these lines:1,6 2,942 3,63 4,8 5,908 7,443 10,12 20,144 21,943 31,976 35,191 40,64 43,986 54,1006 56,997 57,474 69,304 73,201 74,100 145,452 171,1046 185,916 190,1043 222,1038 248,293 290,668 295,337 306,440 394,1060 519,950 672,1009 894,1010 940,975 978,982

X(72) = reflection of X(65) about X(10)X(72) = isogonal conjugate of X(28)X(72) = isotomic conjugate of X(286)X(72) = anticomplement of X(942)X(72) = X(I)-Ceva conjugate of X(J) for these (I,J): (8,10), (63,71), (69,306), (321,37)X(72) = X(I)-cross conjugate of X(J) for these (I,J): (201,10), (228,37)X(72) = crosspoint of X(I) and X(J) for these (I,J): (8,78), (63,69), (306,307)

X(73) CROSSPOINT OF INCENTER AND CIRCUMCENTER

Trilinears (cos B + cos C) cos A : (cos C + cos A) cos B : (cos A + cos B) cos CBarycentrics (cos B + cos C) sin 2A : (cos C + cos A) sin 2B : (cos A + cos B) sin 2C

X(73) lies on these lines:1,4 3,212 6,41 21,651 35,74 36,54 37,836 42,65 55,64 57,386 66,976 68,1060 69,77 72,201 102,947 228,408 284,951 290,336 1036,1037 1057,1059

X(73) = isogonal conjugate of X(29)X(73) = X(1)-Ceva conjugate of X(65)X(73) = X(228)-cross conjugate of X(71)X(73) = crosspoint of X(I) and X(J) for these (I,J): (1,3), (77,222), (226,307)X(73) = X(1)-Hirst inverse of X(243)

X(74) ISOGONAL CONJUGATE OF EULER INFINITY POINT

Trilinears 1/(cos A - 2 cos B cos C) : 1/(cos B - 2 cos C cos A) : 1/(cos C - 2 cos A cos B)

= f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a/[2a4 - (b2 - c2)2 - a2(b2 + c2)]

Barycentrics a/(cos A - 2 cos B cos C) : b/(cos B - 2 cos C cos A) : c/(cos C - 2 cos A cos B)

As the isogonal conjugate of the point in which the Euler line meets the line at infinity, X(74) lies on the circumcircle; its antipode is X(110). X(74) is the cevapoint of the isodynamic points.

X(74) lies on these lines:2,113 3,110 4,107 6,112 20,68 24,64 30,265 35,73 54,185 65,108 67,935 69,99 71,101 72,100 98,690 187,248 477,523 511,691 512,842 550,930

X(74) = reflection of X(I) about X(J) for these (I,J): (4,125), (110,3), (146,113)X(74) = isogonal conjugate of X(30)X(74) = complement of X(146)X(74) = anticomplement of X(113)X(74) = cevapoint of X(I) and X(J) for these (I,J): (15,16), (50,184)X(74) = X(I)-cross conjugate of X(J) for these (I,J): (186,54), (526,110)

X(75) ISOTOMIC CONJUGATE OF INCENTER

Trilinears 1/a2 : 1/b2 : 1/c2 Barycentrics 1/a : 1/b : 1/c

This is the center X(37) of the anticomplementary triangle.

X(75) lies on these lines:1,86 2,37 6,239 7,8 9,190 10,76 19,27 21,272 31,82 32,746 38,310 43,872 48,336 77,664 99,261 100,675 101,767 141,334 144,391 158,240 194,1107 225,264 234,556 257,698 280,309 299,554 523,876 537,668 689,745 700,971 753,789 758,994 799,897 811,1099

X(75) = reflection of X(192) about X(37)X(75) = isogonal conjugate of X(31)X(75) = isotomic conjugate of X(1)X(75) = complement of X(192)X(75) = anticomplement of X(37)

X(75) = X(I)-Ceva conjugate of X(J) for these (I,J): (76,312), (274,2), (310,76), (314,69)X(75) = cevapoint of X(I) and X(J) for these (I,J): (1,63), (2,8), (7,347), (10,321), (244,514)

X(75) = X(I)-cross conjugate of X(J) for these (I,J):(1,92), (2,85), (7,309), (8,312), (10,2), (38,1), (63,304), (244,514), (307,69), (321,76), (347,322), (522,190)

X(75) = crosspoint of X(I) and X(J) for these (I,J): (2,330), (274,310)X(75) = X(I)-Hirst inverse of X(J) for these (I,J): (2,350), (334,335)X(75) = X(83)-aleph conjugate of X(31)

X(76) = 3rd BROCARD POINT

Trilinears 1/a3 : 1/b3 : 1/c3

= csc(A - ω) : csc(B - ω) : csc(C - ω)

Barycentrics 1/a2 : 1/b2 : 1/c2

X(76) lies on these lines:1,350 2,39 3,98 4,69 5,262 6,83 8,668 10,75 13,299 14,298 17,303 18,302 31,734 32,384 85,226 95,96 100,767 115,626 141,698 275,276 297,343 321,561 335,871 338,599 485,491 486,492 524,598 689,755 693,764 761,789 826,882

X(76) = reflection of X(194) about X(39)X(76) = isogonal conjugate of X(32)X(76) = isotomic conjugate of X(6)X(76) = complement of X(194)X(76) = anticomplement of X(39)X(76) = X(I)-Ceva conjugate of X(J) for these (I,J): (308,2), (310,75)X(76) = cevapoint of X(I) and X(J) for these (I,J): (2,69), (6,22), (75,312), (311,343), (313,321), (339,525)X(76) = X(I)-cross conjugate of X(J) for these (I,J): (2,264), (69,305), (141,2), (321,75), (343,69), (525,99)

X(77)

Trilinears 1/(1 + sec A) : 1/(1 + sec B) : 1/(1 + sec C)= cos A sec2(A/2) : cos B sec2(B/2) : cos C sec2(C/2)= (b2 + c2 - a2)/(b + c - a) : (c2 + a2 - b2)/(c + a - b) : (a2 + b2 - c2)/(a + b - c)

Barycentrics a/(1 + sec A) : b/(1 + sec B) : c/(1 + sec C)

X(77) lies on these lines:1,7 2,189 6,241 9,651 29,34 40,947 55,1037 56,1036 57,81 63,219 65,969 69,73 75,664 102,934 283,603 309,318 738,951 988,1106 999,1057

X(77) = isogonal conjugate of X(33)X(77) = isotomic conjugate of X(318)X(77) = X(I)-Ceva conjugate of X(J) for these (I,J): (85,57), (86,7), (348,63)X(77) = cevapoint of X(I) and X(J) for these (I,J): (1,223), (3,222)X(77) = X(I)-cross conjugate of X(J) for these (I,J): (3,63), (73,222)

X(78)

Trilinears 1/(1 - sec A) : 1/(1 - sec B) : 1/(1 - sec C)= cos A csc2(A/2) : cos B csc2(B/2) : cos C csc2(C/2)= (b2 + c2 - a2)(b + c - a) : (c2 + a2 - b2)(c + a - b) : (a2 + b2 - c2)(a + b - c)

Barycentrics a/(1 - sec A) : b/(1 - sec B) : c/(1 - sec C)

X(78) lies on these lines:1,2 3,63 4,908 9,21 20,329 29,33 37,965 38,988 40,100 46,758 55,960 56,480 57,404 69,73 101,205 207,653 210,958 212,283 220,949 226,377 271,394 273,322 280,282 345,1040 392,1057 474,942 517,945 644,728 999,1059

X(78) = isogonal conjugate of X(34)X(78) = isotomic conjugate of X(273)X(78) = X(I)-Ceva conjugate of X(J) for these (I,J): (69,63), (312,9), (332,345)X(78) = X(I)-cross conjugate of X(J) for these (I,J): (3,271), (72,8), (212,9), (219,63)X(78) = crosspoint of X(69) and X(345)

X(79)

Trilinears 1/(1 + 2 cos A) : 1/(1 + 2 cos B) : 1/(1 + 2 cos C)= f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc/(b2 + c2 - a2 + bc)= g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A/2)(sin 3B/2)(sin 3C/2)

Barycentrics h(a,b,c) : h(b,c,a) : h(c,a,b), where h(a,b,c) = 1/(b2 + c2 - a2 + bc)

X(79) lies on these lines:1,30 8,758 9,46 12,484 21,36 33,1063 34,1061 35,226 57,90 65,80 104,946 314,320 388,1000

X(79) = reflection of X(191) about X(442)X(79) = isogonal conjugate of X(35)X(79) = isotomic conjugate of X(319)X(79) = cevapoint of X(481) and X(482)

X(80) REFLECTION OF INCENTER ABOUT FEUERBACH POINT

Trilinears 1/(1 - 2 cos A) : 1/(1 - 2 cos B) : 1/(1 - 2 cos C)= f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc/(b2 + c2 - a2 - bc)

Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = 1/(b2 + c2 - a2 - bc)

X(80) lies on these lines:1,5 2,214 7,150 8,149 9,528 10,21 30,484 33,1061 34,1063 36,104 40,90 46,84 65,79 313,314 497,1000 499,944 516,655 519,908 943,950

X(80) = midpoint between X(8) and X(149)X(80) = reflection of X(I) about X(J) for these (I,J): (1,11), (100,10)X(80) = isogonal conjugate of X(36)X(80) = isotomic conjugate of X(320)X(80) = inverse of X(1) in the Furhmann circleX(80) = anticomplement of X(214)X(80) = cevapoint of X(10) and X(519)X(80) = X(I)-cross conjugate of X(J) for these (I,J): (44,2), (517,1)

X(81) CEVAPOINT OF INCENTER AND SYMMEDIAN POINT

Trilinears 1/(b + c) : 1/(c + a) : 1/(a + b)Barycentrics a/(b + c) : b/(c + a) : c/(a + b)

X(81) lies on these lines:1,21 2,6 7,27 8,1010 19,969 28,60 29,189 32,980 42,100 43,750 55,1002 56,959 57,77 65,961 88,662 99,739 105,110 145,1043 226,651 239,274 314,321 377,387 386,404 411,581 593,757 715,932 859,957 941,967 982,985 1019,1022 1051,1054 1098,1104

X(81) = isogonal conjugate of X(37)X(81) = isotomic conjugate of X(321)X(81) = X(I)-Ceva conjugate of X(J) for these (I,J): (7,229), (86,21), (286,28)X(81) = cevapoint of X(I) and X(J) for these (I,J): (1,6), (57,222), (58,284)X(81) = X(I)-cross conjugate of X(J) for these (I,J): (1,86), (3,272), (6,58), (57,27), (284,21)X(81) = crosspoint of X(274) and X(286)

X(82)

Trilinears 1/(b2 + c2) : 1/(c2 + a2) : 1/(a2 + b2)= sin A csc(A + ω) : sin B csc(B + ω) : sin C csc(C + ω)

Barycentrics a/(b2 + c2) : b/(c2 + a2) : c/(a2 + b2)

X(82) lies on these lines: 1,560 10,83 31,75 37,251 58,596 689,715 759,827

X(82) = isogonal conjugate of X(38) X(82) = cevapoint of X(1) and X(31)

X(83) CEVAPOINT OF CENTROID AND SYMMEDIAN POINT

Trilinears bc/(b2 + c2) : ca/(c2 + a2) : ab/(a2 + b2)= csc(A + ω) : csc(B + ω) : csc(C + ω)

Barycentrics 1/(b2 + c2) : 1/(c2 + a2) : 1/(a2 + b2)

X(83) lies on these lines:2,32 3,262 4,182 5,98 6,76 10,82 17,624 18,623 39,99 213,239 217,287 275,297 597,671 689,729

X(83) = isogonal conjugate of X(39)X(83) = isotomic conjugate of X(141)X(83) = cevapoint of X(2) and X(6)X(83) = X(I)-cross conjugate of X(J) for these (I,J): (2,308), (6,251), (512,99)

X(84)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = 1/(cos B + cos C - cos A - 1)Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

Let A',B',C' be the excenters. The perpendiculars from B' to A'B and from C' to A'C meet in a point A". Points B" and C" are determined cyclically. The hexyl triangle, A"B"C", is perspective to ABC, and X(84) is the perspector.

X(84) lies on these lines: 1,221 3,9 4,57 7,946 8,20 21,285 33,603 36,90 46,80 58,990 171,989 256,988 294,580 309,314 581,941 944,1000

X(84) = isogonal conjugate of X(40)X(84) = isotomic conjugate of X(322)X(84) = X(I)-Ceva conjugate of X(J) for these (I,J): (189,282), (280,1)X(84) = X(I)-cross conjugate of X(J) for these (I,J): (19,57), (56,1) X(84) = X(280)-aleph conjugate of X(84)

X(85) ISOTOMIC CONJUGATE OF X(9)

Trilinears b2c2/(b + c - a) : c2a2/(c + a - b) : a2b2/(a + b - c)= tan(A/2) cos2A : tan(B/2) cos2B : tan(C/2) cos2C

Barycentrics bc/(b + c - a) : ca/(c + a - b) : ab/(a + b - c)

X(85) lies on these lines:1,664 2,241 7,8 12,120 29,34 56,870 57,274 76,226 92,331 109,767 150,355 178,508 264,309

X(85) = isogonal conjugate of X(41)X(85) = isotomic conjugate of X(9)X(85) = X(274)-Ceva conjugate of X(348)

X(85) = cevapoint of X(I) and X(J) for these (I,J): (1,169), (2,7), (57,77), (92,342)X(85) = X(I)-cross conjugate of X(J) for these (I,J): (2,75), (57,273), (92,309), (142,2), (226,7)

X(86) CEVAPOINT OF INCENTER AND CENTROID

Trilinears bc/(b + c) : ca/(c + a) : ab/(a + b)Barycentrics 1/(b + c) : 1/(c + a) : 1/(a + b)

X(86) lies on these lines:1,75 2,6 7,21 10,319 29,34 37,190 58,238 60,272 99,106 110,675 142,284 239,1100 269,1088 283,307 310,350 741,789 870,871

X(86) = isogonal conjugate of X(42)X(86) = isotomic conjugate of X(10)X(86) = X(274)-Ceva conjugate of X(333)X(86) = cevapoint of X(I) and X(J) for these (I,J): (1,2), (7,77), (21,81)X(86) = X(I)-cross conjugate of X(J) for these (I,J): (1,81), (2,274), (7,286), (21,333), (58,27), (513,190)

X(87) X(2)-CROSS CONJUGATE OF X(1)

Trilinears 1/(ab + ac - bc) : 1/(bc + ba - ca) : 1/(ca + cb - ab)Barycentrics a/(ab + ac - bc) : b/(bc + ba - ca) : c/(ca + cb - ab)

X(87) lies on these lines: 1,192 6,43 9,292 10,979 34,242 56,238 58,978 106,932

X(87) = isogonal conjugate of X(43)X(87) = cevapoint of X(2) and X(330)X(87) = X(2)-cross conjugate of X(1)

X(88)

Trilinears 1/(b + c - 2a) : 1/(c + a - 2b) : 1/(a + b - 2c)Barycentrics a/(b + c - 2a) : b/(c + a - 2b) : c/(a + b - 2c)

X(88) lies on these lines: 1,100 2,45 6,89 28,162 44,679 57,651 81,662 105,901 274,799 278,653 279,658 291,660

X(88) = isogonal conjugate of X(44)X(88) = cevapoint of X(I) and X(J) for these (I,J): (1,44), (6,36)X(88) = X(I)-cross conjugate of X(J) for these (I,J): (44,1), (517,7) X(88) = X(I)-aleph conjugate of X(J) for these (I,J): (88,1), (679,88), (903,63), (1022,1052)

X(89)

Trilinears 1/(2b + 2c - a) : 1/(2c + 2a - b) : 1/(2a + 2b - c)Barycentrics a/(2b + 2c - a) : b/(2c + 2a - b) : c/(2a + 2b - c)

X(89) lies on these lines: 1,902 2,44 6,88 649,1022

X(89) = isogonal conjugate of X(45)

X(90) X(3)-CROSS CONJUGATE OF X(1)

Trilinears 1/(cos B + cos C - cos A) : 1/(cos C + cos A - cos B) : 1/(cos A + cos B - cos C)Barycentrics a/(cos B + cos C - cos A) : b/(cos C + cos A - cos B) : c/(cos A + cos B - cos C)

X(90) lies on these lines: 1,155 4,46 9,35 21,224 33,47 36,84 40,80 57,79

X(90) = isogonal conjugate of X(46)X(90) = X(3)-cross conjugate of X(1)

X(91)

Trilinears sec 2A : sec 2B : sec 2CBarycentrics sin A sec 2A : sin B sec 2B : sin C sec 2C

X(91) lies on these lines: 19,920 31,1087 37,498 47,92 63,921 65,68 225,847 255,1109 759,925

X(91) = isogonal conjugate of X(47)X(91) = X(48)-cross conjugate of X(92)

X(92) CEVAPOINT OF INCENTER AND CLAWSON POINT

Trilinears csc 2A : csc 2B : csc 2CBarycentrics sec A : sec B : sec C

X(92) lies on these lines:1,29 2,273 4,8 7,189 19,27 25,242 31,162 38,240 40,412 47,91 55,243 57,653 85,331 100,917 226,342 239,607 255,1087 257,297 264,306 304,561 406,1068 608,894

X(92) = isogonal conjugate of X(48)X(92) = isotomic conjugate of X(63)X(92) = X(I)-Ceva conjugate of X(J) for these (I,J): (85, 342), (264,318), (286,4), (331,273)X(92) = cevapoint of X(I) and X(J) for these (I,J): (1,19), (4,281), (47,48), (196,278)X(92) = X(I)-cross conjugate of X(J) for these (I,J): (1,75), (4,273), (19,158), (48,91),

(226,2), (281,318)X(92) = crosspoint of X(I) and X(J) for these (I,J): (85,309), (264,331)X(92) = X(275)-aleph conjugate of X(47)

X(93)

Trilinears sec 3A : sec 3B : sec 3CBarycentrics sin A sec 3A : sin B sec 3B : sin C sec 3C

X(93) lies on these lines: 4,562 49,94 186,252

X(93) = isogonal conjugate of X(49)X(93) = X(50)-cross conjugate of X(94)

X(94)

Trilinears csc 3A : csc 3B : csc 3CBarycentrics sin A csc 3A : sin B csc 3B : sin C csc 3C

X(94) lies on these lines: 2,300 4,143 23,98 49,93 96,925 275,324

X(94) = isogonal conjugate of X(50)X(94) = isotomic conjugate of X(323)X(94) = cevapoint of X(49) and X(50)X(94) = X(I)-cross conjugate of X(J) for these (I,J): (30,264), (50,93), (265,328)X(94) = X(300)-Hirst inverse of X(301)

X(95) CEVAPOINT OF CENTROID AND CIRCUMCENTER

Trilinears b2c2 sec(B - C) : c2a2 sec(C - A) : a2b2 sec(A - B)Barycentrics bc sec(B - C) : ca sec(C - A) : ab sec(A - B)

X(95) lies on these lines:2,97 3,264 54,69 76,96 99,311 140,340 141,287 160,327 183,305 216,648 307,320

X(95) = isogonal conjugate of X(51)X(95) = isotomic conjugate of X(5)X(95) = anticomplement of X(233)X(95) = X(276)-Ceva conjugate of X(275)X(95) = cevapoint of X(I) and X(J) for these (I,J): (2,3), (6,160), (54,97)X(95) = X(I)-cross conjugate of X(J) for these (I,J): (2,276), (3,97), (54,275), (140,2)

X(96)

Trilinears sec 2A sec(B - C) : sec 2B sec(C - A) : sec 2C sec(A - B)Barycentrics a sec 2A sec(B - C) : b sec 2B sec(C - A) : c sec 2C sec(A - B)

X(96) lies on these lines: 2,54 4,231 24,847 76,95 94,925

X(96) = isogonal conjugate of X(52)X(96) = cevapoint of X(3) and X(68)X(96) = X(3)-cross conjugate of X(54)

X(97)

Trilinears cot A sec(B - C) : cot B sec(C - A) : cot C sec(A - B)Barycentrics cos A sec(B - C) : cos B sec(C - A) : cos C sec(A - B)

X(97) lies on these lines: 2,95 3,54 110,418 216,288 276,401

X(97) = isogonal conjugate of X(53)X(97) = isotomic conjugate of X(324)X(97) = X(95)-Ceva conjugate of X(54)X(97) = X(3)-cross conjugate of X(95)

Centers 98- 112, 74, and 476 lie on the circumcircle. Mappings Λ and Ψ for such points are defined

here:Λ(P,X) = isogonal conjugate of the point where line PX meets the line at infinity; letY = Λ(P,X); Q = isogonal conjugate of P; Y and Z = points where line YQ meets the

circumcircle;then Ψ(P,X) = Z.

X(98) = TARRY POINT

Trilinears sec(A + ω) : sec(B + ω) : sec(C + ω)= f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc/(b4 + c4 - a2b2 - a2c2)

Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = 1/(b4 + c4 - a2b2 - a2c2)

The antipode of X(99), and a point of intersection of the circumcircle and the Kiepert hyperbola. Also, X(98) = Ψ(X(101), X(10)).

J. W. Clawson, "Points on the circumcircle," American Mathematical Monthly 32 (1925) 169-174.

X(98) lies on these lines:2,110 3,76 4,32 5,83 6,262 10,101 13,1080 14,383 20,148 22,925 23,94 25,107 30,671

100,228 109,171 186,935 275,427 376,543 381,598 385,511 468,685 523,842 620,631 804,878

X(98) = midpoint between X(20) and X(148)X(98) = reflection of X(I) about X(J) for these (I,J): (4,115), (99,3), (147,114)X(98) = isogonal conjugate of X(511)X(98) = isotomic conjugate of X(325)X(98) = complement of X(147)X(98) = anticomplement of X(114)X(98) = X(290)-Ceva conjugate of X(287)X(98) = cevapoint of X(I) and X(J) for these (I,J): (2,385), (6,237)X(98) = X(I)-cross conjugate of X(J) for these (I,J): (230,2), (237,6), (248,287), (446,511)X(98) = X(2)-Hirst inverse of X(287)

X(99) = STEINER POINT

Trilinears bc/(b2 - c2) : ca/(c2 - a2) : ab/(a2 - b2)= b2c2 csc(B - C) : c2a2 csc(C - A) : a2b2 csc(A - B)

Barycentrics 1/(b2 - c2) : 1/(c2 - a2) : 1/(a2 - b2)

The antipode of X(98), and a point of intersection of the circumcircle and the Steiner ellipse. Also, X(98) = Ψ(X(6), X(2)).

X(99) lies on these lines:1,741 2,111 3,76 4,114 6,729 13,303 14,302 20,147 21,105 22,305 30,316 31,715 32,194 36,350 38,745 39,83 58,727 69,74 75,261 81,739 86,106 95,311 100,668 101,190 102,332 103,1043 104,314 108,811 109,643 110,690 112,648 141,755 163,825 187,385 249,525 264,378 286,915 298,531 299,530 310,675 476,850 512,805 523,691 524,843 666,919 669,886 670,804 692,785 695,711 813,1016 889,898

X(99) = midpoint between X(I) and X(J) for these (I,J): (20,147), (616,617)X(99) = reflection of X(I) about X(J) for these (I,J): (4,114), (98,3), (148,115), (316,325), (385,187)X(99) = isogonal conjugate of X(512)X(99) = isotomic conjugate of X(523)X(99) = complement of X(148)X(99) = anticomplement of X(115)X(99) = cevapoint of X(I) and X(J) for these (I,J): (2,523), (3,525), (39,512), (100,190)X(99) = X(I)-cross conjugate of X(J) for these (I,J): (3,249), (22,250), (512,83), (523,2), (525,76)

X(100) ANTICOMPLEMENT OF FEUERBACH POINT

Trilinears 1/(b - c) : 1/(c - a) : 1/(a - b)= (a - b)(a - c) : (b - c)(b - a) : (c - a)(c - b)

Barycentrics a/(b - c) : b/(c - a) : c/(a - b)

The antipode of X(104) on the circumcircle; X(100) = Ψ(X(6), X(1)).

X(100) lies on these lines:1,88 2,11 3,8 4,119 6,739 7,1004 9,1005 10,21 20,153 22,197 31,43 32,713 36,519 37,111 40,78 42,81 46,224 56,145 59,521 63,103 72,74 75,675 76,767 92,917 98,228 99,668 101,644 107,823 108,653 109,651 110,643 112,162 144,480 190,659 198,346 213,729 238,899 281,1013 329,972 442,943 484,758 513,765 516,908 517,953 518,840 522,655 560,697 594,1030 645,931 649,660 650,919 658,664 667,898 693,927 731,869 733,893 753,984 756,846 789,874 976,986

X(100) = midpoint between X(20) and X(153)X(100) = reflection of X(I) about X(J) for these (I,J): (1,214), (4,119), (80,10), (104,3), (149,11)X(100) = isogonal conjugate of X(513)X(100) = complement of X(149)X(100) = anticomplement of X(11)X(100) = X(99)-Ceva conjugate of X(190)X(100) = cevapoint of X(I) and X(J) for these (I,J): (1,513), (3,521), (10,522), (142,514), (442,523)X(100) = X(I)-cross conjugate of X(J) for these (I,J): (3,59), (513,1), (521,8), (522,21)X(100) = X(1)-line conjugate of X(244)

X(100) = X(I)-aleph conjugate of X(J) for these (I,J): (1,1052), (100,1), (190,63), (643,411), (666,673), (765,100), (1016,190)

X(101)

Trilinears a/(b - c) : b/(c - a) : c/(a - b)= a(a - b)(a - c) : b(b - c)(b - a) : c(c - a)(c - b)

Barycentrics a2/(b - c) : b2/(c - a) : c2/(a - b)

The antipode of X(103) on the circumcircle; X(101) = Ψ(X(1), X(6)).

X(101) lies on these lines:1,41 2,116 3,103 4,118 6,106 9,48 10,98 19,913 20,152 31,609 32,595 36,672 37,284 40,972 42,111 56,218 58,172 59,657 71,74 75,767 78,205 99,190 100,644 102,198 109,654 110,163 514,664 517,910 522,929 560,713 643,931 649,901 651,934 663,919 667,813 668,789 692,926 733,904 743,869 761,984 765,898

X(101) = midpoint between X(20) and X(152)X(101) = reflection of X(I) about X(J) for these (I,J): (4,118), (103,3), (150,116)X(101) = isogonal conjugate of X(514)X(101) = complement of X(150)X(101) = anticomplement of X(116)X(101) = X(59)-Ceva conjugate of X(55)X(101) = cevapoint of X(354) and X(513)X(101) = X(I)-cross conjugate of X(J) for these (I,J): (55,59), (199,250)X(101) = X(I)-aleph conjugate of X(J) for these (I,J): (100,165), (509,1052), (662,572), (664,169)

X(102)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = 1/[sin B (sec A - sec B) + sin C (sec A - sec C)]= g(a,b,c) : g(b,c,a) : g(c,a,b), whereg(a,b,c) = a/[2a5 + (b + c)a4 - 2(b2 + c2)a3 - (b + c)(b2 - c2)2]

Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), whereg(A,B,C) = (sin A)/[sin B (sec A - sec B) + sin C (sec A - sec C)]

The antipode of X(109) on the circumcircle; also, X(102) = Λ(X(1), X(4)).

X(102) lies on these lines:1,108 2,117 3,109 4,124 19,282 29,107 40,78 73,947 77,934 99,332 101,198 103,928 110,283 112,284 226,1065 516,929

X(102) = midpoint between X(20) and X(153)X(102) = reflection of X(I) about X(J) for these (I,J): (4,124), (109,3), (151,117)X(102) = isogonal conjugate of X(515)X(102) = complement of X(151)X(102) = anticomplement of X(117)

X(103)

Trilinears a/[(a - b) cot C + (a - c) cot B] : b/[(b - c) cot A + (b - a) cot C] : c/[(c - a) cot B + (c - b) cot A]Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2/[(a - b) cot C + (a - c) cot B]

The antipode of X(101) on the circumcircle; X(103) = Ψ(X(101), X(3)).

X(103) lies on these lines:1,934 2,118 3,101 4,116 20,150 27,107 33,57 55,109 58,112 63,100 99,1043 102,928 295,813 376,544 515,929 516,927 572,825 672,919 910,971

X(103) = midpoint between X(20) and X(150)X(103) = reflection of X(I) about X(J) for these (I,J): (4,116), (101,3), (152,118)

X(103) = isogonal conjugate of X(516)X(103) = complement of X(152)X(103) = anticomplement of X(118)

X(104)

Trilinears 1/(-1 + cos B + cos C) : 1/(-1 + cos C + cos A) : 1/(-1 + cos C + cos B)Barycentrics a/(-1 + cos B + cos C) : b/(-1 + cos C + cos A) : c/(-1 + cos C + cos B)

The antipode of X(100) on the circumcircle; X(104) = Λ(X(1), X(3)) = Ψ(X(101), X(9)).

X(104) lies on these lines:1,109 2,119 3,8 4,11 7,934 9,48 20,149 21,110 28,107 36,80 55,1000 79,946 99,314 105,885 112,1108 256,1064 294,919 355,404 376,528 513,953 517,901 631,958

X(104) = midpoint between X(20) and X(149)X(104) = reflection of X(I) about X(J) for these (I,J): (4,11), (100,3), (153,119)X(104) = isogonal conjugate of X(517)X(104) = complement of X(153)X(104) = anticomplement of X(517) X(104) = cevapoint of X(I) and X(J) for these (I,J): (1,36), (44,55)

X(105)

Trilinears 1/[b2 + c2 - a(b + c)] : 1/[c2 + a2 - b(c + a)] : 1/[a2 + b2 - c(a + b)]Barycentrics a/[b2 + c2 - a(b + c)] : b/[c2 + a2 - b(c + a)] : c/[a2 + b2 - c(a + b)]

A center on the circumcircle; X(105) = Λ(X(1), X(6)) = Ψ(X(101), X(1)).

X(105 lies on these lines:1,41 2,11 3,277 6,1002 21,99 25,108 28,112 31,57 56,279 81,110 88,901 104,885 106,1022 165,1054 238,291 330,932 513,840 644,1083 659,884 666,898 825,985 910,919 961,1104

X(105) = isogonal conjugate of X(518)X(105) = anticomplement of X(120)X(105) = cevapoint of X(1) and X(238)X(105) = X(1)-Hirst inverse of X(294)

X(106)

Trilinears a/(2a - b - c) : b/(2b - c - a) : c/(2c - a - b)Barycentrics a2/(2a - b - c) : b2/(2b - c - a) : c2/(2c - a - b)

A center on the circumcircle; X(106) = Λ(X(1), X(2)) = Ψ(X(101), X(6)).

X(106) lies on these lines:1,88 2,121 6,101 34,108 36,901 56,109 58,110 86,99 87,932 105,1022 238,898 269,934 292,813 614,998 663,840 789,870 833,977 919,1055

X(106) = isogonal conjugate of X(519)X(106) = anticomplement of X(121)X(106) = X(36)-cross conjugate of X(58)

X(107)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = 1/[cos A (sin 2B - sin 2C)= g(a,b,c) : g(b,c,a) : g(c,a,b) where g(a,b,c) = bc/[(b2 - c2)(b2 + c2 - a2)2]

Barycentrics 1/[(b2 - c2)(b2 + c2 - a2)2] : 1/[(c2 - a2)(c2 + a2 - b2)2] : 1/[(a2 - b2)(a2 + b2 - c2)2]

A center on the circumcircle; X(107) = Ψ(X(6), X(4)).

X(107) lies on these lines:2,122 4,74 24,1093 25,98 27,103 28,104 29,102 51,275 100,823 109,162 110,648 111,393 158,759 186,477 250,687 450,511 468,842 741,1096

X(107) = reflection of X(4) about X(133)X(107) = isogonal conjugate of X(520)X(107) = anticomplement of X(122)X(107) = cevapoint of X(4) and X(523)X(107) = X(I)-cross conjugate of X(J) for these (I,J): (24,250), (108,162), (523,4)X(107) = trilinear pole of line X(4)X(6)

X(108)

Trilinears a/(sec B - sec C) : b/(sec C - sec A): c/(sec A - sec B)= g(a,b,c) : g(b,c,a) : g(c,a,b) where g(a,b,c) = 1/[(b - c)(b + c - a)(b2 + c2 - a2)]

Barycentrics a2/(sec B - sec C) : b2/(sec C - sec A): c2/(sec A - sec B)

A center on the circumcircle; X(108) = Ψ(X(3), X(1)) = Ψ(X(1), X(4)).

X(108) lies on these lines:1,102 2,123 4,11 7,1013 12,451 24,915 25,105 28,225 33,57 34,106 40,207 55,196 65,74 99,811 100,653 109,1020 110,162 204,223 273,675 318,404 331,767 388,406 429,961 608,739 648,931

X(108) = isogonal conjugate of X(521)X(108) = anticomplement of X(123)X(108) = X(162)-Ceva conjugate of X(109)X(108) = cevapoint of X(I) and X(J) for these (I,J): (56,513), (429,523)X(108) = X(513)-cross conjugate of X(4)X(108) = crosspoint of X(107) and X(162)

X(109)

Trilinears a/(cos B - cos C) : b/(cos C - cos A): c/(cos A - cos B)= g(a,b,c) : g(b,c,a) : g(c,a,b) where g(a,b,c) = a/[(b - c)(b + c - a)]

Barycentrics a2/(cos B - cos C) : b2/(cos C - cos A): c2/(cos A - cos B)

The antipode of X(102) on the circumcircle; X(109) = Λ(X(1), X(3)).

X(109) lies on these lines:1,104 2,124 3,102 4,117 7,675 20,151 31,57 34,46 35,73 36,953 40,255 55,103 56,106 58,65 59,901 85,767 98,171 99,643 100,651 101,654 107,162 108,1020 112,163 165,212 191,201 278,917 284,296 478,573 579,608 604,739 649,919 658,927 662,931 840,902

X(109) = midpoint between X(20) and X(151)X(109) = reflection of X(I) about X(J) for these (I,J): (4,117), (102,3)X(109) = isogonal conjugate of X(522)X(109) = anticomplement of X(124)X(109) = X(I)-Ceva conjugate of X(J) for these (I,J): (59,56), (162,108)X(109) = cevapoint of X(65) and X(513)X(109) = X(I)-cross conjugate of X(J) for these (I,J): (56,59), (513,58)X(109) = crosspoint of X(110) and X(162)X(109) = X(I)-aleph conjugate of X(J) for these (I,J): (100,1079), (162,580), (651,223)

X(110) = FOCUS OF KIEPERT PARABOLA

Trilinears csc(B - C) : csc(C - A) : csc(A - B)= a/(b2 - c2) : b/(c2 - a2) : c/(a2 - b2)

Barycentrics a2/(b2 - c2) : b2/(c2 - a2) : c2/(a2 - b2)

The antipode of X(74) on the circumcircle, and the isogonal conjugate of the isotomic conjugate of X(99). Also, X(110) = Ψ(X(6), X(3)) = Feuerbach point of the tangential triangle.

J. W. Clawson, "Points on the circumcircle," American Mathematical Monthly 32 (1925) 169-174.

Roland H. Eddy and R. Fritsch, "The conics of Ludwig Kiepert: a comprehensive lesson in the geometry of the triangle," Mathematics Magazine 67 (1994) 188-205.

X(110) lies on these lines:1,60 2,98 3,74 4,113 5,49 6,111 11,215 20,146 21,104 22,154 23,323 24,155 27,917 28,915 30,477 31,593 32,729 39,755 58,106 65,229 67,141 69,206 81,105 86,675 97,418 99,690 100,643 101,163 102,283 107,648 108,162 143,195 187,352 190,835 249,512 250,520 251,694 274,767 324,436 351,526 353,574 373,575 376,541 476,523 525,935 560,715 595,849 668,839 669,805 670,689 681,823 685,850 789,799 859,953

X(110) = midpoint between X(I) and X(J) for these (I,J): (3,399), (20,146), (23,323)X(110) = reflection of X(I) about X(J) for these (I,J): (4,113), (67,141), (74,3), (265,5)X(110) = isogonal conjugate of X(523)X(110) = isotomic conjugate of X(850)X(110) = inverse of X(2) in the Brocard circleX(110) = anticomplement of X(125)X(110) = X(I)-Ceva conjugate of X(J) for these (I,J): (249,6), (250,3)X(110) = cevapoint of X(I) and X(J) for these (I,J): (3,520), (5,523), (6,512), (141,525)

X(110) = X(I)-cross conjugate of X(J) for these (I,J):(1,59), (3,250), (6,249), (109,162), (351,111), (512,6), (520,3), (523,54), (526,74)

X(110) = X(I)-Hirst inverse of X(J) for these (I,J): (1,245), (2,125), (3,246), (4,247)

Let X = X(110) and let V be the vector-sum XA + XB + XC; then V = X(265)X(399).

Let W be the vector-sum XA' + XB' + XC', where A'B'C' is the pedal triangle of X;then W = X(125)X(110) = X(265)X(113) = X(113,399).

X(111) = PARRY POINT

Trilinears a/(2a2 - b2 - c2) : b/(2b2 - c2 - a2) : c/(2c2 - a2 - b2)Barycentrics a2/(2a2 - b2 - c2) : b2/(2b2 - c2 - a2) : c2/(2c2 - a2 - b2)

A point on the circumcircle; X(111) = Λ(X(2), X(6)).

X(111) lies on these lines:2,99 6,110 23,187 25,112 37,100 42,101 107,393 182,353 230,476 251,827 308,689 352,511 385,892 468,935 512,843 647,842 694,805 931,941

X(111) = isogonal conjugate of X(524)X(111) = inverse of X(353) in the Brocard circleX(111) = anticomplement of X(126)X(111) = cevapoint of X(6) and X(187)X(111) = X(I)-cross conjugate of X(J) for these (I,J): (23,251), (187,6), (351,110)

X(112)

Trilinears a/(sin 2B - sin 2C) : b/(sin 2C - sin 2A) : c/(sin 2A - sin 2B)= f(a,b,c) : f(b,c,a) : f(c,a,b) where f(a,b,c) = a/[(b2 - c2)(b2 + c2 - a2)]

Barycentrics a2/(sin 2B - sin 2C) : b2/(sin 2C - sin 2A) : c2/(sin 2A - sin 2B)

A point on the circumcircle; X(112) = Ψ(X(4), X(6)).

X(112) lies on these lines:2,127 4,32 6,74 19,759 25,111 27,675 28,105 33,609 50,477 54,217 58,103 99,648

100,162 102,284 104,1108 109,163 186,187 230,403 250,691 251,427 286,767 376,577 393,571 523,935 789,811

X(112) = reflection of X(4) about X(132)X(112) = isogonal conjugate of X(525)X(112) = anticomplement of X(127)X(112) = X(I)-Ceva conjugate of X(J) for these (I,J): (249,24), (250,25)X(112) = cevapoint of X(I) and X(J) for these (I,J): (32,512), (427,523)X(112) = X(I)-cross conjugate of X(J) for these (I,J): (25,250), (512,4), (523,251)

Centers 113-139 lie on the nine-point circle.

X(113) = JERABEK ANTIPODE

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), wheref(A,B,C) = [1 - cos A (sin 2B + sin 2C)][cos A - 2 cos B cos C]

Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A)f(A,B,C)

The antipode on the nine-point circle of X(125); also, X(74) of the medial triangle, and X(104) of the orthic triangle.

X(113) lies on these lines: 2,74 3,122 4,110 5,125 6,13 11,942 52,135 114,690 123,960 127,141 137,546

X(113) = midpoint between X(I) and X(J) for these (I,J): (4,110), (74,146), (265,399)X(113) = reflection of X(125) about X(5)X(113) = X(4)-Ceva conjugate of X(30)X(113) = crosspoint of X(4) and X(403)

Let X = X(113) and let V be the vector-sum XA + XB + XC; then V = X(113)X(146).

X(114) = KIEPERT ANTIPODE

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc[b sec(B + ω) + c sec(C + ω)]Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = b sec(B + ω) + c sec(C + ω)

The antipode on the nine-point circle of X(115); also, X(98) of the medial triangle, and X(103) of the orthic triangle.

X(114) lies on these lines: 2,98 3,127 4,99 5,39 25,135 52,211 113,690 132,684 136,427 325,511 381,543

X(114) = midpoint between X(I) and X(J) for these (I,J): (4,99), (98,147)X(114) = reflection of X(115) about X(5)X(114) = X(I)-Ceva conjugate of X(J) for these (I,J): (2,230), (4,511)X(114) = crosspoint of X(2) and X(325)

Let X = X(114) and let V be the vector-sum XA + XB + XC; then V = X(114)X(147).

X(115) = CENTER OF KIEPERT HYPERBOLA

Trilinears bc(b2 - c2)2 : ca(c2 - a2)2 : ab(a2 - b2)2

Barycentrics (b2 - c2)2 : (c2 - a2)2 : (a2 - b2)2

On the nine-point circle; also, X(99) of the medial triangle, and X(101) of the orthic triangle.

Roland H. Eddy and R. Fritsch, "The conics of Ludwig Kiepert: a comprehensive lesson in the geometry of the triangle," Mathematics Magazine 67 (1994) 188-205.

X(115) lies on these lines:2,99 4,32 5,39 6,13 11,1015 30,187 50,231 53,133 76,626 116,1086 120,442 125,245 127,338 128,233 129,389 131,216 232,403 316,385 325,538 395,530 396,531 593,1029 804,1084

X(115) = midpoint between X(I) and X(J) for these (I,J): (4,98), (13,14), (99,148), (316,385)X(115) = reflection of X(I) about X(J) for these (I,J): (114,5), (187,230)X(115) = isogonal conjugate of X(249)X(115) = inverse of X(6) in the orthocentroidal circleX(115) = X(I)-Ceva conjugate of X(J) for these (I,J): (2,523), (4,512), (338,125)X(115) = crosspoint of X(I) and X(J) for these (I,J): (2,523), (68,525)X(115) = X(2)-Hirst inverse of X(148)

Let X = X(115) and let V be the vector-sum XA + XB + XC; then V = X(115)X(148).

X(116)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b) wheref(a,b,c) = bc/[(b4 + c4 - a(b3 + c3) - bc(b2 + c2) + abc(b + c)]

Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b) whereg(a,b,c) = (b2 + c2 + bc - ab - ac)(b - c)2

A point on the nine-point circle; also X(101) of the medial triangle.

X(116) lies on these lines: 2,101 4,103 5,118 10,120 115,1086 119,142 121,141 124,928

X(116) = midpoint between X(I) and X(J) for these (I,J): (4,103), (101,150)X(116) = reflection of X(118) about X(5)X(116) = X(4)-Ceva conjugate of X(514)

Let X = X(116) and let V be the vector-sum XA + XB + XC; then V = X(116)X(150).

X(117)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = g(b,c,a) + g(b,c,a), andg(b,c,a) = b2c/[c(sec B - sec C) + a(sec B - sec A)]

Barycentrics h(a,b,c) : h(b,c,a) : h(c,a,b) where h(a,b,c) = af(a,b,c)

A point on the nine-point circle; also X(102) of the medial triangle.

X(117) lies on these lines: 2,102 4,109 5,124 10,123 11,65 118,928 136,407

X(117) = midpoint between X(I) and X(J) for these (I,J): (4,109), (102,151)X(117) = reflection of X(124) about X(5)X(117) = X(4)-Ceva conjugate of X(515)

Let X = X(117) and let V be the vector-sum XA + XB + XC; then V = X(117)X(151).

X(118)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = g(b,c,a) + g(b,c,a), andg(b,c,a) = b3c/[(b - c) cot A + (b - a) cot C]

Barycentrics h(a,b,c) : h(b,c,a) : h(c,a,b) where h(a,b,c) = af(a,b,c)

A point on the nine-point circle; also X(103) of the medial triangle.

X(118) lies on these lines: 2,103 4,101 5,116 11,226 117,928 122,440 136,430 381,544 516,910

X(118) = midpoint between X(I) and X(J) for these (I,J): (4,101), (103,152)X(118) = reflection of X(116) about X(5)X(118) = X(4)-Ceva conjugate of X(516)

Let X = X(118) and let V be the vector-sum XA + XB + XC; then V = X(118)X(152).

X(119) = FEUERBACH ANTIPODE

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), wheref(A,B,C) = (csc A)(-1 + cos B + cos C)[sin 2B + sin 2C + 2(-1 + cos A)(sin B + sin C)]

Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B) whereg(A,B,C) = (-1 + cos B + cos C)[sin 2B + sin 2C + 2(-1 + cos A)(sin B + sin C)]

Antipode on the nine-point circle of X(11); also, X(104) of the medial triangle.

X(119) lies on these lines:1,5 2,104 3,123 4,100 10,124 116,142 125,442 135,431 136,429 214,515 381,528 517,908

X(119) = midpoint between X(I) and X(J) for these (I,J): (4,100), (104,153)X(119) = reflection of X(11) about X(5)X(119) = X(4)-Ceva conjugate of X(517)

Let X = X(119) and let V be the vector-sum XA + XB + XC; then V = X(119)X(153).

X(120)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), wheref(a,b,c) = bc[2abc - (b + c)(a2 + (b - c)2][b2 + c2 - ab -ac]

Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b) where g(a,b,c) = [2abc - (b + c)(a2 + (b - c)2][b2 + c2 - ab -ac]

A point on the nine-point circle; also X(105) of the medial triangle.

X(120) lies on these lines: 2,11 10,116 12,85 115,442

X(120) = X(4)-Ceva conjugate of X(518)

X(121)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), wheref(a,b,c) = bc(b + c - 2a)[b3 + c3 + a(b2 + c2) - 2bc(b + c)]

Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b) where g(a,b,c) = (b + c - 2a)[b3 + c3 + a(b2 + c2) - 2bc(b + c)]

A point on the nine-point circle; also X(106) of the medial triangle.

X(121) lies on these lines: 2,106 10,11 116,141

X(121) = X(4)-Ceva conjugate of X(519)

X(122)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), wheref(a,b,c) = (b2 - c2)2(cos A - cos B cos C) cot2A

Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b) where g(a,b,c) = a(b2 - c2)2(cos A - cos B cos C) cot2A

A point on the nine-point circle; also X(107) of the medial triangle.

X(122) lies on these lines: 2,107 3,113 5,133 118,440 125,684 138,233

X(122) = reflection of X(133) about X(5)X(122) = X(I)-Ceva conjugate of X(J) for these (I,J): (4,520), (253,525)X(122) = crosspoint of X(253) and X(525)

X(123)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), wheref(A,B,C) = (csc A)(sec B - sec C)[(sec A)(sin2B - sin2C) + sin C tan C - sin B tan B]

Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B) whereg(A,B,C) = (sec B - sec C)[(sec A)(sin2B - sin2) + sin C tan C - sin B tan B]

A point on the nine-point circle; also X(108) of the medial triangle.

X(123) lies on these lines: 2,108 3,119 10,117 113,960

X(123) = X(4)-Ceva conjugate of X(521)

X(124)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), wheref(a,b,c) = bc(b + c - a)(b - c)2[(b + c)(b2 + c2 - a2 - bc) + abc]

Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), whereg(a,b,c) = (b + c - a)(b - c)2 [(b + c)(b2 + c2 - a2 - bc) + abc]

A point on the nine-point circle; also X(109) of the medial triangle.

X(124) lies on these lines: 2,109 4,102 5,117 10,119 116,928

X(124) = midpoint between X(4) and X(102)X(124) = reflection of X(117) about X(5)X(124) = X(4)-Ceva conjugate of X(522)

X(125) = CENTER OF JERABEK HYPERBOLA

Trilinears cos A sin2(B - C) : cos B sin2(C - A)] : cos C sin2(A - B)= (sec A)(c cos C - b cos B)2 : (sec B)(a cos A - c cos C)2 : (sec C)(b cos B - a cos A)2

= bc(b2 + c2 - a2)(b2 - c2)2 : ca(c2 + a2 - b2)(c2 - a2)2 : ab(a2 + b2 - c2)(a2 - b2)2

Barycentrics (sin 2A)[sin(B - C)]2 : (sin 2B)[sin(C - A)]2 : (sin 2C)[sin(A - B)]2

On the nine-point circle; also, X(110) of the medial triangle and X(100) of the orthic triangle, if ABC is acute.

Roland H. Eddy and R. Fritsch, "The conics of Ludwig Kiepert: a comprehensive lesson in the geometry of the triangle," Mathematics Magazine 67 (1994) 188-205.

X(125) lies on these lines:2,98 3,131 4,74 5,113 6,67 51,132 68,1092 69,895 115,245 119,442 122,684 126,141 128,140 136,338 381,541 511,858

X(125) = midpoint between X(I) and X(J) for these (I,J): (3,265), (4,74), (6,67)X(125) = reflection of X(113) about X(5)X(125) = isogonal conjugate of X(250)X(125) = inverse of X(184) in the Brocard circleX(125) = complement of X(110)X(125) = X(I)-Ceva conjugate of X(J) for these (I,J): (4,523), (66,512), (68,520), (69,525), (338,115)X(125) = crosspoint of X(I) and X(J) for these (I,J): (4,523), (69,525), (338,339)X(125) = X(2)-line conjugate of X(110)

Let X = X(125) and let V be the vector-sum XA + XB + XC; then V = X(399)X(113) = X(113)X(265).

X(126)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), wheref(a,b,c) = bc(2a2 - b2 - c2)[b4 + c4 + a2(b2 + c2) - 4b2c2]

Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b) where g(a,b,c) = (2a2 - b2 - c2)[b4 + c4 + a2(b2 + c2) - 4b2c2]

A point on the nine-point circle; also X(111) of the medial triangle.

X(126) lies on these lines: 2,99 125,141 625,858 X(126) = X(4)-Ceva conjugate of X(524)

X(127)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), wheref(a,b,c) = bc(sin 2B - sin 2C)[(b2 - c2)sin 2A - b2sin 2B + c2sin 2C]

Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), whereg(a,b,c) = (sin 2B - sin 2C)[(b2 - c2)sin 2A - b2sin 2B + c2sin 2C]

A point on the nine-point circle; also X(112) of the medial triangle.

X(127) lies on these lines: 2,112 3,114 5,132 113,141 115,338 133,381 125,140

X(127) = reflection of X(132) about X(5)X(127) = X(4)-Ceva conjugate of X(525)

X(128)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), wheref(A,B,C) = (sec A)(cos 2B + cos 2C)(1 + 2 cos 2A)(cos 2A + 2 cos 2B cos 2C)

Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B) where g(A,B,C) = (sin A) f(A,B,C)

A point on the nine-point circle; also X(74) of the orthic triangle.

X(128) lies on these lines: 5,137 52,134 53,139 115,233 125,140

X(128) = reflection of X(137) about X(5)X(128) = X(2)-Ceva conjugate of X(231)

X(129)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), wheref(A,B,C) = (sec A)(sin 2A)(sin 2B + sin 2C) s(A,B,C) t(A,B,C),s(A,B,C) = sin4(2B) + sin4(2C) - sin2(2A) sin2(2B) - sin2(2A) sin2(2C),t(A,B,C) = sin4(2A) + sin2(2A) u(A,B,C) + v(A,B,C),u(A,B,C) = sin 2B sin 2C - sin2(2B) - sin2(2C),v(A,B,C) = (sin 2B sin 2C)(sin 2B - sin 2C)2

Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B) where g(A,B,C) = (sin A) f(A,B,C)

A point on the nine-point circle; also X(98) of the orthic triangle.

X(129) lies on these lines: 5,130 51,137 52,139 115,389

X(129) = reflection of X(130) about X(5)

X(130)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), wheref(A,B,C) = (sin A)(sin 2B + sin 2C)[(sin 2B - sin 2C)2][sin2(2A) + sin 2B sin 2C]

Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)

A point on the nine-point circle; also X(99) of the orthic triangle.

X(130) lies on these lines: 5,129 51,138

X(130) = reflection of X(129) about X(5)

X(131)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), wheref(A,B,C) = (sec A)[2T - S(sec 2B + sec 2C)](T - S sec 2A),S = sin 2A + sin 2B + sin 2C, T = tan 2A + tan 2B + tan 2C

Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)

A point on the nine-point circle; also X(102) of the orthic triangle.

X(131) lies on these lines: 3,125 4,135 5,136 115,216

X(131) = reflection of X(136) about X(5)

X(132)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), wheref(A,B,C) = (sec A) u(A,B,C) v(A,B,C),u(A,B,C) = [sin2(2A) + (sin 2B - sin 2C)2 + (sin 2A)(sin 2A - sin 2B - sin 2C)],v(A,B,C) = [sin2(2B) + sin2(2C) - (sin 2A sin 2B) - (sin 2A sin 2C)]

Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)

A point on the nine-point circle; also X(105) of the orthic triangle.

X(132) lies on these lines: 2,107 4,32 5,127 25,136 51,125 114,684 137,428 147,648

X(132) = midpoint between X(4) and X(112)X(132) = reflection of X(127) about X(5)X(132) = X(2)-Ceva conjugate of X(232)X(132) = X(4)-line conjugate of X(248)

X(133)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), wheref(A,B,C) = (sec A) u(A,B,C) v(A,B,C),u(A,B,C) = (sin 2B - sin 2C)2 + sin 2A sin 2B - sin 2A sin 2C - 2 sin 2B sin 2C),v(A,B,C) = 2 sin 2A - sin 2B - sin 2C)]

Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)

A point on the nine-point circle; also X(106) of the orthic triangle.

X(133) lies on these lines: 4,74 5,122 53,115 127,381 136,235

X(133) = midpoint between X(4) and X(107)X(133) = reflection of X(122) about X(5)

X(134)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), wheref(A,B,C) = (sec A) u(A,B,C) [v(B,C,A) - v(C,B,A)],u(A,B,C) = (sin 2A)[sin2(2B) - sin2(2C)][sin2(2B) + sin2(2C) - sin2(2A)]2,v(B,C,A) = sin 2C [sin2(2A) - sin2(2B)][sin2(2A) + sin2(2B) - sin2(2C)]2

Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)

A point on the nine-point circle; also X(107) of the orthic triangle.

X(134) lies on this line: 52,128

X(135)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), wheref(A,B,C) = (sec A)[(sin 2B)/(sec 2C - sec 2A) + (sin 2C)/(sec 2A - sec 2B)]

Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), whereg(A,B,C) = (tan A)[(sin 2B)/(sec 2C - sec 2A) + (sin 2C)/(sec 2A - sec 2B)]

A point on the nine-point circle; also X(108) of the orthic triangle.

X(135) lies on these lines: 4,131 25,114 52,113 119,431

X(136)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), wheref(A,B,C) = (sec A)[(sin 2B - sin 2C)2](sin 2B + sin 2C - sin 2A) u(A,B,C),u(A,B,C) = [sin2(2B) + sin2(2C) - sin2(2A)]

Barycentrics af(A,B,C) : bf(B,C,A) : cf(C,A,B)

A point on the nine-point circle; also X(109) of the orthic triangle.

X(136) lies on these lines:2,925 4,110 5,131 25,132 68,254 114,427 117,407 118,430 119,429 125,338 127,868 133,235

X(136) = reflection of X(131) about X(5)X(136) = complement of X(925)X(136) = X(254)-Ceva conjugate of X(523)

X(137)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), wheref(A,B,C) = (sec A)(sin 2B + sin 2C)[(sin 2B - sin 2C)2] u(A,B,C),u(A,B,C) = [sin2(2A) - sin2(2B) - sin2(2C) - (sin 2B sin 2C)]

Barycentrics af(A,B,C) : bf(B,C,A) : cf(C,A,B)

A point on the nine-point circle; also X(110) of the orthic triangle.

X(137) lies on these lines: 5,128 51,129 53,138 113,546 132,428

X(137) = reflection of X(128) about X(5) X(137) = complement of X(930)

X(138)

Trilinears (v + w) sec A : (w + u) sec B : (u + v) sec C, whereu = u(A,B,C) = (sin 2A)/(2 sin22A - sin22B - sin22C), v = u(B,C,A), w = u(C,A,B)

Barycentrics (v + w) tan A : (w + u) tan B : (u + v) tan C

A point on the nine-point circle; also X(111) of the orthic triangle.

X(138) lies on these lines: 51,130 53,137 122,233

X(139)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), wheref(A,B,C) = (sec A)(sin 2B + sin 2C)[(sin 2B - sin 2C)2][sin2(2B) + sin2(2C) - sin2(2A)] u(A,B,C),u(A,B,C) = (sin 2B)4 + (sin 2C)4 - (sin 2A)4 + (sin 2B sin 2C)[sin2(2B) + sin2(2C) - sin2(2A)]

Barycentrics af(A,B,C) : bf(B,C,A) : cf(C,A,B)

A point on the nine-point circle; also X(112) of the orthic triangle.

X(139) lies on these lines: 52,129 53,128

Centers 113- 170 113- 127, 140- 143: centers of the medial triangle

128- 139: centers of the orthic triangle144- 153: centers of the anticomplementary triangle154- 157, 159- 163: centers of the tangential triangle

164- 170: centers of the excentral triangle

X(140) = Midpoint of X(3) and X(5)

Trilinears 2 cos A + cos(B - C) : 2 cos B + cos(C - A) : 2 cos C + cos(A - B)= 1/(cos A + 2 sin B sin C) : 1/(cos B + 2 sin C sin A) : 1/(cos C + 2 sin A sin B)= f(a,b,c) : f(b,c,a) : f(c,a,b), wheref(a,b,c) = bc[b cos( C - A ) + c cos (B - A)]

Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b) where g(a,b,c) = b cos( C - A ) + c cos (B - A)

A center on the Euler line, the crosspoint of the two Napoleon points, also, X(5) of the medial triangle.

X(140) lies on these lines:2,3 10,214 11,35 12,36 15,18 16,17 39,230 54,252 55,496 56,495 61,395 62,396 95,340 125,128 141,182 143,511 195,323 298,628 299,627 302,633 303,634 343,569 371,615 372,590 524,575 576,597 601,748 602,750 618,630 619,629

X(140) = midpoint between X(I) and X(J) for these (I,J): (3,5), (141,182)X(140) = complement of X(5)X(140) = X(2)-Ceva conjugate of X(233)X(140) = crosspoint of X(I) and X(J) for these (I,J): (2,95), (17,18)

Let X = X(140) and let V be the vector-sum XA + XB + XC; then V = X(140)X(3) = X(143,389) = X(5,140).

X(141) COMPLEMENT OF SYMMEDIAN POINT

Trilinears bc(b2 + c2) : ca(c2 + a2) : ab(a2 + b2)f(A,B,C) = csc2A sin(A + ω) : csc2B sin(B + ω) : csc2C sin(C + ω)

Barycentrics b2 + c2 : c2 + a2 : a2 + b2

X(6) of the medial triangle.

X(141) lies on these lines:2,6 3,66 5,211 10,142 37,742 39,732 45,344 53,264 67,110 75,334 76,698 95,287 99,755 113,127 116,121 125,126 140,182 239,319 241,307 308,670 311,338 317,458 320,894 384,1031 441,577 486,591 498,611 499,613 523,882 542,549 575,629 997,1060

X(141) = midpoint between X(I) and X(J) for these (I,J): (6,69), (66,159), (67,110)X(141) = reflection of X(182) about X(140)X(141) = isogonal conjugate of X(251)X(141) = isotomic conjugate of X(83)X(141) = inverse of X(625) in the nine-point circleX(141) = complement of X(6)X(141) = X(I)-Ceva conjugate of X(J) for these (I,J): (2,39), (67,524), (110,525)X(141) = X(39)-cross conjugate of X(427)X(141) = crosspoint of X(2) and X(76)

Let X = X(141) and let V be the vector-sum XA + XB + XC; then V = X(141)X(69).

Let W be the vector-sum XA' + XB' + XC', where A'B'C' is the pedal triangle of X; then W = X(143)X(140).

X(142) COMPLEMENT OF MITTENPUNKT

Trilinears b + c - [(b - c)2]/a : c + a - [(c - a)2]/b : a + b - [(a - b)2]/cBarycentrics bc[ab + ac - (b - c)2] : ca[bc + ba - (c - a)2] : ab[ca + cb - (a - b)2]

X(9) of the medial triangle. Also, X(142) is the centroid of the set {X(1), X(4), X(7), X(40)}.

X(142) lies on these lines: 1,277 2,7 3,516 5,971 10,141 37,1086 86,284 116,119 214,528 269,948 377,950 474,954

X(142) = midpoint between X(7) and X(9)X(142) = complement of X(9)X(142) = X(100)-Ceva conjugate of X(514)X(142) = crosspoint of X(2) and X(85)

Let X = X(142) and let V be the vector-sum XA + XB + XC; then V = X(142)X(7).

X(143)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (sec A)([cos(2C - 2A) + cos(2A - 2B)]

Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (tan A)([cos(2C - 2A) + cos(2A - 2B)]

X(5) of the orthic triangle.

X(143) lies on these lines: 4,94 5,51 6,26 25,156 30,389 110,195 140,511 324,565

X(143) = midpoint between X(5) and X(52)X(143) = isogonal conjugate of X(252)

X(144) ANTICOMPLEMENT OF X(7)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where Barycentrics tan B/2 + tan C/2 - tan A/2 : tan C/2 + tan A/2 - tan B/2 : tan A/2 + tan B/2 - tan C/2

X(7) of the anticomplementary triangle.

X(144) lies on these lines:

2,7 8,516 20,72 21,954 69,190 75,391 100,480 145,192 219,347 220,279 320,344

X(144) = reflection of X(I) about X(J) for these (I,J): (7,9), (145,390)X(144) = anticomplement of X(7)X(144) = X(8)-Ceva conjugate of X(2)

X(145) ANTICOMPLEMENT OF NAGEL POINT

Trilinears bc(3a - b - c) : ca(3b - c - a) : ab(3c - a - b)Barycentrics 3a - b - c : 3b - c -a : 3c - a - b

X(8) of the anticomplementary triangle.

X(145) lies on these lines: 1,2 4,149 6,346 20,517 21,956 37,391 56,100 72,452 81,1043 144,192 218,644 279,664 329,950 330,1002 377,1056 404,999 515,962

X(145) = reflection of X(I) about X(J) for these (I,J): (8,1), (144,390)X(145) = anticomplement of X(8)X(145) = X(7)-Ceva conjugate of X(2)

X(146)

Trilinears bc(-avw + bwu + cuv) : ca(-bwu + cuv + avw) : ab(-cuv + avw + bwu), whereu = u(A,B,C) = cos A - 2 cos B cos C, v = u(B,C,A), w = u(C,A,B)

Barycentrics -avw + bwu + cuv : -bwu + cuv + avw : -cuv + avw + bwu

X(74) of the anticomplementary triangle.

X(146) lies on these lines: 2,74 4,94 20,110 30,323 147,690 148,193

X(146) = reflection of X(I) about X(J) for these (I,J): (20,110), (74,113)

X(147) TARRY POINT OF ANTICOMPLEMENTARY TRIANGLE

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), wheref(a,b,c) = bc[a8 + (b2 + c2)a6 - (2b4 + 3b2c2 + 2c4)a4

+ (b6 + b4c2 + b2c4 + c6)a2 - b8 + b6c2 + b2c6 - c8]

Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(98) of the anticomplementary triangle.

X(147) lies on these lines: 1,150 2,98 4,148 20,99 132,648 146,690 684,804

X(147) = reflection of X(I) about X(J) for these (I,J): (20,99), (98,114), (148,4)X(147) = X(325)-Ceva conjugate of X(2)

X(148) STEINER POINT OF ANTICOMPLEMENTARY TRIANGLE

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), wheref(a,b,c) = bc[a4 - (b2 - c2)2 + b2c2 - a2b2 - a2c2]

Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a4 - (b2 - c2)2 + b2c2 - a2b2 - a2c2

X(99) of the anticomplementary triangle.

X(148) lies on these lines: 2,99 4,147 13,617 20,98 30,385 146,193 316,538

X(148) = reflection of X(I) about X(J) for these (I,J): (20,98), (99,115), (147,4)X(148) = X(523)-Ceva conjugate of X(2)X(148) = X(2)-Hirst inverse of X(115)

X(149)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), wheref(a,b,c) = bc[b3 + c3 - a3 + (a2 - bc)(b + c) + a(bc - b2 - c2)]

Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), whereg(a,b,c) = b3 + c3 - a3 + (a2 - bc)(b + c) + a(bc - b2 - c2)

X(100) of the anticomplementary triangle.

X(149) lies on these lines: 2,11 4,145 8,80 20,104 151,962 377,1058 404,496

X(149) = reflection of X(I) about X(J) for these (I,J): (8,80), (20,104), (100,11), (153,4)

X(150)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), wheref(a,b,c) = bc[b4 + c4 - a4 + a(bc2 +cb2 - b3 - c3) - bc(a2 + b2 + c2) + (b + c)a3]

Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), whereg(a,b,c) = b4 + c4 - a4 + a(bc2 +cb2 - b3 - c3) - bc(a2 + b2 + c2) + (b + c)a3

X(101) of the anticomplementary triangle.

X(150) lies on these lines: 1,147 2,101 4,152 7,80 20,103 69,668 85,355 295,334 348,944 664,952

X(150) = reflection of X(I) about X(J) for these (I,J): (20,103), (101,116), (152,4)

X(151)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), wheref(a,b,c) = (by + cz - ax)/a, where x : y : z = X(102)

Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(102) of the anticomplementary triangle.

X(151) lies on these lines: 2,102 20,109 149,962 152,928

X(151) = reflection of X(I) about X(J) for these (I,J): (20,109), (102,117)

X(152)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), wheref(a,b,c) = (by + cz - ax)/a, where x : y : z = X(103)

Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(103) of the anticomplementary triangle.

X(152) lies on these lines: 2,103 4,150 20,101 151,928

X(152) = reflection of X(I) about X(J) for these (I,J): (20,101), (103,118), (150,4)

X(153)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), wheref(a,b,c) = (by + cz - ax)/a, where x : y : z = X(104)

Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(104) of the anticomplementary triangle.

X(153) lies on these lines: 2,104 4,145 7,80 11,388 20,100 515,908

X(153) = reflection of X(I) about X(J) for these (I,J): (20,100), (104,119), (149,4)

X(154) X(3)-CEVA CONJUGATE OF X(6)

Trilinears (cos A - cos B cos C)a2 : (cos B - cos C cos A)b2 : (cos C - cos A cos B)c2

= a(tan B + tan C - tan A) : b(tan C + tan A - tan B): c(tan A + tan B - tan C)

Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin2 A)(tan B + tan C - tan A)

X(2) of the tangential triangle.

X(154) lies on these lines:

3,64 6,25 22,110 26,155 31,56 48,55 160,418 197,692 198,212 205,220 237,682

X(154) = isogonal conjugate of X(253)X(154) = X(3)-Ceva conjugate of X(6)

Let X = X(154) and let V be the vector-sum XA + XB + XC; then V = X(64)X(20) = X(66)X(159).

X(155) EIGENCENTER OF ORTHIC TRIANGLE

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (cos A)[cos2B + cos2C - cos2A]Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A)f(A,B,C)

X(4) of the tangential triangle.

X(155) lies on these lines:

1,90 3,49 4,254 5,6 20,323 24,110 25,52 26,154 30,1078 159,511 195,381 382,399 450,1075 648,1093 651,1068

X(155) = reflection of X(I) about X(J) for these (I,J): (26,156), (68,5)X(155) = isogonal conjugate of X(254)X(155) = eigencenter of cevian triangle of X(4)X(155) = eigencenter of anticevian triangle of X(3)X(155) = X(4)-Ceva conjugate of X(3)

Let W be the vector-sum XA' + XB' + XC', where A'B'C' is the pedal triangle of X(155); then W = X(68)X(4) = X(3)X(155).

X(156)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a[b2y/v + c2z/w - a2x/u],u = u(A,B,C) = sin 2A, v = u(B,C,A), w = u(C,A,B);x = x(A,B,C) = u2(v2 + w2) - (v2 - w2)2, y = x(B,C,A), z = x(C,A,B)

Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(5) of the tangential triangle.

X(156) lies on these lines: 3,74 4,49 5,184 25,143 26,154 54,381 546,578 550,1092

X(156) = midpoint between X(26) and X(155)

X(157)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a[b3cos B + c3cos C - a3cos A]= g(a,b,c) : g(b,c,a) : g(c,a,b), whereg(a,b,c) = a[a6 - b6 - c6 - a4(b2 + c2) + b4(c2 + a2) + c4(a2 + b2)]

Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(6) of the tangential triangle.

X(157) lies on these lines: 3,66 6,248 22,183 25,53 161,418 206,216

X(157) = X(264)-Ceva conjugate of X(6)

X(158)

Trilinears sec2A : sec2B : sec2CBarycentrics sec A tan A : sec B tan B : sec C tan C

X(158) lies on these lines:

1,29 3,243 4,65 10,318 37,281 46,412 47,162 75,240 107,759 225,1093 255,775 286,969 823,897 920,921

X(158) = isogonal conjugate of X(255) = isogonal conjugate of X(326)X(158) = X(I)-cross conjugate of X(J) for these (I,J): (19,92), (225,4)X(158) = X(I)-aleph conjugate of X(J) for these (I,J): (821,158), (1105,255)

X(159)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), wheref(a,b,c) = a[(a2 + b2 + c2)sin 2A + (c2 - b2 - a2)sin 2B + (b2 - c2 - a2)sin 2C]

Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(9) of the tangential triangle.

X(159) lies on these lines: 3,66 6,25 22,69 23,193 155,511 197,200

X(159) = reflection of X(I) about X(J) for these (I,J): (6,206), (66,141)X(159) = X(I)-Ceva conjugate of X(J) for these (I,J): (22,3), (69,6)

Let W be the vector-sum XA' + XB' + XC', where A'B'C' is the pedal triangle of X(159); then W = X(64)X(20) = X(66)X(159).

X(160)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), wheref(a,b,c) = a[(a2 + b2)sin 2A + (c2 - a2)sin 2B +(b2 - c2)sin 2C

Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(37) of the tangential triangle.

X(160) lies on these lines: 3,66 6,237 22,325 95,327 154,418 206,57

X(160) = X(95)-Ceva conjugate of X(6)

X(161)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), wheref(a,b,c) = a[(a2 + b2 + c2)sin2(2A) + (c2 - b2 - a2)sin2(2B) +(b2 - c2 - a2)sin2(2C)

Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(63) of the tangential triangle.

X(161) lies on these lines: 6,25 22,343 26,68 157,418

X(161) = X(68)-Ceva conjugate of X(6)

X(162)

Trilinears 1/(sin 2B - sin 2C) : 1/(sin 2C - sin 2A) : 1/(sin 2A - sin 2B)= f(a,b,c) : f(b,c,a) : f(c,a,b) , where f(a,b,c) = 1/[(b2 - c2)(b2 + c2 - a2)]

Barycentrics a/(sin 2B - sin 2C) : b/(sin 2C - sin 2A) : c/(sin 2A - sin 2B)

X(100) of the tangential triangle.

X(162) lies on these lines: 4,270 6,1013 19,897 27,673 28,88 29,58 31,92 47,158 63,204 100,112 107,109 108,110 190,643 238,415 240,896 242,422 255,1099 412,580 799,811

X(162) = isogonal conjugate of X(656)X(162) = X(250)-Ceva conjugate of X(270)X(162) = cevapoint of X(I) and X(J) for this (I,J): (108,109)

X(162) = X(I)-cross conjugate of X(J) for these (I,J): (108,107), (109,110)X(162) = X(I)-aleph conjugate of X(J) for these (I,J): (28,1052), (107,920), (162,1), (648,63)X(162) = trilinear pole of line X(1)X(19)

X(163)

Trilinears (sin 2A)/(sin 2B - sin 2C) : (sin 2B)/(sin 2C - sin 2A) : (sin 2C)/(sin 2A - sin 2B)= a2/(b2 - c2) : b2/(c2 - a2) : c2/(a2 - b2)

Barycentrics a3/(b2 - c2) : b3/(c2 - a2) : c3/(a2 - b2)

X(101) of the tangential triangle.

X(163) lies on these lines: 1,293 19,563 31,923 32,849 48,1094 99,825 101,110 109,112 284,909 643,1018 692,906 798,1101 813,827

X(163) = X(I)-aleph conjugate of X(J) for these (I,J): (648,19), (662,610)

X(164) INCENTER OF EXCENTRAL TRIANGLE

Trilinears sin 2B + sin 2C - sin 2A : sin 2C + sin 2A - sin 2B : sin 2A + sin 2B - sin 2CBarycentrics a(sin 2B + sin 2C - sin 2A) : b(sin 2C + sin 2A - sin 2B) : c(sin 2A + sin 2B - sin 2C)

X(1) of the excentral triangle.

X(164) lies on these lines: 1,258 9,168 40,188 57,177 165,167 173,504 361,503 362,845

X(164) = isogonal conjugate of X(505)X(164) = X(188)-Ceva conjugate of X(1)X(164) = X(I)-aleph conjugate of X(J) for these (I,J): (1,361), (2,362), (9,844), (188,164), (366,173)

X(165) = CENTROID OF THE EXCENTRAL TRIANGLE

Trilinears tan(B/2) + tan(C/2) - tan(A/2) : tan(C/2) + tan(A/2) - tan(B/2) : tan(2/A) + tan(B/2) - tan(C/2)Barycentrics a[tan(B/2) + tan(C/2) - tan(A/2)] : b[tan(C/2) + tan(A/2) - tan(B/2)] : c[tan(2/A) + tan(B/2) - tan(C/2)]

X(165) is the centroid of the triangle with vertices X(1), X(8), X(20), as well as the triangle with vertices X(4), X(20), X(40).

X(165) lies on these lines:1,3 2,516 9,910 10,20 42,991 43,573 63,100 71,610 105,1054 109,212 164,167 166,168

210,971 255,1103 355,550 376,515 380,579 411,936 572,1051 580,601 612,990 614,902 631,946 750,968

X(165) = X(9)-Ceva conjugate of X(1)

X(165) = X(I)-aleph conjugate of X(J) for these (I,J):(2,169), (9,165), (21,572), (100,101), (188,9), (259,43), (365,978), (366,57), (650,1053)

Let X = X(165) and let V be the vector-sum XA + XB + XC; then V = X(1)X(20) = X(4)X(40) = X(382)X(355).

Let W be the vector-sum XA' + XB' + XC', where A'B'C' is the pedal triangle of X; then W = X(392)X(20).

X(166) GERGONNE POINT OF EXCENTRAL TRIANGLE

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), wheref(A,B,C) = (tan A/2)/(cos B/2 + cos C/2 - cos A/2)

Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A)f(A,B,C)

X(7) of the excentral triangle.

X(166) lies on these lines: 165,168 167,188X(166) = X(9)-aleph conjugate of X(167)

X(167) NAGEL POINT OF EXCENTRAL TRIANGLE

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), wheref(A,B,C) = s(B,C,A)t(B,C,A) + s(C,A,B)t(C,A,B) + s(A,B,C)t(A,B,C),s(A,B,C) = sin(A/2), t(A,B,C) = cos B/2 + cos C/2 - cos A/2

Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A)f(A,B,C)

X(8) of the excentral triangle.

X(167) lies on these lines: 1,174 164,165 166,188

X(167) = X(9)-aleph conjugate of X(166)

X(168) MITTENPUNKT OF EXCENTRAL TRIANGLE

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), wheref(A,B,C) = sin A - sin B - sin C + 2[cos A/2 + sin(B/2 - A/2) + sin(C/2 - A/2)]

Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A)f(A,B,C)

X(9) of the excentral triangle.

X(168) lies on these lines: 1,173 9,164 165,166

X(168) = X(188)-aleph conjugate of X(363)

X(169)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), wheref(A,B,C) = - (sin A )cos2(A/2) + (sin B)cos2(B/2) + (sin C)cos2(C/2)

Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A)f(A,B,C)

X(32) of the excentral triangle.

X(169) lies on these lines: 1,41 4,9 6,942 46,672 57,277 63,379 65,218 220,517 572,610

X(169) = X(85)-Ceva conjugate of X(1)

X(169) = X(I)-aleph conjugate of X(J) for these (I,J):(2,165), (85,169), (86,572), (174,43), (188,170), (508,1), (514,1053), (664,101)

X(170)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), wheref(A,B,C) = - (tan A/2)sec2(A/2) + (tan B/2)sec2(B/2) + (tan C/2)sec2(C/2)

Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A)f(A,B,C)

X(76) of the excentral triangle.

X(170) lies on these lines: 1,7 43,218

X(170) = X(220)-Ceva conjugate of X(1)X(170) = X(I)-aleph conjugate of X(J) for these (I,J): (9,9), (55,43), (188,169), (220,170), (644,1018)

X(171)

Trilinears a2 + bc : b2 + ca : c2 + abBarycentrics a3 + abc : b3 + abc : c3 + abc

X(171) lies on these lines: 1,3 2,31 4,601 6,43 7,983 10,58 37,846 42,81 47,498 63,612 72,1046 84,989 98,109 181,511 222,611 292,893 319,757 385,894 388,603 474,978 602,631 756,896

X(171) = isogonal conjugate of X(256)X(171) = X(292)-Ceva conjugate of X(238)

X(172)

Trilinears a3 + abc : b3 + abc : c3 + abcBarycentrics a4 + bca2 : b4 + cab2 : c3 + abc2

X(172) lies on these lines:1,32 6,41 12,230 21,37 35,187 36,39 42,199 58,101 65,248 350,384 577,1038 694,904 699,932

X(172) = isogonal conjugate of X(257)

X(173) = CONGRUENT ISOSCELIZERS POINT

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = cos B/2 + cos C/2 - cos A/2Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A)f(A,B,C)

Let P(B)Q(C) be an isoscelizer: let P(B) on sideline AC and Q(C) on AB be equidistant from A, so that AP(B)Q(C) is an isosceles triangle. Line P(B)-to-Q(C), P(C)-to-Q(A), P(A)-to-Q(B) concur in X(173). (P. Yff, unpublished notes, 1989)

X(173) lies on these lines: 1,168 9,177 57,174 164,504 180,483 503,844

X(173) = isogonal conjugate of X(258)X(173) = X(174)-Ceva conjugate of X(1)

X(173) = X(I)-aleph conjugate of X(J) for these (I,J):(1,503), (2,504), (174,173), (188,845), (366,164), (507,1), (508,362), (509,361)

X(174) = YFF CENTER OF CONGRUENCE

Trilinears sec A/2 : sec B/2 : sec C/2f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = [bc/(b + c - a)]1/2

Barycentrics sin A/2 : sin B/2 : sin C/2

In notes dated 1987, Yff raises a question concerning certain triangles lying within ABC: can three isoscelizers (as defined in connection with X(173), P(B)Q(C), P(C)Q(A), P(A)Q(B) be constructed so that the four triangles P(A)Q(A)A, P(B)Q(B)B, P(C)Q(C)C, ABC are congruent? After proving that the answer is yes, Yff moves the three isoscelizers in such a way that the three outer triangles, P(A)Q(A)A, P(B)Q(B)B, P(C)Q(C)C stay congruent and the inner triangle, ABC, shrinks to X(174).

X(174) lies on these lines: 1,167 2,236 7,234 57,173 175,483 188,266

X(174) = isogonal conjugate of X(259)X(174) = X(508)-Ceva conjugate of X(188)X(174) = cevapoint of X(I) and X(J) for these (I,J): (1,173), (259,266)X(174) = X(I)-cross conjugate of X(J) for these (I,J): (177,7), (259,188)

X(175) = ISOPERIMETRIC POINT

Trilinears -1 + sec A/2 cos B/2 cos C/2 : -1 + sec B/2 cos C/2 cos A/2 : -1 + sec C/2 cos A/2 cos B/2Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A)(-1 + sec A/2 cos B/2 cos C/2)

If a + b + c > 4R + r, where R and r denote the circumradius and inradius, respectively, then there exists a point X for which the perimeters of triangles XBC, XCA, XAB are equal. Veldkamp proved that X = X(175), and Yff, in unpublished notes, proved that X(175) is the center of the outer Soddy circle. See also the 1st and 2nd Eppstein points, X(481), X(482).

Clark Kimberling and R. W. Wagner, Problem E 3020 and Solution, American Mathematical Monthly 93 (1986) 650-652 [proposed 1983].

G. R. Veldkamp, "The isoperimetric point and the point(s) of equal detour," American Mathematical Monthly 92 (1985) 546-558.

X(175) lies on these lines: 1,7 174,483 490,664

X(175) = X(8)-Ceva conjugate of X(176)

X(176) = EQUAL DETOUR POINT

Trilinears 1 + sec A/2 cos B/2 cos C/2 : 1 + sec B/2 cos C/2 cos A/2 : 1 + sec C/2 cos A/2 cos B/2Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A)(1 + sec A/2 cos B/2 cos C/2)

If X is a point not between A and B, we make a detour of magnitude |AX| + |XB| - |AB| if we walk from A to B via X; then a point has the equal detour property if the magnitues of the three detours, A to B via X, B to C via X, and C to A via X, are equal; X(176) is the only such point unless ABC has an angle greater than 2*arcsin(4/5), and then X(175) also has the equal detour property. Yff found that X(176) is also is the center of the inner Soddy circle. The following construction was found by Elkies: call two circles within ABC companion circles if they are the incircles of two triangles formed by dividing ABC into two smaller triangles by passing a line through one of the vertices and some point on the opposite side; chain of circles O(1), O(2), ... such that O(n),O(n+1) are companion incircles for every n consists of at most six distinct circles; there is a unique chain consisting of only three distinct circles; and for this chain, the three subdividing lines concur in X(176). Centers X(175) and X(176) are harmonic conjugates with respect to X(1) and X(7).

G. R. Veldkamp, "The isoperimetric point and the point(s) of equal detour," American Mathematical Monthly 92 (1985) 546-558.

Noam D. Elkies and Jiro Fukuta, Problem E 3236 and Solution, American Mathematical Monthly 97 (1990) 529-531 [proposed 1987].

X(176) lies on these lines: 1,7 489,664

X(176) = X(8)-Ceva conjugate of X(175)

X(177) = 1st MID-ARC POINT

Trilinears (cos B/2 + cos C/2) sec A/2 : (cos C/2 + cos A/2) sec B/2 : (cos A/2 + cos B/2) sec C/2Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A)(cos B/2 + cos C/2) sec A/2

Let A',B',C' be the first points of intersection of the angle bisectors of ABC with its incircle. The tangents at A', B', C' form a triangle A"B"C", and the lines AA",BB",CC" concur in X(177). Also, X(177) = X(1) of the intouch triangle.

Clark Kimberling and G. R. Veldkamp, Problem 1160 and Solution, Crux Mathematicorum 13 (1987) 298-299 [proposed 1986].

X(177) lies on these lines: 1,167 7,555 8,556 9,173 57,164

X(177) = isogonal conjugate of X(260)X(177) = X(7)-Ceva conjugate of X(234)X(177) = crosspoint of X(7) and X(174)

X(178) = 2nd MID-ARC POINT

Trilinears (cos B/2 + cos C/2) csc A : (cos C/2 + cos A/2) csc B : (cos A/2 + cos B/2) csc CBarycentrics cos B/2 + cos C/2 : cos C/2 + cos A/2 : cos A/2 + cos B/2

Let A',B',C' be the first points of intersection of the angle bisectors of ABC with its incircle. Let A",B",C" be the midpoints of segments BC,CA,AB, respectively. The lines A'A",B'B",C'C" concur in X(178).

Clark Kimberling, Problem 804, Nieuw Archikef voor Wiskunde 6 (1988) 170.

X(178) lies on these lines: 2,188 8,236 85,508

X(178) = complement of X(188)X(178) = crosspoint of X(2) and X(508)

X(179) = 1st AJIMA-MALFATTI POINT

Trilinears sec4(A/4) : sec4(B/4) : sec4(C/4)Barycentrics sin A sec4(A/4) : sin B sec4(B/4) : sin C sec4(C/4)

The famous Malfatti Problem is to construct three circles O(A), O(B), O(C) inside ABC such that each is externally tangent to the other two, O(A) is tangent to lines AB and AC, O(B) is tangent to BC and BA, and O(C) is tangent to CA and CB. Let A' = O(B)^ O(C), B' = O(C)^O(A), C' = O(A)^O(B). The lines AA',BB',CC' concur in X(179). Trilinears are found in Yff's unpublished notes. See also the Yff-Malfatti Point, X(400), having trilinears csc4(A/4) : csc4(B/4) : csc4(C/4), and the references for historical notes.

H. Fukagawa and D. Pedoe, Japanese Temple Geometry Problems (San Gaku), The Charles Babbage Research Centre, Winnipeg, Canada, 1989.

Michael Goldberg, "On the original Malfatti problem," Mathematics Magazine, 40 (1967) 241-247.

Clark Kimberling and I. G. MacDonald, Problem E 3251 and Solution, American Mathematical Monthly 97 (1990) 612-613.

X(180) = 2nd AJIMA-MALFATTI POINT

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = 1/t(B,C,A) + 1/t(C,B,A) - 1/t(C,A,B),t(A,B,C) = 1 + 2(sec A/4 cos B/4 cos C/4)2

Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A)f(A,B,C)

Let A",B",C" be the excenters of ABC, and let A',B',C' be as in the construction of X(179). The lines A'A",B'B",B'B" concur in X(180). Trilinears are found in Yff's unpublished notes. See X(179).

X(180) lies on this line: 173,483

X(181) = APOLLONIUS POINT

Trilinears a(b + c)2/(b + c - a) : b(c + a)2/(c + a - b) : c(a + b)2/(a + b - c)= a2cos2(B/2 - C/2) : b2cos2(C/2 - A/2) : c2cos2(A/2 - B/2)

Barycentrics a3cos2(B/2 - C/2) : b3cos2(C/2 - A/2) : c3cos2(A/2 - B/2)

Let O(A),O(B),O(C) be the excircles. Apollonius's Problem includes the construction of the circle O tangent to the three excircles and encompassing them. Let A' = O^O(A), B'=O^O(B), C'=O^O(C). The lines AA',BB',CC' concur in X(181). Yff derived trilinears in 1992.

Clark Kimberling, Shiko Iwata, and Hidetosi Fukagawa, Problem 1091 and Solution, Crux Mathematicorum 13 (1987) 128-129; 217-218. [proposed 1985].

X(181) lies on these lines:1,970 6,197 8,959 10,12 31,51 42,228 43,57 44,375 55,573 56,386 171,511 373,748

X(181) = isogonal conjugate of X(261)

X(182) = MIDPOINT OF BROCARD DIAMETER

Trilinears cos(A - ω) : cos(B - ω) : cos(C - ω)Barycentrics sin A cos(A - ω) : sin B cos(B - ω) : sin C cos(C - ω)

Midpoint of the Brocard diameter (the segment X(3)-to-X(6)); also the center of the 1st Lemoine circle, and the center of the Brocard circle.

X(182) lies on these lines:1,983 2,98 3,6 4,83 5,206 22,51 54,69 55,613 56,611 111,353 140,141 524,549 692,1001 952,996

X(182) = midpoint between X(3) and X(6)X(182) = reflection of X(141) about X(140)X(182) = isogonal conjugate of X(262)X(182) = isotomic conjugate of X(327)

Let W be the vector-sum XA' + XB' + XC', where A'B'C' is the pedal triangle of X(182); then W = X(5)X(3) = X(4)X(5).

X(183)

Trilinears b2c2cos(A - ω) : c2a2cos(B - ω) : a2b2cos(C - ω)Barycentrics csc A cos(A - ω) : csc B cos(B - ω) : csc C cos(C - ω)

X(183) lies on these lines:2,6 3,76 5,315 22,157 25,264 55,350 95,305 187,1003 274,474 316,381 317,427 383,621 538,574 622,1080 668,956

X(183) = isogonal conjugate of X(263)X(183) = isotomic conjugate of X(262)

X(184) = INVERSE OF X(125) IN THE BROCARD CIRCLE

Trilinears a2cos A : b2cos B : c2cos CBarycentrics a3cos A : b3cos B : c3cos C

X(184) lies on these lines:2,98 3,49 4,54 5,156 6,25 22,511 23,576 24,389 26,52 22,511 31,604 32,211 48,212

55,215 157,570 160,571 199,573 205,213 251,263 351,686 381,567 397,463 398,462 418,577 572,1011 647,878

X(184) = isogonal conjugate of X(264)X(184) = inverse of X(125) in the Brocard circleX(184) = X(I)-Ceva conjugate of X(J) for these (I,J): (6,32), (54,6), (74,50)X(184) = X(217)-cross conjugate of X(6)X(184) = crosspoint of X(3) and X(6)X(184) = X(32)-Hirst inverse of X(237)

Let W be the vector-sum XA' + XB' + XC', where A'B'C' is the pedal triangle of X(184); then W = X(343)X(22) = X(427)X(184).

X(185) = NAGEL POINT OF THE ORTHIC TRIANGLE

Trilinears (cos A)[1 - cos A cos(B - C)] : (cos B)[1 - cos B cos(C - A)] : (cos C)[1 - cos C cos(A - B)]Barycentrics (sin 2A)[1 - cos A cos(B - C)] : (sin 2B)[1 - cos B cos(C - A)] : (sin 2C)[1 - cos C cos(A - B)]

X(185) lies on these lines:1,296 3,49 4,51 5,113 6,64 20,193 25,1078 30,52 39,217 54,74 72,916 287,384 378,578 382,568 411,970 648,1105

X(185) = reflection of X(4) about X(389)X(185) = isogonal conjugate of X(1105)X(185) = X(I)-Ceva conjugate of X(J) for these (I,J): (3,417), (4,235)X(185) = crosspoint of X(3) and X(4)

X(186) = INVERSE OF X(4) IN CIRCUMCIRCLE

Trilinears 4 cos A - sec A : 4 cos B - sec B : 4 cos C - sec C= sin 3A csc 2A : sin 3B csc 2B : sin 3C csc 2C

Barycentrics (sin A)(4 cos A - sec A) : (sin B)(4 cos B - sec B) : (sin C)(4 cos C - sec C)

X(186) lies on these lines: 2,3 54,389 93,252 98,935 107,477 112,187 249,250

X(186) = reflection of X(I) about X(J) for these (I,J): (4,403), (403,468)X(186) = isogonal conjugate of X(265)X(186) = isotomic conjugate of X(328)X(186) = inverse of X(4) in the circumcircleX(186) = X(340)-Ceva conjugate of X(323)X(186) = X(50)-cross conjugate of X(323)X(186) = crosspoint of X(54) and X(74)

Let X = X(186) and let V be the vector-sum XA + XB + XC; then V = X(4)X(23).

X(187) = INVERSE OF X(6) IN CIRCUMCIRCLE (SCHOUTE CENTER)

Trilinears a(2a2 - b2 - c2) : b(2b2 - c2 - a2) : c(2c2 - a2 - b2)Barycentrics a2(2a2 - b2 - c2) : b2(2b2 - c2 - a2) : c2(2c2 - a2 - b2)

X(187) lies on these lines:2,316 3,6 23,111 30,115 35,172 36,1015 74,248 99,385 110,352 112,186 183,1003 237,351 249,323 325,620 395,531 396,530 729,805

X(187) = midpoint between X(I) and X(J) for these (I,J): (15,16), (99,385)X(187) = reflection of X(115) about X(230)X(187) = isogonal conjugate of X(671)X(187) = inverse of X(6) in the circumcircleX(187) = inverse of X(574) inthe Brocard circleX(187) = complement of X(316)X(187) = anticomplement of X(625)X(187) = X(111)-Ceva conjugate of X(6)X(187) = crosspoint of X(I) and X(J) for these (I,J): (2,67), (6,111), (468,524)

X(188) = 2nd MID-ARC POINT OF ANTICOMPLEMENTARY TRIANGLE

Trilinears csc A/2 : csc B/2 : csc C/2= [a/(b + c - a)]1/2 : [b/(c + a - b)]1/2 : [c/(a + b - c)]1/2

Barycentrics sin A csc A/2 : sin B csc B/2 : sin C csc C/2

X(188) lies on these lines: 1,361 2,178 9,173 40,164 166,167 174,266

X(188) = isogonal conjugate of X(266)X(188) = isotomic conjugate of X(508)X(188) = anticomplement of X(178)X(188) = X(I)-Ceva conjugate of X(J) for these (I,J): (2,236), (508,174)X(188) = cevapoint of X(1) and X(164)X(188) = X(259)-cross conjugate of X(174)

X(189) = CYCLOCEVIAN CONJUGATE OF X(8)

Trilinears (cos B + cos C - cos A - 1)/a : (cos C + cos A - cos B - 1)/b : (cos A + cos B - cos C - 1)/cBarycentrics 1/(cos B + cos C - cos A - 1) : 1/(cos C + cos A - cos B - 1) : 1/(cos A + cos B - cos C - 1)

X(189) lies on these lines: 2,77 7,92 8,20 29,81 69,309 222,281

X(189) = isogonal conjugate of X(198)X(189) = isotomic conjugate of X(329)X(189) = cyclocevian conjugate of X(8)X(189) = anticomplement of X(223)X(189) = X(309)-Ceva conjugate of X(280)X(189) = cevapoint of X(84) and X(282)X(189) = X(I)-cross conjugate of X(J) for these (I,J): (4,7), (57,2), (282,280)

X(190) = YFF PARABOLIC POINT

Trilinears bc/(b - c) : ca/(c - a) : ab/(a - b)Barycentrics 1/(b - c) : 1/(c - a) : 1/(a - b

In unpublished notes, Yff has studied the parabola tangent to sidelines BC, CA, AB and having focus X(101). If A',B',C' are the respective points of tangency, then the lines AA', BB', CC' concur in X(190).

X(190) lies on these lines:1,537 2,45 6,192 7,344 8,528 9,75 10,671 37,86 40,341 44,239 63,312 69,144 71,290 72,1043 99,101 100,659 110,835 162,643 191,1089 238,726 320,527 321,333 329,345 350,672 513,660 514,1016 522,666 644,651 646,668 649,889 658,1020 670,799 789,813 872,1045

X(190) = reflection of X(I) about X(J) for these (I,J): (239,44), (335,37)X(190) = isogonal conjugate of X(649)X(190) = isotomic conjugate of X(514)X(190) = anticomplement of X(1086)X(190) = X(99)-Ceva conjugate of X(100)X(190) = cevapoint of X(I) and X(J) for these (I,J): (2,514), (9,522), (37,513), (440,525)X(190) = X(I)-cross conjugate of X(J) for these (I,J): (513,86), (514,2), (522,75)X(190) = X(I)-aleph conjugate of X(J) for these (I,J): (2,1052), (190,1), (645,411), (668,63), (1016,100)X(190) = pole of the line X(1)X(2)

Centers 191- 236 are Ceva conjugates. The P-Ceva conjugate of Q is the perspector

of the cevian triangle of P and the anticevian triangle of Q.

X(191) = X(10)-CEVA CONJUGATE OF X(1)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c - a)(bc + ca + ab) + b3 + c3 - a3

Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c).

X(191) lies on these lines:1,21 9,46 10,267 30,40 35,72 36,960 109,201 190,1089 329,498

X(191) = reflection of X(I) about X(J) for these (I,J): (1,21), (79,442)X(191) = isotomic conjugate of X(267)

X(191) = X(10)-Ceva conjugate of X(1)X(191) = crosspoint of X(I) and X(J) for these (I,J): (10,502)X(191) = X(I)-aleph conjugate of X(J) for these (I,J): (2,2), (8,20), (10,191), (37,1045), (188,3), (366,6)

X(192) = X(1)-CEVA CONJUGATE OF X(2) (EQUAL PARALLELIANS POINT)

Trilinears bc(ca + ab - bc) : ca(ab + bc - ca) : ab(bc + ca - ab)Barycentrics ca + ab - bc : ab + bc - ca : bc + ca - ab

Segments through X(192) parallel to the sidelines with endpoints on the sidelines have equal length. For references as early as 1881, see Hyacinthos message 2929 (Paul Yiu, May 29, 2001). See also

Sabrina Bier, "Equilateral Triangles Intercepted by Oriented Parallelians," Forum Geometricorum 1 (2001) 25-32.

X(192) lies on these lines:1,87 2,37 6,190 7,335 8,256 9,239 55,385 69,742 144,145 315,746 869,1045

X(192) = reflection of X(75) about X(37)X(192) = isotomic conjugate of X(330)X(192) = anticomplement of X(75)X(192) = X(1)-Ceva conjugate of X(2)X(192) = crosspoint of X(1) and X(43)X(192) = X(9)-Hirst inverse of X(239)

X(193) = X(4)-CEVA CONJUGATE OF X(2)

Trilinears (csc A)(cos B + cos C - cos A) : (csc B)(cos C + cos A - cos B) : (csc C)(cos A + cos B - cos C)Barycentrics cos B + cos C - cos A : cos C + cos A - cos B : cos A + cos B - cos C

X(193) lies on these lines:2,6 7,239 8,894 20,185 23,159 44,344 66,895 144,145 146,148 253,287 317,393 330,959 371,488 372,487 608,651

X(193) = reflection of X(69) about X(6)X(193) = anticomplement of X(69)X(193) = X(I)-Ceva conjugate of X(J) for these (I,J): (4,2), (459,439)X(193) = crosspoint of X(4) and X(459)X(193) = X(2)-Hirst inverse of X(230)

Let W be the vector-sum XA' + XB' + XC', where A'B'C' is the pedal triangle of X(193); then W = X(3)X(52) = X(20)X(185).

X(194) = X(6)-CEVA CONJUGATE OF X(2)

Trilinears bc[a2b2 + a2c2 - b2c2] : ca[b2c2 + b2a2 - c2a2] : ab[c2a2 + c2b2 - a2b2]Barycentrics a2b2 + a2c2 - b2c2 : b2c2 + b2a2 - c2a2 : c2a2 + c2b2 - a2b2

X(194) lies on these lines:1,87 2,39 3,385 4,147 6,384 8,730 20,185 32,99 63,239 69,695 75,1107 257,986 315,736

X(194) = reflection of X(76) about X(39)X(194) = anticomplement of X(76)X(194) = eigencenter of cevian triangle of X(6)X(194) = eigencenter of anticevian triangle of X(2)

X(194) = X(6)-Ceva conjugate of X(2)X(194) = X(3)-Hirst inverse of X(385)

X(195) = X(5)-CEVA CONJUGATE OF X(3)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (cos A)(v + w - u),u = u(A,B,C) = cos A cos(B - A) cos(C - A), v = u(B,C,A), w = u(C,A,B)

Barycentrics af(A,B,C) : bf(B,C,A) : cf(C,A,B)

X(195) lies on these lines:3,54 4,399 6,17 49,52 110,143 140,323 155,381 382,1078

Let W be the vector-sum XA' + XB' + XC', where A'B'C' is the pedal triangle of X(195); then W = X(3)X(54) = X(54)X(195).

X(195) = reflection of X(3) about X(54)X(195) = X(5)-Ceva conjugate of X(3)

X(196) = X(7)-CEVA CONJUGATE OF X(4)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (cos B + cos C - cos A - 1) sec A tan A/2Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (cos B + cos C - cos A - 1) tan A tan A/2

X(196) lies on these lines:1,207 2,653 4,65 7,92 19,57 34,937 40,208 55,108 226,281 329,342

X(196) = isogonal conjugate of X(268)X(196) = X(I)-Ceva conjugate of X(J) for these (I,J): (7,4), (92,278)X(196) = cevapoint of X(19) and X(207)X(196) = X(221)-cross conjugate of X(347)

X(197) = X(8)-CEVA CONJUGATE OF X(6)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = a[-a2tan A/2 + b2tan B/2 + c2tan C/2]Barycentrics af(A,B,C) : bf(B,C,A) : cf(C,A,B)

X(197) lies on these lines:3,10 6,181 19,25 22,100 42,48 56,227 159,200

X(197) = X(8)-Ceva conjugate of X(6)

X(198) = X(9)-CEVA CONJUGATE OF X(6)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = a(cos B + cos C - cos A - 1)Barycentrics af(A,B,C) : bf(B,C,A) : cf(C,A,B)

X(198) lies on these lines: 3,9 6,41 19,25 45,1030 64,71 100,346 101,102 154,212 208,227 218,579 284,859 478,577 958,966

X(198) = isogonal conjugate of X(189)X(198) = X(I)-Ceva conjugate of X(J) for these (I,J): (3,55), (9,6), (223,221)X(198) = crosspoint of X(40) and X(223)

X(199) = X(10)-CEVA CONJUGATE OF X(6)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a[b4 + c4 - a4 + (b2 + c2 - a2)(bc + ca + ab )]Barycentrics af(A,B,C) : bf(B,C,A) : cf(C,A,B)

X(199) lies on these lines: 2,3 42,172 51,572 55,1030 184,573

X(199) = X(10)-Ceva conjugate of X(6)X(199) = crosspoint of X(101) and X(250)

X(200) = X(8)-CEVA CONJUGATE OF X(9)

Trilinears cot2(A/2) : cot(B/2) : cot(C/2)= (b + c - a)2 : (c + a - b) 2 : (a + b - c)2

Barycentrics a(b + c - a)2 : b(c + a - b) 2 : c(a + b - c)2

X(200) lies on these lines:1,2 3,963 9,55 33,281 40,64 46,1004 57,518 63,100 69,269 159,197 219,282 220,728 255,271 318,1089 319,326 329,516 341,1043 756,968

X(200) = isogonal conjugate of X(269) X(200) = isotomic conjugate of X(1088)X(200) = X(8)-Ceva conjugate of X(9)X(200) = cevapoint of X(220) and X(480)X(200) = X(220)-cross conjugate of X(9)X(200) = crosspoint of X(8) and X(346)

X(201) = X(10)-CEVA CONJUGATE OF X(12)

Trilinears (cos A)[1 + cos(B - C)] : (cos B)[1 + cos(C - A)] : (cos C)[1 + cos(A - B)]Barycentrics (sin 2A)[1 + cos(B - C)] : (sin 2B)[1 + cos(C - A)] : (sin 2C)[1 + cos(A - B)]

X(201) lies on these lines:1,212 9,34 10,225 12,756 33,40 37,65 38,56 55,774 57,975 63,603 72,73 109,191 210,227 220,221 255,1060 337,348 388,984 601,920

X(201) = isogonal conjugate of X(270)X(201) = X(10)-Ceva conjugate of X(12)X(201) = crosspoint of X(10) and X(72)

X(202) = X(1)-CEVA CONJUGATE OF X(15)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = u(v + w - u),u = u(A,B,C) = sin(A + π/3), v = u(B,C,A), w = u(C,A,B)

Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(202) lies on these lines:1,62 6,101 11,13 12,18 15,36 16,55 17,499 56,61 395,495 397,496

X(202) = X(1)-Ceva conjugate of X(15)

X(203) = X(1)-CEVA CONJUGATE OF X(16)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = u(v + w - u),u = u(A,B,C) = sin(A - π/3), v = u(B,C,A), w = u(C,A,B)

Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(203) lies on these lines:1,61 6,101 11,14 12,17 15,55 16,36 18,499 56,62 396,495 398,496

X(203) = X(1)-Ceva conjugate of X(16)

X(204) = X(1)-CEVA CONJUGATE OF X(19)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (tan A)(tan B + tan C - tan A)Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(204) lies on these lines: 6,33 19,31 25,34 55,1033 63,162 108,223 207,221

X(204) = X(1)-Ceva conjugate of X(19)

X(205) = X(9)-CEVA CONJUGATE OF X(31)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2[b2tan B/2 + c2tan C/2 - a2tan A/2]Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(205) lies on these lines: 25,41 37,48 78,101 154,220 184,213

X(205) = X(9)-Ceva conjugate of X(31)

X(206) = X(2)-CEVA CONJUGATE OF X(32)

Trilinears a3(b4 + c4 - a4) : b3(c4 + a4 - b4) : c3(a4 + b4 - c4)Barycentrics a4(b4 + c4 - a4) : b4(c4 + a4 - b4) : c4(a4 + b4 - c4)

This is also X(66) of the medial triangle.

X(206) lies on these lines:2,66 5,182 6,25 26,511 69,110 157,216 160,577 219,692 237,571

X(206) = midpoint between X(6) and X(159)X(206) = complement of X(66)X(206) = X(2)-Ceva conjugate of X(32)X(206) = crosspoint of X(2) and X(315)

Let W be the vector-sum XA' + XB' + XC', where A'B'C' is the pedal triangle of X(206); then W = X(66)X(141) = X(141)X(159).

X(207) = X(1)-CEVA CONJUGATE OF X(34)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (1 - sec A)(sec B + sec C - sec A - 1)Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(207) lies on these lines: 1,196 19,56 33,64 34,1042 40,108 78,653 204,221

X(207) = X(I)-Ceva conjugate of X(J) for these (I,J): (1,34), (196,19)

X(208) = X(4)-CEVA CONJUGATE OF X(34)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (1 - sec A)(cos B + cos C - cos A - 1)Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(208) lies on these lines:1,102 4,57 19,225 25,34 33,64 40,196 198,227 226,406 318,653

X(208) = isogonal conjugate of X(271)X(208) = X(I)-Ceva conjugate of X(J) for these (I,J): (4,34), (57,19), (342,223)

X(209) = X(4)-CEVA CONJUGATE OF X(37)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (sin B + sin C)[sin A + sin(A - B) + sin(A - C)]Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(209) lies on these lines: 6,31 10,12 44,51 306,518

X(209) = isogonal conjugate of X(272)X(209) = X(4)-Ceva conjugate of X(37)

X(210) = X(10)-CEVA CONJUGATE OF X(37)

Trilinears (b + c)(b + c - a) : (c + a)(c + a - b) : (a + b)(a + b - c)Barycentrics a(b + c)(b + c - a) : b(c + a)(c + a - b) : c(a + b)(a + b - c)

X(210) lies on these lines:2,354 6,612 8,312 9,55 10,12 31,44 33,220 37,42 38,899 43,984 45,968 51,374 56,936 63,1004 78,958 165,971 201,227 213,762 381,517 392,519 430,594 869,1107 956,997 976,1104

X(210) = reflection of X(I) about X(J) for these (I,J): (51,375), (354,2)X(210) = isogonal conjugate of X(1014)

X(210) = X(10)-Ceva conjugate of X(37)X(210) = crosspoint of X(8) and X(9)

Let X = X(210) and let V be the vector-sum XA + XB + XC; then V = X(65)X(8) = X(1)X(72).

X(211) = X(4)-CEVA CONJUGATE OF X(39)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C)= sin(A + ω)[cos B sin(B + ω) + cos C sin(C + ω) - cos A sin(A + ω)]

Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(211) lies on these lines: 5,141 32,184 52,114

X(211) = X(4)-Ceva conjugate of X(39)

X(212) = X(9)-CEVA CONJUGATE OF X(41)

Trilinears (cos A)(1 + cos A) : (cos B)(1 + cos B) : (cos C)(1 + cos C)= (cos A)cos2(A/2) : (cos B)cos2(B/2) : (cos C)cos2(C/2)

Barycentrics (sin 2A)(1 + cos A) : (sin 2B)(1 + cos B) : (sin 2C)(1 + cos C)

X(212) lies on these lines:1,201 3,73 6,31 9,33 11,748 34,40 35,47 48,184 56,939 63,1040 78,283 109,165 154,198 238,497 312,643 582,942

X(212) = isogonal conjugate of X(273)X(212) = X(I)-Ceva conjugate of X(J) for these (I,J): (3,48), (9,41), (283,219)X(212) = X(228)-cross conjugate of X(55)X(212) = crosspoint of X(I) and X(J) for these (I,J): (3,219), (9,78)

X(213) = X(6)-CEVA CONJUGATE OF X(42)

Trilinears (b + c)a2 : (c + a)b2 : (a + b)c2

Barycentrics (b + c)a3 : (c + a)b3 : (a + b)c3

X(213) lies on these lines: 1,6 8,981 31,32 39,672 58,101 63,980 83,239 100,729 184,205 274,894 607,1096 667,875 692,923

X(213) = isogonal conjugate of X(274)X(213) = X(I)-Ceva conjugate of X(J) for these (I,J): (6,42), (37,228)X(213) = crosspoint of X(6) and X(31)

X(214) = X(2)-CEVA CONJUGATE OF X(44)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c - 2a)(b2 + c2 - a2 - bc)Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(214) lies on these lines: 1,88 2,80 9,48 10,140 11,442 36,758 44,1017 119,515 142,528 535,908 662,759 1015,1100

X(214) = midpoint between X(1) and X(100)X(214) = complement of X(80)X(214) = X(2)-Ceva conjugate of X(44)X(214) = crosspoint of X(2) and X(320)

X(215) = X(1)-CEVA CONJUGATE OF X(50)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (sin 3A)(sin 3B + sin 3C - sin 3A)Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(215) lies on these lines: 1,49 11,110 12,54 55,184

X(215) = X(I)-Ceva conjugate of X(J) for these (I,J): (1,50)

X(216) = X(5)-CEVA CONJUGATE OF X(51)

Trilinears sin 2A cos(B - C) : sin 2B cos(C - A) : sin 2C cos(A - B)Barycentrics (sin A)(sin 2A)cos(B - C) : sin B sin 2B cos(C - A) : sin C sin 2C cos(A - B)

X(216) lies on these lines:2,232 3,6 5,53 51,418 95,648 97,288 115,131 157,206 395,465 373,852 395,465 396,466 631,1075 1015,1060

X(216) = isogonal conjugate of X(275)X(216) = isotomic conjugate of X(276)X(216) = inverse of X(577) in the Brocard circleX(216) = complement of X(264)X(216) = X(I)-Ceva conjugate of X(J) for these (I,J): (2,5), (3,418), (5,51), (324,52)X(216) = cevapoint of X(217) and X(418)X(216) = X(217)-cross conjugate of X(51)X(216) = crosspoint of X(I) and X(J) for these (I,J): (2,3), (5,343)

X(217) = X(6)-CEVA CONJUGATE OF X(51)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (sin3A) cos A cos(B - C)Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(217) lies on these lines: 4,6 32,184 39,185 54,112 83,287 232,389

X(217) = isogonal conjugate of X(276)X(217) = X(I)-Ceva conjugate of X(J) for these (I,J): (6,51), (216,418)X(217) = crosspoint of X(I) and X(J) for these (I,J): (6,184), (51,216)

X(218) = X(7)-CEVA CONJUGATE OF X(55)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), wheref(A,B,C) = cos2(A/2) [cos4(B/2) + cos4(C/2)- cos4(A/2)]

Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(218) lies on these lines:1,6 3,41 4,294 7,277 32,906 43,170 46,910 56,101 65,169 145,644 198,579 222,241 279,651

X(218) = isogonal conjugate of X(277)X(218) = eigencenter of cevian triangle of X(7)X(218) = eigencenter of anticevian triangle of X(55)X(218) = X(7)-Ceva conjugate of X(55)

X(219) = X(8)-CEVA CONJUGATE OF X(55)

Trilinears cos A cos A/2 : cos B cos B/2 : cos C cos C/2= (sin A)/(1 - sec A) : (sin B)/(1 - sec B) : (sin C)/(1 - sec C)= 1/(csc A - 2 csc 2A) : 1/(csc B - 2 csc 2B) : 1/(csc C - 2 csc 2C)

Barycentrics sin 2A cos A/2 : sin 2B cos B/2 : sin 2C cos C/2

X(219) lies on these lines:1,6 3,48 8,29 10,965 19,517 40,610 41,1036 55,284 56,579 63,77 101,102 144,347 200,282 206,692 255,268 278,329 332,345 346,644 572,947 577,906 604,672

X(219) = isogonal conjugate of X(278)X(219) = isotomic conjugate of X(331)X(219) = X(I)-Ceva conjugate of X(J) for these (I,J): (8,55), (63,3), (283,212)X(219) = X(I)-cross conjugate of X(J) for these (I,J): (48,268), (71,9), (212,3)X(219) = crosspoint of X(I) and X(J) for these (I,J): (8,345), (64,78)

Let W be the vector-sum XA' + XB' + XC', where A'B'C' is the pedal triangle of X(219); then W = X(19)X(219).

X(220) = X(9)-CEVA CONJUGATE OF X(55)

Trilinears csc A/2 cos3(A/2) : csc B/2 cos3(B/2) : csc C/2 cos3(C/2)Barycentrics sin A csc A/2 cos3(A/2) : sin B csc B/2 cos3(B/2) : sin C csc C/2 cos3(C/2)

X(220) lies on these lines:1,6 3,101 8,294 33,210 40,910 41,55 48,963 63,241 64,71 78,949 144,279 154,205 169,517 200,728 201,221 268,577 277,1086 281,594 329,948 346,1043

X(220) = isogonal conjugate of X(279)X(220) = X(I)-Ceva conjugate of X(J) for these (I,J): (9,55), (200,480)X(220) = cevapoint of X(1) and X(170)X(220) = crosspoint of X(9) and X(200)

X(221) = X(1)-CEVA CONJUGATE OF X(56)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (sin2A/2)(cos B + cos C - cos A - 1)Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(221) lies on these lines:1,84 3,102 6,19 8,651 31,56 40,223 55,64 201,220 204,207 960,1038

X(221) = isogonal conjugate of X(280)X(221) = X(I)-Ceva conjugate of X(J) for these (I,J): (1,56), (222,6), (223,198)X(221) = crosspoint of X(I) and X(J) for these (I,J): (1,40), (196,347)

X(222) = X(7)-CEVA CONJUGATE OF X(56)

Trilinears cos A tan A/2 : cos B tan B/2 : cos C tan C/2= 1/(csc A + 2 csc 2A) : 1/(csc B + 2 csc 2B) : 1/(csc A + 2 csc 2C)= a(b2 + c2 - a2)/(b + c - a) : b(c2 + a2 - b2)/(c + a - b) : c(a2 + b2 - c2)/(a + b - c)

Barycentrics a2/(1 + sec A) : b2/(1 + sec B) : c2/(1 + sec C)

X(222) lies on these lines:1,84 2,651 3,73 6,57 7,27 33,971 34,942 46,227 55,103 56,58 63,77 72,1038 171,611 189,281 218,241 226,478 268,1073 581,1035 601,1066 613,982 912,1060

X(222) = isogonal conjugate of X(281)X(222) = X(I)-Ceva conjugate of X(J) for these (I,J): (7,56), (77,3), (81,57)X(222) = cevapoint of X(6) and X(221)X(222) = X(I)-cross conjugate of X(J) for these (I,J): (48,3), (73,77)X(222) = crosspoint of X(7) and X(348)

Let W be the vector-sum XA' + XB' + XC', where A'B'C' is the pedal triangle of X(222); then W = X(33)X(222).

X(223) = X(2)-CEVA CONJUGATE OF X(57)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (tan A/2)(cos B + cos C - cos A - 1]Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(223) lies on these lines:1,4 2,77 3,1035 6,57 9,1073 40,221 56,937 63,651 108,204 109,165 312,664 329,347 380,608 580,603 936,1038

X(223) = isogonal conjugate of X(282)>BR> X(223) = complement of X(189)X(223) = X(I)-Ceva conjugate of X(J) for these (I,J): (2,57), (77,1), (342,208), (347,40)X(223) = cevapoint of X(198) and X(221)X(223) = X(I)-cross conjugate of X(J) for these (I,J): (198,40), (227,347)X(223) = crosspoint of X(2) and X(329)X(223) = X(I)-aleph conjugate of X(J) for these (I,J): (63,1079), (77,223), (81,580), (174,46), (508,19), (651,109)

X(224) = X(7)-CEVA CONJUGATE OF X(63)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), wheref(A,B,C) = [cot B cos2(B/2) + cot C (cot C/2)2 - cot A (cot C/2)2]cot A

Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(224) lies on these lines: 1,377 3,63 8,914 21,90 46,100 65,1004 X(224) = X(7)-Ceva conjugate of X(63)

X(225) = X(4)-CEVA CONJUGATE OF X(65)

Trilinears (sec A)(cos B + cos C) : (sec B)(cos C + cos A) : (sec C)(cos A + cos B)Barycentrics (tan A)(cos B + cos C) : (tan B)(cos C + cos A) : (tan C)(cos A + cos B)

X(225) lies on these lines:1,4 3,1074 7,969 10,201 12,37 19,208 28,108 46,254 65,407 75,264 91,847 158,1093 377,1038 412,775 653,897

X(225) = isogonal conjugate of X(283)X(225) = isotomic conjugate of X(332)X(225) = X(4)-Ceva conjugate of X(65)X(225) = X(407)-cross conjugate of X(4)X(225) = crosspoint of X(I) and X(J) for these (I,J): (4,158), (273,278)

X(226) = X(7)-CEVA CONJUGATE OF X(65)

Trilinears (csc A)(cos B + cos C) : (csc B)(cos C + cos A) : (csc C)(cos A + cos B)= bc(b + c)/(b + c - a) : ca(c + a)/(c + a - b) : ab(a + b)/(a + b - c)

Barycentrics (b + c)/(b + c - a) : (c + a)/(c + a - b) : (a + b)/(a + b - c)

This center is also X(63) of the medial triangle.

X(226) lies on these lines:1,4 2,7 5,912 10,12 11,118 13,1082 14,554 27,284 29,951 35,79 36,1006 37,440 41,379 46,498 55,516 56,405 76,85 78,377 81,651 92,342 98,109 102,1065 196,281 208,406 222,478 228,851 262,982 273,469 306,321 443,936 481,485 482,486 495,517 535,551 664,671 975,1038 990,1040

X(226) = isogonal conjugate of X(284)X(226) = isotomic conjugate of X(333)X(226) = complement of X(63)X(226) = X(I)-Ceva conjugate of X(J) for these (I,J): (7,65), (349,307)X(226) = cevapoint of X(37) and X(65)X(226) = X(I)-cross conjugate of X(J) for these (I,J): (37,10), (73,307)X(226) = crosspoint of X(2) and X(92)

X(227) = X(10)-CEVA CONJUGATE OF X(65)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c)(cos B + cos C - cos A - 1)tan 2ABarycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(227) lies on these lines:12,37 34,55 40,221 42,65 46,222 56,197 198,208 201,210 322,347 607,910

X(227) = isogonal conjugate of X(285)X(227) = X(10)-Ceva conjugate of X(65)X(227) = crosspoint of X(223) and X(347)

X(228) = X(3)-CEVA CONJUGATE OF X(71)

Trilinears (sin 2A)(sin B + sin C) : (sin 2B)(sin C + sin A) : (sin 2C)(sin A + sin B)Barycentrics (sin A sin 2A)(sin B + sin C) : (sin B sin 2B)(sin C + sin A) : (sin C sin 2C)(sin A + sin B)

X(228) lies on these lines:3,63 9,1011 12,407 19,25 28,943 31,32 35,846 42,181 48,184 73,408 98,100 226,851

X(228) = isogonal conjugate of X(286)X(228) = X(I)-Ceva conjugate of X(J) for these (I,J): (3,71), (37,213), (55,42)X(228) = crosspoint of X(I) and X(J) for these (I,J): (3,48), (37,72), (55,212), (71,73)

X(229) = X(7)-CEVA CONJUGATE OF X(81)

Trilinears f(a,b,c) : f(b,c,a : f(c,a,b), where f(a,b,c) = (v + w - u)/(b + c),u = u(a,b,c) = a(b + c - a)/(b + c), v = u(b,c,a), w = u(c,a,b)

Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(229) lies on these lines: 1,267 21,36 28,60 58,244 65,110 593,1104

X(229) = midpoint between X(1) and X(267)X(229) = X(7)-Ceva conjugate of X(81)

X(230) = X(2)-CEVA CONJUGATE OF X(114)

Trilinears f(a,b,c) : f(b,c,a : f(c,a,b), wheref(a,b,c) = bc[a2(2a2 - b2 - c2) + (b2 - c2)2]

Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(230) lies on these lines:2,6 5,32 12,172 25,53 30,115 39,140 50,858 111,476 112,403 231,232 393,459 538,620 549,574 625,754

X(230) = midpoint between X(I) and X(J) for these (I,J): (115,187), (325,385), (395,396)X(230) = complement of X(325)X(230) = X(2)-Ceva conjugate of X(114)X(230) = crosspoint of X(2) and X(98)X(230) = X(2)-Hirst inverse of X(193)

Let X = X(230) and let V be the vector-sum XA + XB + XC; then V = X(230)X(385) = X(265)X(399).

X(231) = X(2)-CEVA CONJUGATE OF X(128)

Trilinears f(a,b,c) : f(b,c,a : f(c,a,b), where f(a,b,c) = u(-au + bv + cw), u : v : w = X(128)Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(231) lies on these lines: 4,96 6,17 50,115 230,232

X(231) = X(2)-Ceva conjugate of X(128)

X(232) = X(2)-CEVA CONJUGATE OF X(132)

Trilinears tan A cos(A + ω) : tan B cos(B + ω) : tan C cos(C = ω)Barycentrics sin A tan A cos(A + ω) : sin B tan B cos(B + ω) : sin C tan C cos(C = ω)

X(232) lies on these lines:2,216 4,39 6,25 19,444 22,577 23,250 24,32 53,427 112,186 115,403 217,389 230,231 297,325 378,574 385,648 459,800

X(232) = isogonal conjugate of X(287)X(232) = X(I)-Ceva conjugate of X(J) for these (I,J): (2,132), (297,511)X(232) = X(237)-cross conjugate of X(511)X(232) = X(6)-Hirst inverse of X(25)

X(233) = X(2)-CEVA CONJUGATE OF X(140)

Trilinears f(a,b,c) : f(b,c,a : f(c,a,b), where f(a,b,c) = [b cos(C - A) + c cos(B - A)]cos(B - C)Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(233) lies on these lines: 2,95 5,53 6,17 115,128 122,138

X(233) = isogonal conjugate of X(288)X(233) = complement of X(95)X(233) = X(2)-Ceva conjugate of X(140)X(233) = crosspoint of X(2) and X(5)

X(234) = X(7)-CEVA CONJUGATE OF X(177)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (cos B/2 + cos C/2)(cos B/2 cos C/2)2

Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(234) lies on these lines: 2,178 7,174 57,362 75,556 555,1088

X(234) = X(7)-Ceva conjugate of X(177)

X(235) = X(4)-CEVA CONJUGATE OF X(185)

Trilinears f(a,b,c) : f(b,c,a : f(c,a,b), where f(a,b,c) = u(-u cos A + v cos B + w cos C), where u : v : w = X(185)Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(235) lies on these lines: 2,3 11,34 12,33 52,113 133,136

X(235) = midpoint between X(4) and X(24)X(235) = X(4)-Ceva conjugate of X(185)

X(236) = X(2)-CEVA CONJUGATE OF X(188)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (csc A/2)(cos B/2 + cos C/2 - cos A/2)Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(236) lies on these lines: 2,174 8,178 9,173

X(236) = isogonal conjugate of X(289)X(236) = X(2)-Ceva conjugate of X(188)

Centers 237- 248 are line conjugates. The P-line conjugate of Q is the point

where line PQ meets the polar of the isogonal conjugate of Q.

X(237) = X(3)-LINE CONJUGATE OF X(2)

Trilinears a2cos(A + ω) : b2cos(B + ω)2cos(C + ω)Barycentrics a3cos(A + ω) : b3cos(B + ω) : c3cos(C + ω)

X(237) lies on these lines: 2,3 6,160 31,904 32,184 39,51 154,682 187,351 206,571

X(237) = isogonal conjugate of X(290)X(237) = X(98)-Ceva conjugate of X(6)X(237) = crosspoint of X(I) and X(J) for these (I,J): (6,98), (232,511)X(237) = X(32)-Hirst inverse of X(184)X(237) = X(3)-line conjugate of X(2)

X(238) = X(1)-LINE CONJUGATE OF X(37)

Trilinears a2 - bc : b2 - ca : c2 - abBarycentrics a3 - abc : b3 - abc : c3 - abc

X(238) lies on these lines:1,6 2,31 3,978 4,602 8,983 10,82 21,256 36,513 43,55 47,499 56,87 58,86 63,614 100,899 105,291 106,898 162,415 190,726 212,497 239,740 242,419 244,896 516,673 517,1052 519,765 580,946 601,631 942,1046 992,1009 993,995 1006,1064

X(238) = isogonal conjugate of X(291)X(238) = isotomic conjugate of X(334)X(238) = X(I)-Ceva conjugate of X(J) for these (I,J): (105,1), (292,171)X(238) = X(I)-Hirst inverse of X(J) for these (I,J): (1,6), (43,55)

X(238) = X(1)-line conjugate of X(37)X(238) = X(105)-aleph conjugate of X(238)

Let X = X(238) and let V be the vector-sum XA + XB + XC; then V = X(320)X(1).

X(239) = X(1)-LINE CONJUGATE OF X(42)

Trilinears bc(a2 - bc) : ca(b2 - ca) : ab(c2 - ab)Barycentrics a2 - bc : b2 - ca : c2 - ab

X(239) lies on these lines:1,2 6,75 7,193 9,192 44,190 57,330 63,194 81,274 83,213 86,1100 92,607 141,319 238,740 241,664 257,333 294,666 318,458 320,524 335,518 514,649 1043,1104

X(239) = isogonal conjugate of X(292) = isotomic conjugate of X(335) X(239) = reflection of X(190) about X(44)X(239) = crosspoint of X(256) and X(291)X(239) = X(I)-Hirst inverse of X(J) for these (I,J): (1,2), (9,192)X(239) = X(1)-line conjugate of X(42)

X(240) = X(1)-LINE CONJUGATE OF X(48)

Trilinears sec A cos(A + ω) : sec B cos(B + ω) : sec C cos(C + ω)Barycentrics tan A cos(A + ω) : tan B cos(B + ω) : tan C cos(C + ω)

X(240) lies on these lines: 1,19 4,256 38,92 63,1096 75,158 162,896 278,982 281,984 522,656 607,611 608,613

X(240) = isogonal conjugate of X(293)X(240) = isotomic conjugate of X(336)X(240) = X(1)-Hirst inverse of X(19)X(240) = X(31-line conjugate of X(48)

X(241) = X(1)-LINE CONJUGATE OF X(55)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), wheref(A,B,C) = cos4B/2 - [cos2(A/2)][cos2(B/2) +cos2(C/2)] + cos4(C/2)

Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(241) lies on these lines: 1,3 2,85 6,77 7,37 9,269 44,651 63,220 141,307 218,222 239,664 277,278 294,910 347,1108 514,650 960,1042

X(241) = isogonal conjugate of X(294)X(241) = X(1)-Hirst inverse of X(57)X(241) = X(1)-line conjugate of X(55)

X(242) = X(4)-LINE CONJUGATE OF X(71)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (sec A)[sin2A - sin B sin C]Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(242) lies on these lines: 4,9 25,92 28,261 29,257 34,87 162,422 238,419 278,459 915,929

X(242) = isogonal conjugate of X(295)X(242) = isotomic conjugate of X(337)X(242) = X(4)-Hirst inverse of X(19)X(242) = X(4)-line conjugate of X(71)

X(243) = X(4)-LINE CONJUGATE OF X(73)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (sec A)[cos2A - cos B cos C]Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(243) lies on these lines: 1,4 3,158 55,92 65,412 318,958 411,821 425,662 522,652 920,1075 1040,1096

X(243) = isogonal conjugate of X(296)X(243) = X(1)-Hirst inverse of X(4)X(243) = X(1)-line conjugate of X(73)

X(244) = X(1)-LINE CONJUGATE OF X(100)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = [1 - cos(B - C)]sin2(A/2)Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(244) lies on these lines: 1,88 2,38 11,867 31,57 34,1106 42,354 58,229 63,748 238,896 474,976 518,899 596,1089 665,866

X(244) = X(I)-Ceva conjugate of X(J) for these (I,J): (1,513), (75,514)X(244) = crosspoint of X(1) and X(513)X(244) = X(1)-line conjugate of X(100)

X(245) = X(1)-LINE CONJUGATE OF X(110)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), wheref(A,B,C) = csc2(C - A) + csc(C - B) [csc(C - A) -csc(B - A)] + csc2(A - B)

Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(245) lies on these lines: 1,60 115,125

X(245) = X(1)-line conjugate of X(110)

X(246) = X(3)-LINE CONJUGATE OF X(110)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), wheref(A,B,C) = csc(B-A)[cos A csc(B - A) + cos C csc(B - C)] - csc(C - A) u(A,B,C),u(A,B,C) = [cos A csc(C - A) + cos B csc(C - B)]

Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(246) lies on these lines: 3,74 115,125

X(246) = X(3)-line conjugate of X(110)

X(247) = X(4)-LINE CONJUGATE OF X(110)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), wheref(A,B,C) = csc(B-A)[sec A csc(B - A) + sec C csc(B - C)] - csc(C - A) u(A,B,C),u(A,B,C) = [sec A csc(C - A) + sec B csc(C - B)]

Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(247) lies on these lines: 4,110 115,125

X(247) = X(4)-line conjugate of X(110)

X(248) = X(4)-LINE CONJUGATE OF X(132)

Trilinears sin 2A sec(A + ω) : sin 2B sec(B + ω) : sin 2C sec(C + ω)Barycentrics sin A sin 2A sec(A + ω) : sin B sin 2B sec(B + ω) : sin C sin 2C sec(C + ω)

X(248) lies on these lines:4,32 6,157 39,54 50,67 65,172 66,571 69,287 72,293 74,187 290,385 682,695

X(248) = isogonal conjugate of X(297)X(248) = crosspoint of X(98) and X(287)X(248) = X(4)-line conjugate of X(132)

Centers 249- 297 are isogonal conjugates of previously listed centers.

X(249)

Trilinears (csc A)cos2(B - C) : (csc B)cos2(C - A) : (csc C)cos2(A - B)= a/(b2 - c2)2 : b/(c2 - a2)2 : c/(a2 - b2)2

Barycentrics cos2(B - C) : cos2(C - A) : cos2(A - B)

X(249) lies on these lines: 99,525 110,512 186,250 187,323 297,316 648,687 805,827 849,1110

X(249) = isogonal conjugate of X(115)X(249) = isotomic conjugate of X(338)X(249) = cevapoint of X(I) and X(J) for these (I,J): (6,110), (24,112)X(249) = X(I)-cross conjugate of X(J) for these (I,J): (3,99), (6,110)

Let X = X(249) and let V be the vector-sum XA + XB + XC; then V = X(316)X(323).

X(250)

Trilinears (sec A)csc2(B - C) : (sec B)csc2(C - A) : (sec C)csc2(A - B)= (a2sec A)/(b2 - c2)2 : (b2sec B)/(c2 - a2)2 : (c2sec C)/(a2 - b2)2

Barycentrics (tan A)csc2(B - C) : (tan B)csc2(C - A) : (tan C)csc2(A - B)

X(250) lies on these lines: 23,232 107,687 110,520 112,691 186,249 325,340 476,933 523,648 827,935

X(250) = isogonal conjugate of X(125)X(250) = isotomic conjugate of X(339)X(250) = cevapoint of X(I) and X(J) for these (I,J): (3,110), (25,112), (162,270)X(250) = X(I)-cross conjugate of X(J) for these (I,J): (3,110), (22,99), (24,107), (25,112), (199,101)

Let X = X(250) and let V be the vector-sum XA + XB + XC; then V = X(340)X(23).

X(251)

Trilinears a2csc(A + ω) : b2csc(B + ω) : c2csc(C + ω)= a/(b2 + c2) : b/(c2 + a2) : c/(a2 + b2)

Barycentrics a3csc(A + ω) : b3csc(B + ω) : c3csc(C + ω)

X(251) lies on these lines: 2,32 6,22 37,82 110,694 112,427 184,263 308,385 609,614 689,699

X(251) = isogonal conjugate of X(141)X(251) = cevapoint of X(6) and X(32)X(251) = X(I)-cross conjugate of X(J) for these (I,J): (6,83), (23,111), (523,112)

X(252)

Trilinears (cos A csc 2A)/f(A,B,C) : (cos B csc 2B)/f(B,C,A) : (cos C csc 2C)/f(C,A,B), wheref(A,B,C) = [(v + w)2][u4 + v4 + w4 - u2(2 v2 + 2w2 - vw) - vw(v2 + w2)], whereu = sin(2A), v = sin(2B), w = sin(2C).

Barycentrics 1/f(A,B,C) : 1/f(B,C,A) : 1/f(C,A,B)

X(252) lies on these lines: 3,930 54,140 93,186

X(252) = isogonal conjugate of X(143)

X(253) X(4)-CROSS CONJUGATE OF X(2)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (csc A)/(tan B + tan C - tan A)g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (csc2A)/(cos A - cos B cos C)

Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = 1/(tan B + tan C - tan A)

X(253) lies on these lines: 2,1073 7,280 8,307 20,64 193,287 306,329 318,342 322,341

X(253) = isogonal conjugate of X(154)X(253) = isotomic conjugate of X(20)X(253) = cyclocevian conjugate of X(69)X(253) = cevapoint of X(122) and X(525)X(253) = X(I)-cross conjugate of X(J) for these (I,J): (4,2), (122,525)

X(254) X(3)-CROSS CONJUGATE OF X(4)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (sec A)/(cos2B + cos2C - cos2A)Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (tan A)/(cos2B + cos2C - cos2A)

X(254) lies on these lines: 2,847 4,155 24,393 46,225 68,136

X(254) = isogonal conjugate of X(155)X(254) = cevapoint of X(136) and X(523)X(254) = X(3)-cross conjugate of X(4)

X(255)

Trilinears cos2A : cos2B : cos2CBarycentrics sin A cos2A : sin B cos2B : sin C cos2C

X(255) lies on these lines: 1,21 3,73 35,991 36,1106 40,109 48,563 55,601 56,602 57,580 91,1109 92,1087 158,775 162,1099 165,1103 200,271 201,1060 219,268 293,304 326,1102 411,651 498,750 499,748

X(255) = isogonal conjugate of X(158)X(255) = X(I)-Ceva conjugate of X(J) for these (I,J): (63,48), (283,3)X(255) = crosspoint of X(63) and X(326)X(255) = X(I)-aleph conjugate of X(J) for these (I,J): (775,255), (1105,158)

Let W be the vector-sum XA' + XB' + XC', where A'B'C' is the pedal triangle of X(255); then W = X(225)X(255).

X(256)

Trilinears 1/(a2 + bc) : 1/(b2 + ca) : 1/(c2 + ab)Barycentrics a/(a2 + bc) : b/(b2 + ca) : c/(c2 + ab)

X(256) lies on these lines: 1,511 3,987 4,240 7,982 8,192 9,43 21,238 37,694 40,989 55,983 84,988 104,1064 291,894 314,350 573,981

X(256) = isogonal conjugate of X(171)X(256) = X(239)-cross conjugate of X(291)

X(257)

Trilinears 1/(a3 + abc) : 1/(b3 + abc) : 1/(c3 + abc)Barycentrics a/(a3 + abc) : b/(b3 + abc) : c/(c3 + abc)

X(257) lies on these lines: 1,385 8,192 29,242 65,894 75,698 92,297 194,986 239,333 330,982 335,694

X(257) = isogonal conjugate of X(172)X(527) = isotomic conjugate of X(894)X(257) = X(350)-cross conjugate of X(335)

X(258) = CONGRUENT INCIRCLES ISOSCELIZER POINT

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = 1/( cos B/2 + cos C/2 - cos A/2)= g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = 1 + sin(B/2) + sin(C/2) - sin(A/2)

Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

In Yff's isoscelizer configuration, if X = X(258), then the isosceles triangles Ta, Tb, Tc have congruent incircles.

X(258) lies on these lines: 1,164 57,173 259,289

X(258) = isogonal conjugate of X(173)X(258) = X(259)-cross conjugate of X(1)X(258) = X(366)-aleph conjugate of X(363)

X(259)

Trilinears cos A/2 : cos B/2 : cos C/2= [a(b + c - a)]1/2 : [b(c + a - b)]1/2 : [c(a + b - c)]1/2

Barycentrics sin A cos A/2 : sin B cos B/2 : sin C cos C/2

X(259) lies on these lines: 1,168 258,289 260,266

X(259) = isogonal conjugate of X(174)X(259) = X(I)-Ceva conjugate of X(J) for these (I,J): (174,266), (260,55)X(259) = cevapoint of X(1) and X(503)X(259) = crosspoint of X(I) and X(J) for these (I,J): (1,258), (174,188)

X(260)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (cos A/2)/[cos B/2 + cos C/2 - cos A/2)]Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(260) lies on these lines: 1,3 259,266

X(260) = isogonal conjugate of X(177)X(260) = cevapoint of X(55) and X(259)

X(261)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = [(csc A)(sec(B/2 - C/2))]2

Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(261) lies on these lines:2,593 9,645 21,314 28,242 58,86 75,99 272,310 284,332 317,406 319,502 552,873 572,662

X(261) = isogonal conjugate of X(181)X(261) = isotomic conjugate of X(12)X(261) = cevapoint of X(21) and X(333)

X(262)

Trilinears sec(A - ω) : sec(B - ω) : sec(C - ω)Barycentrics sin A sec(A - ω) : sin B sec(B - ω) : sin C sec(C - ω)

X(262) lies on these lines: 2,51 3,83 4,39 5,76 6,98 13,383 14,1080 25,275 30,598 226,982 381,671 385,576

X(262) = isogonal conjugate of X(182)X(262) = isotomic conjugate of X(183)

Let X = X(262) and let V be the vector-sum XA + XB + XC; then V = X(76)X(4).

X(263)

Trilinears a2sec(A - ω) : b2sec(B - ω) : c2sec(C - ω)Barycentrics a3sec(A - ω) : b3sec(B - ω) : c3sec(C - ω)

X(263) lies on these lines: 2,51 6,160 69,308 184,251

X(263) = isogonal conjugate of X(183)

X(264) ISOTOMIC CONJUGATE OF CIRCUMCENTER

Trilinears csc A csc 2A : csc B csc 2B : csc C csc 2C= sec A csc2A : sec B csc2B : sec C csc2C= tan A csc(A - ω) : tan B csc(B - ω) : tan C csc(C - ω)

Barycentrics csc 2A : csc 2B : csc 2C

X(264) lies on these lines:2,216 3,95 4,69 5,1093 6,287 25,183 33,350 53,141 75,225 85,309 92,306 99,378 274,475 281,344 298,472 299,473 300,302 301,303 305,325 339,381 379,823 401,577

X(264) = isogonal conjugate of X(184)X(264) = isotomic conjugate of X(3)X(264) = anticomplement of X(216)X(264) = X(276)-Ceva conjugate of X(2)X(264) = cevapoint of X(I) and X(J) for these (I,J): (2,4), (5,324), (6,157), (92,318), (273,342), (338,523), (491,492)

X(264) = X(I)-cross conjugate of X(J) for these (I,J): (2,76), (5,2), (30,94), (92,331), (427,4), (442,321)

X(265)

Trilinears sin 2A csc 3A : sin 2B csc 3B : sin 2C csc 3C= 1/(4 cos A - sec A) : 1/(4 cos B - sec B) : 1/(4 cos C sec C)

Barycentrics sin A sin 2A csc 3A : sin B sin 2B csc 3B : sin C sin 2C csc 3C

X(265) lies on these lines: 3,125 4,94 5,49 6,13 30,74 64,382 65,79 67,511 69,328 290,316 300,621 301,622

X(265) = reflection of X(I) about X(J) for these (I,J): (3,125), (110,5), (399,113)X(265) = isogonal conjugate of X(186)X(265) = isotomic conjugate of X(340)X(265) = cevapoint of X(5) and X(30)X(265) = crosspoint of X(94) and X(328)

X(266)

Trilinears sin A/2 : sin B/2 : sin C/2= [bc(b + c - a)]1/2 : [ca(c + a - b)]1/2 : [ab(a + b - c)]1/2

Barycentrics sin A sin A/2 : sin B sin B/2 : sin C sin C/2

X(266) lies on these lines:1,164 56,289 174,188 259,260 361,978

X(266) = isogonal conjugate of X(188)X(266) = eigencenter of cevian triangle of X(174) X(266) = eigencenter of anticevian triangle of X(259)X(266) = X(174)-Ceva conjugate of X(259)X(266) = cevapoint of X(1) and X(361)X(266) = X(6)-cross conjugate of X(289)X(266) = crosspoint of X(1) and X(505)

X(267)

Trilinears f(a,b,c) : f(b,c,a) : f(ca,b), wheref(a,b,c) = 1/[b3 + c3 - a3 + (b + c - a)(bc + ca + ab)]

Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(267) lies on these lines: 1,229 10,191 35,37

X(267) = reflection of X(1) about X(229)X(267) = isogonal conjugate of X(191)X(267) = cevapoint of X(58) and X(501)X(267) = X(58)-cross conjugate of X(1)

X(268)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (cos A cot A/2)/(-1 - cos A + cos B + cos C)Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(268) lies on these lines: 3,9 21,280 219,255 220,577 222,1073 281,1012

X(268) = isogonal conjugate of X(196)X(268) = X(I)-cross conjugate of X(J) for these (I,J): (48,219), (55,3)

X(269)

Trilinears tan2A/2 : tan2B/2 : tan2C/2Barycentrics sin A tan2A/2 : sin B tan2B/2 : sin C tan2C/2

X(269) lies on these lines: 1,7 3,939 6,57 9,241 46,1103 56,738 69,200 86,1088 106,934 142,948 273,1111 292,1020 307,936 320,326 479,614

X(269) = isogonal conjugate of X(200)X(269) = isotomic conjugate of X(341)X(269) = X(279)-Ceva conjugate of X(57)X(269) = X(56)-cross conjugate of X(57)X(269) = crosspoint of X(279) and X(479)

X(270)

Trilinears (sec A)/[1 + cos(B - C)] : (sec B)/[1 + cos(C - A)] : (sec C)/[1 + cos (A - B)]Barycentrics (tan A)/[1 + cos(B - C)] : (tan B)/[1 + cos(C - A)] : (tan C)/[1 + cos (A - B)]

X(270) lies on these lines: 4,162 27,58 28,60 29,283 759,933

X(270) = isogonal conjugate of X(201)X(270) = X(250)-Ceva conjugate of X(162)X(270) = cevapoint of X(28) and X(58)X(270) = X(58)-cross conjugate of X(60)

X(271)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (cot A cot A/2)/(-1 - cos A + cos B + cos C)Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(271) lies on these lines: 2,1034 8,20 78,394 200,255 282,283

X(271) = isogonal conjugate of X(208)X(271) = isotomic conjugate of X(342)X(271) = X(I)-cross conjugate of X(J) for these (I,J): (3,78), (9,63)

X(272)

Trilinears f(A,B,C)/(b + c) : f(B,C,A)/(c + a) : f(C,A,B)/(a +b), wheref(A,B,C) = 1/[sin A + sin(A - B) + sin(A - C)]

Barycentrics af(A,B,C)/(b + c) : bf(B,C,A)/(c + a) : cf(C,A,B)/(a +b)

X(272) lies on these lines: 2,284 7,58 21,75 28,273 60,86 261,310 1014,1088

X(272) = isogonal conjugate of X(209)X(272) = X(3)-cross conjugate of X(81)

X(273)

Trilinears sec A sec2(A/2) : sec B sec2(B/2) : sec C sec2(C/2)= (1- sec A)csc2A : (1 - sec B)csc2B : (1 - sec C)csc2C

Barycentrics tan A sec2(A/2) : tan B sec2(B/2) : tan C sec2(C/2)

X(273) lies on these lines: 2,92 4,7 19,653 27,57 28,272 29,34 53,1086 75,225 78,322 108,675 226,469 269,1111 317,320 458,894

X(273) = isogonal conjugate of X(212)X(273) = isotomic conjugate of X(78)X(273) = X(I)-Ceva conjugate of X(J) for these (I,J): (264,342), (286,7), (331,92)X(273) = cevapoint of X(I) and X(J) for these (I,J): (4,278), (34,57)X(273) = X(I)-cross conjugate of X(J) for these (I,J): (4,92), (57,85), (225,278)

X(274)

Trilinears b2c2/(b + c) : c2a2/(c + a) : a2b2/(a + b)= [a csc(A - ω)]/(b + c) : [b csc(B - ω)]/(c + a) :[c csc(C - ω)]/(a + b)

Barycentrics bc/(b + c) : ca/(c + a) : ab/(a + b)

X(274) lies on these lines:1,75 2,39 7,959 10,291 21,99 28,242 57,85 58,870 69,443 81,239 88,799 110,767 183,474 213,894 264,475 278,331 315,377 325,442 961,1014

X(274) = isogonal conjugate of X(213)X(274) = isotomic conjugate of X(37)X(274) = X(310)-Ceva conjugate of X(314)X(274) = cevapoint of X(I) and X(J) for these (I,J): (2,75), (85,348), (86,333)X(274) = X(I)-cross conjugate of X(J) for these (I,J): (2,86), (75,310), (81,286), (333,314)

X(275) CEVAPOINT OF ORTHOCENTER AND SYMMEDIAN POINT

Trilinears csc 2A sec(B - C) : csc 2B sec(C - A) : csc 2C sec(A - B)Barycentrics sec A sec(B - C) : sec B sec(C - A) : sec C sec(A - B)

X(275) lies on these lines:2,95 4,54 13,472 14,473 17,471 18,470 25,262 51,107 53,288 76,276 83,297 94,324 98,427

X(275) = isogonal conjugate of X(216)X(275) = isotomic conjugate of X(343)X(275) = X(276)-Ceva conjugate of X(95)X(275) = cevapoint of X(4) and X(6)X(275) = X(I)-cross conjugate of X(J) for these (I,J): (6,54), (54,95)

X(276)

Trilinears a3sec A sec(B - C) : b3sec B sec(C - A) : c3sec C sec(A - B)Barycentrics a4sec A sec(B - C) : b4sec B sec(C - A) : c4sec C sec(A - B)

X(276) lies on these lines: 3,95 4,327 54,290 76,275 97,401

X(276) = isogonal conjugate of X(217)X(276) = isotomic conjugate of X(216)X(276) = cevapoint of X(I) and X(J) for these (I,J): (2,264), (95,275)X(276) = X(I)-cross conjugate of X(J) for these (I,J): (2,95), (401,290)

X(277)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), wheref(A,B,C) = [sec2(A/2)]/[- cos4A/2 + cos4B/2 + cos4C/2]Barycentrics af(A,B,C) : bf(B,C,A) : cf(C,A,B)

X(277) lies on these lines: 1,142 3,105 7,218 57,169 220,1086 241,278 942,1002

X(277) = isogonal conjugate of X(218)X(277) = isotomic conjugate of X(345)X(277) = X(55)-cross conjugate of X(7)

X(278)

Trilinears sec A tan A/2 : sec B tan B/2 : sec C tan C/2= csc A - 2 csc 2A : csc B - 2 csc 2B : csc C - 2 csc 2C= (1 - sec A)/a : (1 - sec B)/b : (1 - sec C)/c

Barycentrics tan A tan A/2 : tan B tan B/2 : tan C tan C/2

= 1 - sec A : 1 - sec B : 1 - sec C

X(278) lies on these lines:1,4 2,92 7,27 19,57 25,105 28,56 65,387 88,653 109,917 219,329 240,982 241,277 242,459 274,331 354,955 393,1108 412,962 443,1038 614,1096

X(278) = isogonal conjugate of X(219)X(278) = isotomic conjugate of X(345)X(278) = X(I)-Ceva conjugate of X(J) for these (I,J): (27,57), (92,196), (273,4), (331,7)X(278) = cevapoint of X(19) and X(34)X(278) = X(I)-cross conjugate of X(J) for these (I,J): (19,4), (56,7), (225,273)

X(279)

Trilinears csc A tan2A/2 : csc B tan2B/2 : csc C tan2C/2Barycentrics tan2A/2 : tan2B/2 : tan2C/2

X(279) lies on these lines: 1,7 2,85 28,1014 56,105 57,479 65,1002 144,220 145,664 304,346 942,955 985,1106

X(279) = isogonal conjugate of X(220)X(279) = isotomic conjugate of X(346)X(279) = cevapoint of X(57) and X(269)X(279) = X(I)-cross conjugate of X(J) for these (I,J): (57,7), (269,479)

X(280) X(1)-CROSS CONJUGATE OF X(8)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (csc2A/2)/(-1 - cos A + cos B + cos C)Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(280) lies on these lines: 2,318 7,253 8,20 21,268 75,309 78,282 285,1043 341,345

X(280) = isogonal conjugate of X(221)X(280) = isotomic conjugate of X(347)X(280) = X(309)-Ceva conjugate of X(189)X(280) = cevapoint of X(1) and X(84)X(280) = X(I)-cross conjugate of X(J) for these (I,J): (1,8), (281,2), (282,189)

X(281)

Trilinears sec A cot A/2 : sec B cot B/2 : sec C cot C/2= csc A + 2 csc 2A : csc B + 2 csc 2B : csc C + 2 csc 2C= (1 + sec A)/a : (1 + sec B)/b : (1 + sec C)/c

Barycentrics tan A cot A/2 : tan B cot B/2 : tan C cot C/2

= 1 + sec A : 1 + sec B : 1 + sec C

X(281) lies on these lines:1,282 2,92 4,9 7,653 8,29 28,958 33,200 37,158 45,53 48,944 100,1013 189,222 196,226 220,594 240,984 264,344 268,1012 318,346 380,950 451,1068 515,610 612,1096

X(281) = isogonal conjugate of X(222)X(281) = isotomic conjugate of X(348)X(281) = complement of X(347)X(281) = X(I)-Ceva conjugate of X(J) for these (I,J): (29,33), (92,4)X(281) = X(I)-cross conjugate of X(J) for these (I,J): (33,4), (37,9), (55,8)X(281) = crosspoint of X(I) and X(J) for these (I,J): (2,280), (92,318)

X(282) X(6)-CROSS CONJUGATE OF X(9)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (cot A/2)/(-1 - cos A + cos B + cos C)Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(282) lies on these lines: 1,281 2,77 3,9 19,102 48,947 78,280 200,219 271,283 380,1036

X(282) = isogonal conjugate of X(223)X(282) = X(189)-Ceva conjugate of X(84)X(282) = X(I)-cross conjugate of X(J) for these (I,J): (6,9), (33,1)X(282) = crosspoint of X(189) and X(280)

X(283)

Trilinears (cos A)/(cos B + cos C) : (cos B)/(cos C + cos A) : (cos C)/(cos A + cos B)Barycentrics (sin 2A)/(cos B + cos C) : (sin 2B)/(cos C + cos A) : (sin 2C)/(cos A + cos B)

X(283) lies on these lines: 1,21 2,580 3,49 29,270 60,284 77,603 78,212 86,307 102,110 271,282 474,582 643,1043 859,945 1010,1065

X(283) = isogonal conjugate of X(225)X(283) = X(333)-Ceva conjugate of X(284)X(283) = cevapoint of X(I) and X(J) for these (I,J): (3,255), (212,219)X(283) = X(3)-cross conjugate of X(21)X(283) = crosspoint of X(332) and X(333)

Let W be the vector-sum XA' + XB' + XC', where A'B'C' is the pedal triangle of X(283); then W = X(407)X(283).

X(284)

Trilinears (sin A)/(cos B + cos C) : (sin B)/(cos C + cos A) : (sin C)/(cos A + cos B)Barycentrics a2/(cos B + cos C) : b2/(cos C + cos A) : c2/(cos A + cos B)

X(284) lies on these lines:1,19 2,272 3,6 9,21 27,226 29,950 35,71 37,101 55,219 57,77 60,283 73,951 86,142 102,112 109,296 163,909 198,859 261,332 405,965 515,1065 942,1100

X(284) = isogonal conjugate of X(226)X(284) = isotomic conjugate of X(349)X(284) = inverse of X(579) in the Brocard circleX(284) = X(I)-Ceva conjugate of X(J) for these (I,J): (81,58), (333,283)X(284) = cevapoint of X(I) and X(J) for these (I,J): (6,48), (41,55)X(284) = X(55)-cross conjugate of X(21)X(284) = crosspoint of X(I) and X(J) for these (I,J): (21,81), (29,333)

X(285)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), wheref(A,B,C) = 1/[(cos B + cos C)(-1 - cos A + cos B + cos C)]

Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(285) lies on these lines: 21,84 29,81 271,282 280,1043

X(285) = isogonal conjugate of X(227)X(285) = X(58)-cross conjugate of X(21)

X(286)

Trilinears (csc 2A)/(sin B + sin C) : (csc 2B)/(sin C + sin A) : (csc 2C)/(sin A + sin B)Barycentrics (sec A)/(sin B + sin C) : (sec B)/(sin C + sin A) : (sec C)/(sin A + sin B)

X(286) lies on these lines: 4,69 7,331 19,27 28,242 29,34 99,915 112,767 158,969 322,1043

X(286) = isogonal conjugate of X(228)X(286) = isotomic conjugate of X(72)X(286) = cevapoint of X(I) and X(J) for these (I,J): (4,92), (7,273), (27,29), (28,81)X(286) = X(I)-cross conjugate of X(J) for these (I,J): (4,27), (7,86), (81,274)

X(287)

Trilinears cot A sec(A + ω) : cot B sec(B + ω) : cot C sec(C + ω) Barycentrics cos A sec(A + ω) : cos B sec(B + ω) : cos C sec(C + ω)

X(287) lies on these lines:2,98 6,264 69,248 83,217 95,141 185,384 193,253 293,306 297,685 305,394 401,511 651,894 879,895

X(287) = isogonal conjugate of X(232)X(287) = isotomic conjugate of X(297)X(287) = X(290)-Ceva conjugate of X(98)X(287) = cevapoint of X(2) and X(401)X(287) = X(248)-cross conjugate of X(98)X(287) = X(2)-Hirst inverse of X(98)

X(288)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = [sec(B - C)]/[b cos(C - A) + c cos(B - A)]Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(288) lies on these lines: 51,54 53,275 97,216

X(288) = isogonal conjugate of X(233)X(288) = cevapoint of X(6) and X(54)

X(289)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (sin A/2)/(cos B/2 + cos C/2 - cos A/2)Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(289) lies on these lines: 1,363 56,266 258,259

X(289) = isogonal conjugate of X(236)X(289) = X(6)-cross conjugate of X(266)

X(290)

Trilinears csc2A sec(A + ω) : csc2B sec(B + ω) : csc2C sec(C + ω)Barycentrics csc A sec(A + ω) : csc B sec(B + ω) : csc C sec(C + ω)

X(290) lies on these lines:2,327 3,76 6,264 54,276 66,317 67,340 68,315 69,670 71,190 72,668 73,336 248,385 265,316 308,311 892,895

X(290) = isogonal conjugate of X(237)X(290) = isotomic conjugate of X(511)X(290) = cevapoint of X(I) and X(J) for these (I,J): (2,511), (98,287)X(290) = X(I)-cross conjugate of X(J) for these (I,J): (385,308), (401,276), (511,2)

X(291)

Trilinears 1/(a2 - bc) : 1/(b2 - ca) : 1/(c2 - ab)Barycentrics a/(a2 - bc) : b/(b2 - ca) : c/(c2 - ab)

X(291) lies on these lines: 1,39 2,38 6,985 8,330 10,274 42,81 43,57 88,660 105,238 256,894 337,986 350,726 659,897 876,891

X(291) = isogonal conjugate of X(238)X(291) = isotomic conjugate of X(350)X(291) = X(I)-cross conjugate of X(J) for these (I,J): (239,256), (518,1)X(291) = X(I)-Hirst inverse of X(J) for these (I,J): (1,292), (2,335)

X(292)

Trilinears a/(a2 - bc) : b/(b2 - ca) : c/(c2 - ab)Barycentrics a2/(a2 - bc) : b2/(b2 - ca) : c2/(c2 - ab)

X(292) lies on these lines: 1,39 2,334 6,869 9,87 37,86 44,660 58,101 106,813 171,893 269,1020 659,665

X(292) = isogonal conjugate of X(239)X(292) = X(335)-Ceva conjugate of X(295)X(292) = cevapoint of X(171) and X(238)X(292) = X(1)-Hirst inverse of X(291)

X(293)

Trilinears cos A sec(A + ω) : cos B sec(B + ω) : cos C sec(C + ω)Barycentrics sin 2A sec(A + ω) : sin 2B sec(B + ω) : sin 2C sec(C + ω)

X(293) lies on these lines: 1,163 31,92 72,248 98,109 255,304 287,306

X(293) = isogonal conjugate of X(240)

X(294)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c - a)/(b2 + c2 - ab - ac)Barycentrics af(a,b,c) : bf(b,c,a) :cf(c,a,b)

X(294) lies on these lines: 1,41 2,949 4,218 6,7 8,220 19,1041 84,580 104,919 239,666 241,910 314,645

X(294) = isogonal conjugate of X(241)X(294) = X(1)-Hirst inverse of X(105)

X(295)

Trilinears (cos A)/(a2 - bc) : (cos B)/(b2 - ca) : (cos C)/(c2 - ab)Barycentrics (sin 2A)/(a2 - bc) : (sin 2B)/(b2 - ca) : (sin 2C)/(c2 - ab)

X(295) lies on these lines: 27,335 43,57 58,101 72,337 103,813 150,334 875,926 876,928

X(295) = isogonal conjugate of X(242)X(295) = X(335)-Ceva conjugate of X(292)X(295) = crosspoint of X(335) and X(337)

X(296)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), wheref(A,B,C) = (cos A)/[cos2A - cos B cos C]Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), whereg(A,B,C) = (sin 2A)/[cos2A - cos B cos C]

X(296) lies on these lines: 1,185 3,820 29,65 109,284

X(296) = isogonal conjugate of X(243)

X(297) X(2)-HIRST INVERSE OF X(4)

Trilinears csc 2A cos(A + ω) : csc 2B cos(B + ω) : csc 2C cos(C + ω)Barycentrics sec A cos(A + ω) : sec B cos(B + ω) : sec C cos(C + ω)

X(297) lies on these lines:2,3 6,317 53,141 69,393 76,343 83,275 92,257 232,325 249,316 287,685 315,394 340,524 525,850

X(297) = isogonal conjugate of X(248)X(297) = isotomic conjugate of X(287)X(297) = inverse of X(458) in orthocentroidal circleX(297) = complement of X(401)X(297) = anticomplement of X(441)X(297) = cevapoint of X(232) and X(511)X(297) = X(511)-cross conjugate of X(325)X(297) = X(2)-Hirst inverse of X(4)

Centers 298- 350 are isotomic conjugates of previously listed centers.

X(298) ISOTOMIC CONJUGATE OF 1st ISOGONIC CENTER

Trilinears csc2A sin(A + π/3) : csc2B sin(B + π/3) : csc2C sin(C + π/3)Barycentrics csc A sin(A + π/3) : csc B sin(B + π/3) : csc C sin(C + π/3)

X(298) lies on these lines:2,6 3,617 5,634 13,532 14,76 15,533 18,636 99,531 140,628 264,472 316,530 317,473 319,1082 340,470 381,622 511,1080

X(298) = reflection of X(I) about X(J) for these (I,J): (299,325), (385,395)X(298) = isotomic conjugate of X(13)X(298) = anticomplement of X(396)X(298) = X(300)-Ceva conjugate of X(303)X(298) = X(15)-cross conjugate of X(470)X(298) = X(2)-Hirst inverse of X(299)

X(299) ISOTOMIC CONJUGATE OF 2nd ISOGONIC CENTER

Trilinears csc2A sin(A - π/3) : csc2B sin(B - π/3) : csc2C sin(C - π/3)Barycentrics csc A sin(A - π/3) : csc B sin(B - π/3) : csc C sin(C - π/3)

X(299) lies on these lines:2,6 3,616 5,633 13,76 14,533 16,532 17,635 30,617 75,554 99,530 140,627 264,473 316,531 317,472 319,559 340,471 381,621 383,511

X(299) = reflection of X(I) about X(J) for these (I,J): (298,325), (385,396)X(299) = isotomic conjugate of X(14)X(299) = anticomplement of X(395)X(299) = X(301)-Ceva conjugate of X(302) X(299) = X(16)-cross conjugate of X(471)X(299) = X(2)-Hirst inverse of X(298)

X(300) ISOTOMIC CONJUGATE OF 1st ISODYNAMIC CENTER

Trilinears csc2A csc(A + π/3) : csc2B csc(B + π/3) : csc2C csc(C + π/3)Barycentrics csc A sin(A + π/3) : csc B sin(B + π/3) : csc C sin(C + π/3)

X(300) lies on these lines: 2,94 13,76 264,302 265,621 303,311

X(300) = isotomic conjugate of X(15)X(300) = cevapoint of X(298) and X(303)X(300) = X(94)-Hirst inverse of X(301)

X(301) ISOTOMIC CONJUGATE OF 2nd ISODYNAMIC CENTER

Trilinears csc2A csc(A - π/3) : csc2B csc(B - π/3) : csc2C csc(C - π/3)Barycentrics csc A sin(A - π/3) : csc B sin(B - π/3) : csc C sin(C - π/3)

X(301) lies on these lines: 2,94 14,76 264,303 265,622 302,311

X(301) = isotomic conjugate of X(16)X(301) = cevapoint of X(299) and X(302)X(301) = X(94)-Hirst inverse of X(300)

X(302) ISOTOMIC CONJUGATE OF 1st NAPOLEON POINT

Trilinears csc2A csc(A + π/6) : csc2B csc(B + π/6) : csc2C csc(C + π/6)Barycentrics csc A sin(A + π/6) : csc B sin(B + π/6) : csc C sin(C + π/6)

X(302) lies on these lines:2,6 3,621 5,622 14,99 16,316 18,76 61,629 140,633 264,300 301,311 317,470 381,616 549,617

X(302) = isotomic conjugate of X(17)X(302) = X(301)-Ceva conjugate of X(299)X(302) = X(61)-cross conjugate of X(473)

X(303) ISOTOMIC CONJUGATE OF 2nd NAPOLEON POINT

Trilinears csc2A csc(A - π/6) : csc2B csc(B - π/6) : csc2C csc(C - π/6)Barycentrics csc A sin(A - π/6) : csc B sin(B - π/6) : csc C sin(C - π/6)

X(303) lies on these lines:2,6 3,622 5,621 13,99 15,316 17,76 62,630 140,634 264,301 300,311 317,471 381,617 549,616

X(303) = isotomic conjugate of X(18)X(303) = X(300)-Ceva conjugate of X(298)X(303) = X(62)-cross conjugate of X(472)

X(304)

Trilinears (cot A)csc2A : (cot B)csc2B : (cot C)csc2C= cos A csc(A - ω) : cos B csc(B - ω) : cos C csc(C - ω)

Barycentrics (cos A)csc2A : (cos B)csc2B : (cos C)csc2C

X(304) lies on these lines:1,75 63,1102 69,72 76,85 92,561 255,293 279,346 305,306 309,322 341,1088 345,348

X(304) = isotomic conjugate of X(19)X(304) = cevapoint of X(I) and X(J) for these (I,J): (63,326), (69,345), (312,322)X(304) = X(I)-cross conjugate of X(J) for these (I,J): (63,75), (306,69)

X(305)

Trilinears b4c4cos A : c4a4cos B : a4b4cos C= cot A csc(A - ω) : cot B csc(B - ω) : cot C csc(C - ω)

Barycentrics b3c3cos A : c3a3cos B : a3b3cos C

X(305) lies on these lines:2,39 22,99 25,683 95,183 264,325 287,394 304,306 311,1007 341,1088 350,614

X(305) = isotomic conjugate of X(25)X(305) = X(69)-cross conjugate of X(76)

X(306)

Trilinears (b2c2)(b + c)cos A : (c2a2)(c + a)cos B : (a2b2)(a + b)cos CBarycentrics bc(b + c)cos A : ca(c + a)cos B : ab(a + b)cos C

X(306) lies on these lines:1,2 27,1043 63,69 72,440 92,264 209,518 226,321 253,329 287,293 304,305 319,333

X(306) = isotomic conjugate of X(27)X(306) = X(I)-Ceva conjugate of X(J) for these (I,J): (69, 72), (312,321), (313,10)X(306) = X(I)-cross conjugate of X(J) for these (I,J): (71,10), (72,307), (440,2)X(306) = crosspoint of X(I) and X(J) for these (I,J): (69,304), (312,345)

X(307)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = b2c2(b + c)(cos A)/(b + c - a)Barycentrics af(a,b,c) : bf(b,c,a) :cf(c,a,b)

X(307) lies on these lines: 2,7 8,253 69,73 75,225 86,283 95,320 141,241 269,936 319,664 948,966

X(307) = isotomic conjugate of X(29)X(307) = X(349)-Ceva conjugate of X(226)X(307) = X(I)-cross conjugate of X(J) for these (I,J): (72,306), (73,226)X(307) = crosspoint of X(69) and X(75)

X(308)

Trilinears b3c3/(b2 + c2) : c3a3/(c2 + a2) : a3b3/(a2 + b2)= csc2A csc(A + ω) : csc2B csc(B + ω) : csc2C csc(C + ω)= [csc(A - ω)]/(b2 + c2) : [csc(B - ω)]/(c2 + a2) : [csc(C - ω)]/(a2 + b2)

Barycentrics (b2c2)/(b2 + c2) : (c2a2)/(c2 + a2) : (a2b2)/(a2 + b2)= csc A csc(A + ω) : csc B csc(B + ω) : csc C csc(C + ω)

X(308) lies on these lines: 2,702 6,76 25,183 42,313 69,263 111,689 141,670 251,385 290,311

X(308) = isotomic conjugate of X(39)X(308) = cevapoint of X(2) and X(76)X(308) = X(I)-cross conjugate of X(J) for these (I,J): (2,83), (385,290)

X(309)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (csc2A)/(-1 - cos A + cos B + cos C)Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(309) lies on these lines: 69,189 75,280 77,318 84,314 85,264 304,322

X(309) = isotomic conjugate of X(40)X(309) = cevapoint of X(189) and X(280)X(309) = X(I)-cross conjugate of X(J) for these (I,J): (7,75), (92,85)

X(310)

Trilinears b3c3/(b + c) : c3a3/(c + a) : a3b3/(a + b)Barycentrics b2c2/(b + c) : c2a2/(c + a) : a2b2/(a + b)

X(310) lies on these lines: 2,39 7,314 38,75 86,350 99,675 261,272 321,335 333,673 670,903 871,982

X(310) = isotomic conjugate of X(42)X(310) = cevapoint of X(I) and X(J) for these (I,J): (75,76), (274,314)X(310) = X(75)-cross conjugate of X(274)

X(311)

Trilinears csc2A cos(B - C) : csc2B cos(C - A) : csc2C cos(A - B)Barycentrics csc A cos(B - C) : csc B cos(C - A) : csc C) cos(A - B)

X(311) lies on these lines: 2,570 4,69 22,157 53,324 95,99 141,338 290,308 300,303 301,302 305,1007

X(311) = isotomic conjugate of X(54)X(311) = anticomplement of X(570)X(311) = X(76)-Ceva conjugate of X(343)X(311) = cevapoint of X(5) and X(343)X(311) = X(5)-cross conjugate of X(324)

X(312)

Trilinears (b + c - a)b2c2 : (c + a - b)c2a2 : (a + b - c)a2b2

= (1 + cos A)csc(A - ω) : (1 + cos B)csc(B - ω) : (1 + cos C)csc(C - ω)

Barycentrics bc(b + c - a) : ca(c + a - b) : ab(a + b - c)

X(312) lies on these lines: 1,1089 2,37 8,210 9,314 29,33 63,190 69,189 76,85 92,264 212,643 223,664 726,982 894,940 975,1010

X(312) = isogonal conjugate of X(604)X(312) = isotomic conjugate of X(57)X(312) = X(I)-Ceva conjugate of X(J) for these (I,J): (76,75), (304,322), (314,8)X(312) = cevapoint of X(I) and X(J) for these (I,J): (2,329), (8,346), (9,78), (306,321)X(312) = X(I)-cross conjugate of X(J) for these (I,J): (8,75), (9,318), (306,345), (346,341)

X(313)

Trilinears (b + c)b3c3 : (c + a)c3a3 : (a + b)a3b3

= (b + c)csc(A - ω) : (c + a)csc(B - ω) : (a + b)csc(C - ω)

Barycentrics (b + c)b2c2 : (c + a)c2a2 : (a + b)a2b2

X(313) lies on these lines: 10,75 12,349 42,308 71,190 80,314 92,264 321,594 561,696

X(313) = isotomic conjugate of X(58)X(313) = X(76)-Ceva conjugate of X(321)X(313) = cevapoint of X(10) and X(306)X(313) = X(321)-cross conjugate of X(349)

X(314)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = b2c2(b + c - a)/(b + c)Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = bc(b + c - a)/(b + c)

X(314) lies on these lines:1,75 2,941 4,69 6,9817,310 9,312 21,261 29,1039 58,987 79,320 80,313 81,321 84,309 99,104 256,350 294,645

X(314) = isotomic conjugate of X(65)X(314) = X(310)-Ceva conjugate of X(274)X(314) = cevapoint of X(I) and X(J) for these (I,J): (8,312), (69,75)X(314) = X(I)-cross conjugate of X(J) for these (I,J): (8,333), (69,332), (333,274), (497,29)

X(315)

Trilinears bc(b4 + c4 - a4) : ca(c4 + a4 - b4) : ab(a4 + b4 - c4)Barycentrics b4 + c4 - a4 : c4 + a4 - b4 : a4 + b4 - c4

X(315) lies on these lines:2,32 3,325 4,69 5,183 8,760 20,99 68,290 192,746 194,736 274,377 297,394 343,458 371,491 372,492 631,1007

X(315) = isotomic conjugate of X(66)X(315) = anticomplement of X(32)X(315) = X(I)-cross conjugate of X(J) for these (I,J): (206,2)

X(316) = DROUSSENT PIVOT

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc(b4 + c4 - a4 - b2c2)Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = b4 + c4 - a4 - b2c2

The reflection of X(99) in the polar of X(&6).

Lucien Droussent, "Cubiques circularies anallagmatiques par points réciproques ou isogonaux," Mathesis 62 (1953) 204-215.

X(316) lies on these lines:2,187 4,69 15,303 16,302 30,99 115,385 148,538 183,381 249,297 265,290 298,530 299,531 376,1007 384,626 512,850 524,671 691,858

X(316) = midpoint between X(621) and X(622) X(316) = reflection of X(I) about X(J) for these (I,J): (99,325), (385,115)X(316) = isotomic conjugate of X(67)X(316) = anticomplement of X(187)

X(317)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = sec A cos 2A csc2ABarycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = tan A cos 2A csc2A

X(317) lies on these lines:2,95 4,69 6,297 25,325 53,524 66,290 141,458 183,427 193,393 261,406 273,320 298,473 299,472 302,470 303,471 318,319 459,1007

X(317) = isotomic conjugate of X(68)X(317) = anticomplement of X(577)X(317) = cevapoint of X(52) and X(467)

X(318)

Trilinears (1 + sec A)/a2 : (1 + sec B)/b2 : (1 + sec C)/c2

= sec A csc2A/2 : sec B csc2B/2 : sec C csc2C/2

Barycentrics (1 + sec A)/a : (1 + sec B)/b : (1 + sec C)/c

X(318) lies on these lines:2,280 4,8 10,158 29,33 53,594 63,412 75,225 77,309 108,404 200,1089 208,653 239,458 243,958 253,342 281,346 317,319 475,1068

X(318) = isogonal conjugate of X(603)X(318) = isotomic conjugate of X(77)X(318) = X(264)-Ceva conjugate of X(92)X(318) = cevapoint of X(9) and X(33)X(318) = X(I)-cross conjugate of X(J) for these (I,J): (9,312), (10,8), (281,92)

X(319)

Trilinears (1 + 2 cos A)/a2 : (1 + 2 cos B)/b2 : (1 + 2 cos C)/c2

Barycentrics (1 + 2 cos A)/a : (1 + 2 cos B)/b : (1 + 2 cos C)/c

X(319) lies on these lines: 2,1100 7,8 10,86 80,313 141,239 171,757 200,326 261,502 298,1082 299,559 306,333 307,664 317,318 321,1029 344,391 524,594

X(319) = isotomic conjugate of X(79)X(319) = anticomplement of X(1100)

X(320)

Trilinears (1 - 2 cos A)/a2 : (1 - 2 cos B)/b2 : (1 - 2 cos C)/c2

Barycentrics (1 - 2 cos A)/a : (1 - 2 cos B)/b : (1 - 2 cos C)/c

X(320) lies on these lines:1,752 2,44 7,8 58,86 79,314 95,307 141,894 144,344 190,527 239,524 269,326 273,317 334,660 350,513 519,679

X(320) = isotomic conjugate of X(80)X(320) = X(214)-cross conjugate of X(1)

X(321)

Trilinears (b + c)b2c2 : (c + a)c2a2 : (a + b)a2b2

= a(b + c)csc(A - ω) : b(c + a)csc(B - ω) : c(a + b)csc(C - ω)

Barycentrics bc(b + c) : ca(c + a) : ab(a + b)

X(321) lies on these lines:1,964 2,37 4,8 10,756 38,726 76,561 81,314 83,213 98,100 190,333 226,306 310,335 313,594 319,1029 668,671 693,824

X(321) = isotomic conjugate of X(81)X(321) = X(I)-Ceva conjugate of X(J) for (I,J) = (75,10), (76,313), (312,306)X(321) = cevapoint of X(37) and X(72)X(321) = X(442)-cross conjugate of X(264)X(321) = crosspoint of X(I) and X(J) for these (I,J): (75,76), (313,349)

X(322)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (-1 - cos A + cos B + cos C)csc2ABarycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (-1 - cos A + cos B + cos C)csc A

X(322) lies on these lines: 2,1108 7,8 78,273 92,264 227,347 253,341 286,1043 304,309 326,664

X(322) = isotomic conjugate of X(84)X(322) = anticomplement of X(1108)X(322) = X(304)-Ceva conjugate of X(312)X(322) = X(347)-cross conjugate of X(75)

X(323)

Trilinears sin 3A csc2A : sin 3B csc2B : sin 3C csc2CBarycentrics sin 3A csc A : sin 3B csc B : sin 3C csc C

X(323) lies on these lines: 2,6 20,155 23,110 30,146 140,195 187,249 401,525

X(323) = reflection of X(23) about X(110)X(323) = isotomic conjugate of X(94)X(323) = X(340)-Ceva conjugate of X(186)X(323) = cevapoint of X(6) and X(399)X(323) = X(50)-cross conjugate of X(186)

X(324)

Trilinears bc sec A cos(B - C) : ca sec B cos(C - A) : ab sec C cos(A - B)Barycentrics sec A cos(B - C) : sec B cos(C - A) : sec C cos(A - B)

X(324) lies on these lines: 2,216 4,52 53,311 94,275 110,436 143,565

X(324) = isotomic conjugate of X(97)X(324) = X(264)-Ceva conjugate of X(5)X(324) = cevapoint of X(I) and X(J) for these (I,J): (5,53), (52,216)X(324) = X(5)-cross conjugate of X(311)

X(325)

Trilinears csc2A cos(A + ω) : csc2B cos(B + ω) : csc2C cos(C + ω)= f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc(b4 + c4 - a2b2 - a2c2)

Barycentrics g(a,b,c) : g(b,c,a) : b(c,a,b), where g(a,b,c) = b4 + c4 - a2b2 - a2c2

X(325) lies on these lines:2,6 3,315 5,76 11,350 22,160 25,317 30,99 39,626 114,511 115,538 187,620 232,297 250,340 264,305 274,442 383,622 523,684 621,1080

X(325) = midpoint between X(I) and X(J) for these (I,J): (99,316), (298,299)X(325) = reflection of X(385) about X(230)X(325) = complement of X(385)X(325) = anticomplement of X(230)X(325) = cevapoint of X(2) and X(147)X(325) = X(I)-cross conjugate of X(J) for these (I,J): (114,2), (511,297)X(325) = X(2)-Hirst inverse of X(69)

X(326)

Trilinears cot2A : cot2B : cot2CBarycentrics csc A - sin A : csc B - sin B : csc C - sin C

X(326) lies on these lines: 1,75 48,63 69,73 200,319 255,1102 269,320 322,664 610,662

X(326) = isogonal conjugate of X(1096)X(326) = isotomic conjugate of X(158)X(326) = X(I)-Ceva conjugate of X(J) for these (I,J): (304,63), (332,69)X(326) = X(255)-cross conjugate of X(63)

X(327)

Trilinears csc2A sec(A - ω) : csc2B sec(B - ω) : csc2C sec(C - ω)= sin A csc(2A - 2 ω): sin B csc(2B - 2 ω) : sin C csc(2C - 2 ω)

Barycentrics csc A sec(A - ω) : csc B sec(B - ω) : csc C sec(C - ω)

X(327) lies on these lines: 2,290 4,276 5,76 53,141 69,263 95,160

X(327) = isotomic conjugate of X(182)

X(328)

Trilinears cot A csc 3A : cot B csc 3B : cot C csc 3CBarycentrics cos A csc 3A : cos B csc 3B : cos C csc 3C

X(328) lies on these lines: 2,94 69,265 95,99

X(328) = isotomic conjugate of X(186)X(328) = X(265)-cross conjugate of X(94)

X(329)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (-1 - cos A + cos B + cos C)(csc A)Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = -1 - cos A + cos B + cos C

X(329) lies on these lines:1,452 2,7 4,8 20,78 55,1005 69,189 100,972 190,345 191,498 196,342 200,516 219,278 220,948 223,347 253,306 388,960 392,1056 394,651 405,999 497,518

X(329) = isotomic conjugate of X(189)X(329) = cyclocevian conjugate of X(1034)X(329) = anticomplement of X(57)X(329) = X(I)-Ceva conjugate of X(J) for (I,J) = (69,8), (312,2)X(329) = X(I)-cross conjugate of X(J) for these (I,J): (40,347), (223,2)

X(330)

Trilinears bc/(ab + ac - bc) : ca/(bc + ba - ca) : ab/(ca + cb - ab)Barycentrics 1/(ab + ac - bc) : 1/(bc + ba - ca) : 1/(ca + cb - ab)

X(330) lies on these lines: 1,87 2,1107 8,291 56,385 57,239 76,1015 105,932 145,1002 193,959 257,982

X(330) = isotomic conjugate of X(192)X(330) = X(87)-Ceva conjugate of X(2)X(330) = X(75)-cross conjugate of X(2)

X(331)

Trilinears sec2A csc(2A) : sec2B csc(2B) : sec2C csc(2C)= (1 - sec A)csc(A - ω) : (1 - sec B)csc(B - ω) : (1 - sec C)csc(C - ω)

Barycentrics sec3A : sec3B : sec3C

X(331) lies on these lines: 4,150 7,286 34,870 75,225 85,92 108,767 274,278

X(331) = isotomic conjugate of X(219)X(331) = cevapoint of X(I) and X(J) for these (I,J): (7,278), (92,273)X(331) = X(92)-cross conjugate of X(264)

X(332)

Trilinears (cot A csc A)/(cos B + cos C) : (cot B csc B)/(cos C + cos A) : (cot C csc C)/(cos A + cos B)Barycentrics (cot A)/(cos B + cos C) : (cot B)/(cos C + cos A) : (cot C)/(cos A + cos B)

X(332) lies on these lines: 1,75 3,69 21,1036 99,102 219,345 261,284 1014,1037

X(332) = isotomic conjugate of X(225)X(332) = cevapoint of X(I) and X(J) for these (I,J): (69,326), (78,345)X(332) = X(I)-cross conjugate of X(J) for these (I,J): (69,314), (283,333)

X(333) CEVAPOINT OF X(8) AND X(9)

Trilinears bc(b + c - a)/(b + c) : ca(c + a - b)/(c + a) : ab(a + b - c)/(a + b)Barycentrics (b + c - a)/(b + c) : (c + a - b)/(c + a) : (a + b - c)/(a + b)

X(333) lies on these lines:2,6 8,21 9,312 10,58 19,27 29,270 57,85 190,321 239,257 261,284 306,319 310,673 662,909 740,846 859,956 1021,1024

X(333) = isotomic conjugate of X(226)

X(333) = X(I)-Ceva conjugate of X(J) for these (I,J): (261,21), (274,86)X(333) = cevapoint of X(I) and X(J) for these (I,J): (2,63), (8,9), (283,284)X(333) = X(I)-cross conjugate of X(J) for these (I,J): (8,314), (9,21), (21,86), (283,332), (284,29)X(333) = crosspoint of X(274) and X(314)

X(334)

Trilinears b2c2/(a2 - bc) : c2a2/(b2 - ca) : a2b2/(c2 - ab)Barycentrics bc/(a2 - bc) : ca/(b2 - ca) : ab/(c2 - ab)

X(334) lies on these lines: 2,292 10,274 12,85 75,141 76,1089 150,295 320,660 741,839 767,813

X(334) = isotomic conjugate of X(238)X(334) = X(75)-Hirst inverse of X(335)

X(335)

Trilinears bc/(a2 - bc) : ca/(b2 - ca) : ab/(c2 - ab)Barycentrics 1/(a2 - bc) : 1/(b2 - ca) : 1/(c2 - ab)

X(335) lies on these lines: 1,384 2,38 7,192 27,295 37,86 75,141 76,871 239,518 257,694 310,321 320,742 536,903 675,813 741,835 876,900

X(335) = reflection of X(190) about X(37)X(335) = isotomic conjugate of X(239)X(335) = cevapoint of X(I) and X(J) for these (I,J): (37,518), (292,295)X(335) = X(I)-cross conjugate of X(J) for these (I,J): (295,337), (350,257)X(335) = X(I)-Hirst inverse of X(J) for these (I,J): (2,291), (75,334)

X(336)

Trilinears csc A cot A sec(A + ω) : csc B cot B sec(B + ω) : csc C cot C sec(C + ω) Barycentrics cot A sec(A + ω) : cot B sec(B + ω) : cot C sec(C + ω)

X(336) lies on these lines: 1,811 48,75 73,290 255,293

X(336) = isotomic conjugate of X(240)

X(337)

Trilinears (csc A cot A)/(a2 - bc) : (csc B cot B))/(b2 - ca) : (csc C cot C)/(c2 - ab)Barycentrics (cot A)/(a2 - bc) : (cot B)/(b2 - ca) : (cot C)/(c2 - ab)

X(337) lies on these lines: 12,85 37,86 72,295 201,348 291,986

X(337) = isotomic conjugate of X(242)X(337) = X(295)-cross conjugate of X(335)

X(338) CEVAPOINT OF X(115) AND X(125)

Trilinears (b2 - c2)2/a3 : (c2 - a2)2/b3 : (a2 - b2)2/c3

= csc A sin2(B - C) : csc B sin2(C - A) : csc C sin2(A - B)

Barycentrics (b2 - c2)2/a2 : (c2 - a2)2/b2 : (a2 - b2)2/c2

= csc A sin2(B - C) : csc B sin2(C - A) : csc C sin2(A - B)

X(338) lies on these lines:2,94 4,67 6,264 50,401 76,599 115,127 125,136 141,311

X(338) = isotomic conjugate of X(249)X(338) = X(264)-Ceva conjugate of X(523)X(338) = cevapoint of X(115) and X(125)X(338) = X(125)-cross conjugate of X(339)

X(339)

Trilinears (b2 - c2)2(cos A)/a4 : (c2 - a2)2(cos B)/b4 : (a2 - b2)2(cos C)/c4

= csc A cot A sin2(B - C) : csc B cot B sin2(C - A) : csc C cot C sin2(A - B)

Barycentrics (b2 - c2)2(cos A)/a3 : (c2 - a2)2(cos B)/b3 : (a2 - b2)2(cos C)/c3

= cot A sin2(B - C) : cot B sin2(C - A) : cot C sin2(A - B)

X(339) lies on these lines: 3,76 69,265 115,127 264,381

X(339) = isotomic conjugate of X(250)X(339) = X(76)-Ceva conjugate of X(525)X(339) = X(125)-cross conjugate of X(338)

X(340)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), wheref(A,B,C) = sec A sin 3A csc3A : sec B sin 3B csc3B : sec C sin 3C csc3C

Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), whereg(A,B,C) = sec A sin 3A csc2A : sec B sin 3B csc2B : sec C sin 3C csc2C

X(340) lies on these lines: 4,69 67,290 95,140 250,325 297,524 298,470 299,471 447,540 458,599 520,850

X(340) = isotomic conjugate of X(265)X(340) = cevapoint of X(186) and X(323)

X(341)

Trilinears csc4A/2 : csc4B/2 : csc4C/2Barycentrics sin A csc4A/2 : sin B csc4B/2 : sin C csc4C/2

X(341) lies on these lines: 1,1050 8,210 10,75 40,190 200,1043 253,322 280,345 304,668 305,1088

X(341) = isogonal conjugate of X(1106)X(341) = isotomic conjugate of X(269)X(341) = X(346)-cross conjugate of X(312)

X(342)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (csc 2A tan A/2)/(-1 - cos A + cos B + cos C)Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sec A tan A/2)/(-1 - cos A + cos B + cos C)

X(342) lies on these lines: 4,7 9,653 85,264 92,226 108,1005 196,329 253,318 393,948

X(342) = isotomic conjugate of X(271)X(342) = X(I)-Ceva conjugate of X(J) for these (I,J): (85,92), (264,273)X(342) = cevapoint of X(208) and X(223)

X(343)

Trilinears cot A cos(B - C) : cot B cos(C - A) : cot C cos(A - B)Barycentrics cos A cos(B - C) : cos B cos(C - A) : cos C cos(A - B)

X(343) lies on these lines:2,6 3,68 5,51 22,161 53,311 76,297 140,569 315,458 427,511 470,634 471,633 472,621 473,622

X(343) = isotomic conjugate of X(275)X(343) = X(I)-Ceva conjugate of X(J) for these (I,J): (76,311), (311,5)X(343) = X(216)-cross conjugate of X(5)X(343) = crosspoint of X(69) and X(76)

X(344)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), wheref(A,B,C) = (csc2A/2)[cos4(B/2) + cos4(C/2) - cos4(A/2)]

Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A)f(A,B,C)

X(344) lies on these lines:2,37 7,190 8,480 9,69 44,193 45,141 144,320 264,281 319,391

X(344) = isotomic conjugate of X(277)

X(345)

Trilinears (csc A)/(1 - sec A) : (csc B)/(1 - sec B) : (csc C)/(1 - sec C)Barycentrics 1/(1 - sec A) : 1/(1 - sec B) : 1/(1 - sec C)

X(345) lies on these lines:2,37 8,21 22,100 57,728 63,69 78,1040 190,329 219,332 280,341 304,348 498,1089

X(345) = isogonal conjugate of X(608)X(345) = isotomic conjugate of X(278)X(345) = X(I)-Ceva conjugate of X(J) for these (I,J): (304,69), (332,78)X(345) = X(I)-cross conjugate of X(J) for these (I,J): (78,69), (219,8), (306,312)

X(346)

Trilinears bc(b + c - a)2 : ca(c + a - b)2 : ab(a + b - c)2

= cos(A/2) csc3(A/2) : cos(B/2) csc3(B/2) : cos(C/2) csc3(C/2)

Barycentrics (b + c - a)2 : (c + a - b)2 : (a + b - c)2

X(346) lies on these lines:2,37 6,145 8,9 45,594 69,144 78,280 100,198 219,644 220,1043 253,306 279,304 281,318 573,1018

X(346) = isotomic conjugate of X(279)X(346) = X(312)-Ceva conjugate of X(8)X(346) = X(200)-cross conjugate of X(8)X(346) = crosspoint of X(312) and X(341)

X(347)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (-1 - cos A + cos B + cos C) sec2(A/2)Barycentrics af(A,B,C) : bf(B,C,A) : cf(C,A,B)

X(347) lies on these lines:1,7 2,92 8,253 34,452 37,948 69,664 75,280 144,219 223,329 227,322 241,1108 573,1020

X(347) = isotomic conjugate of X(280)X(347) = anticomplement of X(281)X(347) = X(I)-Ceva conjugate of X(J) for these (I,J): (75,7), (348,2)X(347) = cevapoint of X(40) and X(223)X(347) = X(I)-cross conjugate of X(J) for these (I,J): (40,329), (221,196), (227,223)X(347) = crosspoint of X(75) and X(322)

X(348)

Trilinears cot A sec2(A/2) : cot B sec2(B/2) : cot C sec2(C/2)= (csc A)/(1 + sec A) : (csc B)/(1 + sec B) : (csc C)/(1 + sec C)

Barycentrics 1/(1 + sec A) : 1/(1 + sec B) : 1/(1 + sec C)

X(348) lies on these lines: 2,85 7,21 8,664 69,73 75,280 150,944 201,337 274,278 304,345 499,1111

X(348) = isogonal conjugate of X(607)X(348) = isotomic conjugate of X(281)X(348) = X(274)-Ceva conjugate of X(85)X(348) = cevapoint of X(I) and X(J) for these (I,J): (2,347), (63,77)X(348) = X(222)-cross conjugate of X(7)

X(349)

Trilinears (cos B + cos C)csc3A : (cos C + cos A)csc3B : (cos A + cos B) csc3C= (cos B + cos C)csc(A - ω) : (cos C + cos A)csc(B - ω) : (cos A + cos B)csc(C - ω)

Barycentrics (cos B + cos C)csc2A : (cos C + cos A)csc2B : (cos A + cos B)(csc C/2)2

X(349) lies on these lines: 12,313 73,290 75,225 76,85

X(349) = isotomic conjugate of X(284)X(349) = cevapoint of X(226) and X(307)X(349) = X(321)-cross conjugate of X(313)

X(350)

Trilinears (a2 - bc)b2c2 : (b2 - ca)c2a2 : (c2 - ab)a2b2 Barycentrics bc(a2 - bc) : ca(b2 - ca) : ab(c2 - ab)

X(350) lies on these lines:1,76 2,37 11,325 33,264 36,99 42,308 55,183 69,497 86,310 172,384 190,672 256,314 291,726 305,614 320,513 447,811 519,668 538,1015 889,903

X(350) = isotomic conjugate of X(291)X(350) = crosspoint of X(257) and X(335)X(350) = X(2)-Hirst inverse of X(75)

X(351) = CENTER OF THE PARRY CIRCLE

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b2 - c2)(b2 + c2 - 2a2)Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a2(b2 - c2)(b2 + c2 - 2a2)

X(351) is the center of the Parry circle introduced in TCCT (Art. 8.13) as the circle that passes through X(i) for i = 2, 15, 16, 23, 110, 111, 352, 353.

X(351) lies on these lines: 2,804 110,526 184,686 187,237 694,881 865,888 X(351) = isogonal conjugate of X(892)X(351) = crosspoint of X(110) and X(111)

X(352)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(-a4 - b4 - c4 - 5b2c2 + 4a2b2 + 4a2c2)Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)

A point on the Parry circle; see X(351).

X(352) lies on these lines: 2,6 3,353 110,187 111,511

X(352) = inverse of X(353) in the circumcircle

X(353)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(4a4 - 2b4 - 2c4 - b2c2 - 4a2b2 - 4a2c2)Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)

A point on the Parry circle; see X(351).

X(353) lies on these lines: 3,352 6,23 110,574 111,182

X(353) = inverse of X(352) in the circumcircleX(353) = inverse of X(111) in the Brocard circle

X(354) = WEILL POINT

Trilinears (b - c)2 - ab - ac : (c - a)2 - bc - ba : (a - b)2 - ca - cbBarycentrics a[(b - c)2 - ab - ac] : b[(c - a)2 - bc - ba] : c[(a - b)2 - ca - cb]

X(354) is the centroid of the intouch triangle.

William Gallatly, The Modern Geometry of the Triangle, 2nd edition, Hodgson, London, 1913, page 16.

X(354) lies on these lines: 1,3 2,210 6,374 7,479 11,118 37,38 42,244 44,748 48,584 63,1001 81,105 278,955 373,375 388,938 392,551 516,553

X(354) = reflection of X(I) about X(J) for these (I,J): (210,2)X(354) = X(101)-Ceva conjugate of X(513)X(354) = crosspoint of X(1) and X(7)

Let X = X(354) and let V be the vector-sum XA + XB + XC; then V = X(72)X(1) = X(8)X(65).

X(355) = FUHRMANN CENTER

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a cos A - (b + c)cos(B - C)Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)

The center of the Fuhrmann circle, defined as the circumcircle of the Furhmann triangle A"B"C", where A" is obtained as follows: let A' be the midpoint of the shorter arc having endpoints B and C on the circumcircle of ABC; then A" is the reflection of A' about line BC. Vertices B" and C" are obtained cyclically.

Ross Honsberger, Episodes in Nineteenth and Twentieth Century Euclidean Geometry, Mathematical Association of America, 1995. Chapter 6: The Fuhrmann Circle.

X(355) lies on these lines:1,5 2,944 3,10 4,8 30,40 65,68 85,150 104,404 165,550 381,519 382,516 388,942 938,1056

X(355) = midpoint between X(4) and X(8)X(355) = reflection of X(I) about X(J) for these (I,J): (1,5), (3,10)X(355) = complement of X(944)

X(356) = 1st MORLEY CENTER

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = cos A/3 + 2 cos B/3 cos C/3Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(356) is the centroid of the Morley equilateral triangle. Triangle centers bearing Morley's name possibly do not appear in the pre-1994 literature on Morley's famous theorem. For a discussion of the theorem and extensive list of references, see

C. O. Oakley and J. C. Baker, "The Morley trisector theorem," American Mathematical Monthly 85 (1978) 737-745.

X(356) lies on this line: 357,358

X(357) = 2nd MORLEY CENTER

Trilinears sec A/3 : sec B/3 : sec C/3Barycentrics sin A sec A/3 : sin B sec B/3 : sin C sec C/3

X(357) is the perspector of Morley triangle and ABC.

X(357) lies on this line: 356,358

X(357) = isogonal conjugate of X(358)

X(358) = MORLEY-YFF CENTER

Trilinears cos A/3 : cos B/3 : cos C/3Barycentrics sin A cos A/3 : sin B cos B/3 : sin C cos C/3

X(358) is the perspector of the adjunct Morley triangle and ABC.

X(358) lies on this line: 356,357

X(358) = isogonal conjugate of X(357)

X(359) = HOFSTADTER ONE POINT

Trilinears a/A : b/B : c/CBarycentrics 1/A : 1/B : 1/C

This point is the limit as r approaches 1 of the perspector of the r-Hofstadter triangle and ABC. See X(360) for details.

X(359) = isogonal conjugate of X(360)

X(360) = HOFSTADTER ZERO POINT

Trilinears A/a : B/b : C/cBarycentrics A : B : C

This point is obtained as a limit of perspectors. Let r denote a real number, but not 0 or 1. Using vertex B as a pivot, swing line BC toward vertex A through angle rB and swing line BC about C through angle rC. Let A(r) be the point in which the two swung lines meet. Obtain B(r) and C(r) cyclically. Triangle A(r)B(r)C(r) is the r-Hofstadter triangle; its perspector with ABC is the point given by trilinears

sin(r(A))/sin(A - r(A)) : sin(r(B))/sin(B - r(B)) : sin(r(C))/sin(C - r(C)).

The limit of this point as r approaches 0 is X(360). The two Hofstadter points, X(359) and X(360) are examples of transcendental triangle centers, since they have no trilinear or barycentric representation using only algebraic functions of a,b,c (or sin A, sin B, sin C).

Clark Kimberling, "Hofstadter points," Nieuw Archief voor Wiskunde 12 (1994) 109-114.

X(360) = isogonal conjugate of X(359)

X(361)

Trilinears csc B/2 + csc C/2 - csc A/2 : csc C/2 + csc A/2 - csc B/2 : csc A/2 + csc B/2 - csc C/2Barycentrics f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (sin A)(csc B/2 + csc C/2 - csc A/2)

The isoscelizer equation au(X) = bv(X) = cw(X) has solution X = X(361).

X(361) lies on these lines: 1,188 164,503 266,978

X(361) = X(266)-Ceva conjugate of X(1)

X(362) = CONGRUENT CIRCUMCIRCLES ISOSCELIZER POINT

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), wheref(a,b,c) = b cos B/2 + c cos C/2 - a cos A/2

Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)

The isoscelizer equations u(X)/a = v(X)/b = w(X)/c have solution X = X(362).

X(362) lies on this line: 57,234

X(362) = X(508)-Ceva conjugate of X(1)

X(363) = EQUAL PERIMETERS ISOSCELIZER POINT

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = b/(1 + sin B/2) + c/(1 + sin C/2) - a/(1 + sin A/2) Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)

When X = X(363), the isoscelizer triangles have equal perimeters.

X(363) lies on these lines: 1,289 40,164 165,166

X(364) = WABASH CENTER (EQUAL AREAS ISOSCELIZER POINT)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = b1/2 + c1/2 - a1/2 Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)

When X = X(364), the isoscelizer triangles T(X,a), T(X,b), T(X,c) have equal areas.

X(364) lies on these lines: 1,365 9,366

X(364) = X(366)-Ceva conjugate of X(1)

X(365) = SQUARE ROOT POINT

Trilinears a1/2 : b1/2 : c1/2 Barycentrics a3/2 : b3/2 : c3/2

For a construction of X(365), see the note at X(2), which provides for a construction barycentric square roots which one can easily extend to a construction for trilinear square roots.

X(365) lies on this line: 1,364

X(365) = isogonal conjugate of X(366)

X(366)

Trilinears a-1/2 : b-1/2 : c-1/2 Barycentrics a1/2 : b1/2 : c1/2

See the note at X(365).

X(366) lies on these lines: 2,367 9,364

X(366) = isogonal conjugate of X(365)X(366) = cevapoint of X(1) and X(364)X(366) = X(367)-cross conjugate of X(1)

X(367)

Trilinears b1/2 + c1/2 : c1/2 + a1/2 : a1/2 + b1/2 Barycentrics a1/2(b1/2 + c1/2) : b1/2(c1/2 + a1/2) : c1/2(a1/2 + b1/2)

X(367) lies on these lines: 1,364 2,366

X(367) = crosspoint of X(1) and X(366)

X(368) = EQUI-BROCARD CENTER

Trilinears (reasonable trilinears are sought) Barycentrics (reasonable barycentrics are sought)

The center X for which the triangle XBC, XCA, XAB have equal Brocard angles. Yff proved that X(368) lies on the self-isogonal conjugate cubic with trilinear equation f(a,b,c)u + f(b,c,a)v + f(c,a,b)w = 0, where f(a,b,c) = bc(b2 - c2) and, for variable x : y : z, the cubics u, v, w are given by u(x,y,z) = x(y2 + z2), v = u(y,z,x), w = u(z,x,y).

Parry proved that X(368) lies on the anticomplement of the Kiepert hyperbola, this anticomplement being given by the trilinear equation a2(b2 - c2)(x2) + b2(c2 - a2)(y2) + c2(a2

- b2)(z2) = 0.

X(369) = TRISECTED PERIMETER POINT

Trilinears x : y : z (see below) Barycentrics ax : by : cz

There exist points A', B', C' on segments BC, CA, AB, respectively, such that A'C + CB' = B'A + AC' = C'B + BA' and the lines AA', BB', CC' concur in X(369). Yff found trilinears for X(369) in terms of the unique real root, r, of the cubic polynomial

2t3 - 3(a + b + c)t2 + (a2 + b2 + c2 + 8bc + 8ca + 8ab)t - (cb2 + ac2 + ba2 + 5bc2 + 5ca2 + 5ab2 + 9abc),

as follows: x = bc(r - c + a)(r - a + b). Although x(a,c,b) ≠ x(a,b,c), Yff states that a symmetric but more elaborate form for x can be obtained.

X(370) = EQUILATERAL CEVIAN TRIANGLE POINT

Trilinears (reasonable trilinears are sought) Barycentrics (reasonable barycentrics are sought)

A point P is an equilateral cevian triangle point if the cevian triangle of P is equilateral. Jiang Huanxin introduced this notion in 1997.

Jiang Huanxin and David Goering, Problem 10358* and Solution, "Equilateral cevian triangles," American Mathematical Monthly 104 (1997) 567-570 [proposed 1994].

X(371) = KENMOTU POINT (CONGRUENT SQUARES POINT)

Trilinears cos(A - π/4) : cos(B - π/4) : cos(C - π/4) = cos A + sin A : cos B + sin B : cos C + sin C

Barycentrics sin A cos(A - π/4) : sin B cos(B - π/4) : sin C cos(C - π/4)

There exist three congruent squares U, V, W positioned in ABC as follows: U has opposing vertices on segments AB and AC; V has opposing vertices on segments BC and BA; W has opposing vertices on segments CA and CB, and there is a single point common to U, V, W. The common point, X(371), may have first been published in Kenmotu's Collection of Sangaku Problems in 1840, indicating that its first appearance may have been anonymously inscribed on a wooden board hung up in a Japanese shrine or temple. (The Kenmotu configuration uses only half-squares; i.e., isosceles right triangles). Trilinears were found by John Rigby.

The edgelength of the three squares is 21/2abc/(a2 + b2 + c2 + 4σ), where σ = area(ABC). (Edward Brisse, 2/12/00)

Hidetoshi Fukagawa, Traditional Japanese Mathematics Problems of the 18th and 19th Centuries, forthcoming.

Floor van Lamoen, Vierkanten in een driehoik: 3. Congruente vierkanten

Tony Rothman, with the cooperation of Hidetoshi Fukagawa, Japanese Temple Geometry (feature article in Scientific American

X(371) lies on these lines:2,486 3,6 4,485 25,493 140,615 193,488 315,491 492,641 601,606 602,605

X(371) = reflection of X(372) about X(32)X(371) = isogonal conjugate of X(485)X(371) = inverse of X(372) in the Brocard circleX(371) = complement of X(637)X(371) = anticomplement of X(639)X(371) = X(4)-Ceva conjugate of X(372)

X(372) = HARMONIC CONJUGATE OF X(371) WRT X(3) AND X(6)

Trilinears cos(A + π/4) : cos(B + π/4) : cos(C + π/4) = cos A - sin A : cos B - sin B : cos C - sin C

Barycentrics sin A cos(A + π/4) : sin B cos(B + π/4) : sin C cos(C + π/4)

For details and references, see X(371).

X(372) lies on these lines:2,485 3,6 4,486 25,494 193,487 315,492 601,605 602,606

X(372) = reflection of X(371) about X(32)X(372) = isogonal conjugate of X(486)X(372) = inverse of X(371) in the Brocard circleX(372) = complement of X(638)X(372) = anticomplement of X(640)X(372) = X(4)-Ceva conjugate of X(371)

X(373) = CENTROID OF THE PEDAL TRIANGLE OF THE CENTROID

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 2bc + ac cos C + ab cos B= g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a(b4 + c4 - a2b2 - a2c2 - 6b2c2)

Barycentrics h(a,b,c) : h(b,c,a) : h(c,a,b), where h(a,b,c) = 2abc + ca2cos C + ba2cos B

X(373) lies on these lines: 2,51 5,113 110,575 181,748 216,852 354,375

Let X = X(373) and let V be the vector-sum XA + XB + XC; then V = X(2)X(51)

X(374) = CENTROID OF THE PEDAL TRIANGLE OF X(9)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 2b + 2c - 3a + (c + a)cos C + (b + a)cos BBarycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(374) lies on these lines: 6,354 9,517 44,65 51,210

X(375) = CENTROID OF THE PEDAL TRIANGLE OF X(10)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 2bc(b + c) + ca(c + a)cos C + ab(a + b)cos BBarycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(375) lies on these lines: 44,181 51,210 354,373

X(375) = midpoint between X(51) and X(210)

X(376) = CENTROID OF THE ANTIPEDAL TRIANGLE OF X(2)

Trilinears 5 cos A - cos(B - C) : 5 cos B - cos(C - A) : 5 cos C - cos(A - B)= f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (csc A)(5 sin 2A - sin 2B - sin 2C)

Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = 5 sin 2A - sin 2B - sin 2C

X(376) is the reflection of X(2) about X(3).

X(376) lies on these lines:1,553 2,3 35,388 36,497 40,519 55,1056 56,1058 69,74 98,543 103,544 104,528 110,541 112,577 165,515 316,1007 390,999 476,841 477,691 487,490 488,489 516,551

X(376) = midpoint between X(2) and X(20)X(376) = reflection of X(I) about X(J) for these (I,J): (2,3), (4,2)X(376) = anticomplement of X(381)

Let X = X(376) and let V be the vector-sum XA + XB + XC; then V = X(382)X(3) = X(4)X(20).

X(377)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc(b4 + c4 - a4 - 2b2c2 - 2abc(a + b + c))Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = b4 + c4 - a4 - 2b2c2 - 2abc(a + b + c)

X(377) lies on these lines:1,224 2,3 7,8 10,46 78,226 81,387 142,950 145,1056 149,1058 225,1038 274,315 908,936 1060,1068

X(377) = anticomplement of X(405)

X(378) = HARMONIC CONJUGATE OF X(24) WRT X(3) AND X(4)

Trilinears sec A + 2 cos A : sec B + 2 cos B : sec C + 2 cos C Barycentrics tan A + sin 2A : tan B + sin 2B : tan C + sin 2C

X(378) lies on these lines:1,1063 2,3 6,74 33,36 34,35 54,64 99,264 185,578 232,574 477,935 847,1105

X(378) = reflection of X(I) about X(J) for these (I,J): (4,427), (22,3)X(378) = inverse of X(403) in the orthocentroidal circle

X(379)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc[a5 + (b + c)(bca2 - (bc + ca + ab)(b - c)2)]Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a5 + (b + c)(bca2 - (bc + ca + ab)(b - c)2

X(379) lies on these lines: 2,3 6,7 41,226 63,169 264,823

X(379) = inverse of X(857) in the orthocentroidal circle

X(380)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c - a)[3a3 + (b + c)(3a2 + (b - c)2 + a(b + c))]Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(380) lies on these lines: 1,19 6,40 9,55 165,579 223,608 281,950 282,1036

X(381) = MIDPOINT OF X(2) BETWEEN X(4)

Trilinears 2 cos(B - C) - cos A : 2 cos(C - A) - cos B : 2 cos(A - B) - cos C= cos A + 4 cos B cos C : cos B + 4 cos C cos A : cos C + 4 cos A cos B

Barycentrics a(cos A + 4 cos B cos C) : b(cos B + 4 cos C cos A) : c(cos C + 4 cos A cos B)

X(381) lies on these lines:2,3 6,13 11,999 49,578 51,568 54,156 98,598 114,543 118,544 119,528 125,541 127,133 155,195 183,316 184,567 210,517 262,671 264,339 298,622 299,621 302,616 303,617 355,519 388,496 495,497 511,599 515,551

X(381) = midpoint between X(2) and X(4)X(381) = reflection of X(I) about X(J) for these (I,J): (2,5), (3,2)X(381) = complement of X(376)X(381) = anticomplement of X(549)

Let X = X(381) and let V be the vector-sum XA + XB + XC; then V = X(20)X(3) = X(3)X(4) = X(185)X(52) = X(399)X(146) = X(74)X(265) = X(40)X(355) = X(376,381) = X(4,382).

X(382) = REFLECTION OF CIRCUMCENTER ABOUT ORTHOCENTER

Trilinears cos A - 4 cos B cos C : cos B - 4 cos C cos A : cos C - 4 cos A cos BBarycentrics a(cos A - 4 cos B cos C) : b(cos B - 4 cos C cos A) : c(cos C - 4 cos A cos B)

X(382) lies on these lines: 2,3 64,265 155,399 185,568 195,1078 355,516 952,962

X(382) = reflection of X(I) about X(J) for these (I,J): (3,4), (20,5)X(382) = inverse of X(546) in the orthocentroidal circleX(382) = anticomplement of X(550)

X(383)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), wheref(A,B,C) = csc(B - C) [sin 2B cos(C - ω) sin(C + π/3) - sin 2C cos(B - ω) sin(B + π/3)]

Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(383) lies on these lines: 2,3 13,262 14,98 183,621 299,511 325,622

X(383) = inverse of X(1080) in the orthocentroidal circle

X(384)

Trilinears bc(a4 + b2c2) : ca(b4 + c2a2) : ab(c4 + a2b2)Barycentrics a4 + b2c2 : b4 + c2a2 : c4 + a2b2

A center on the Euler line; contributed by John Conway, email, 1998.

X(384) lies on these lines:1,335 2,3 6,194 32,76 39,83 141,1031 172,350 185,287 316,626 694,695

X(384) = isogonal conjugate of X(695)X(384) = eigencenter of anticevian triangle of X(385)

X(385) = HARMONIC CONJUGATE OF X(384) WRT X(32) AND X(76)

Trilinears bc(a4 - b2c2) : ca(b4 - c2a2) : ab(c4 - a2b2)Barycentrics a4 - b2c2 : b4 - c2a2 : c4 - a2b2

Contributed by John Conway, 1998.

X(385) lies on these lines:1,257 2,6 3,194 23,523 30,148 32,76 55,192 56,330 98,511 99,187 111,892 115,316 171,894 232,648 248,290 251,308 262,576

X(385) = reflection of X(I) about X(J) for these (I,J): (99,187), (298,395), (299,396), (316,115), (325,230)X(385) = isogonal conjugate of X(694)X(385) = anticomplement of X(325)X(385) = X(I)-Ceva conjugate of X(J) for these (I,J): (98,2), (511,401)X(385) = crosspoint of X(290) and X(308)X(385) = X(I)-Hirst inverse of X(J) for these (I,J): (2,6), (3,194)

X(386)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b2 + c2 + bc + ca + ab)Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a2(b2 + c2 + bc + ca + ab)

X(386) lies on these lines:1,2 3,6 31,35 40,1064 55,595 56,181 57,73 65,994 81,404 474,940 758,986 872,984

X(386) = inverse of X(58) in the Brocard circle

X(387)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc[-a4 + 2a2(a + b + c)2 + (b2 - c2)2]Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = -a4 + 2a2(a + b + c)2 + (b2 - c2)2

X(387) lies on these lines:1,2 4,6 20,58 40,579 65,278 81,377 390,595 443,940

X(388)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc[a2 + (b + c)2]/(b + c -a)= 1 + cos B cos C : 1 + cos C cos A : 1 + cos A cos BBarycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a2 + (b + c)2]/(b + c - a)

X(388) lies on these lines:1,4 2,12 3,495 5,999 7,8 10,57 11,153 20,55 29,1037 35,376 36,498 79,1000 108,406 171,603 201,984 329,960 354,938 355,942 381,496 442,956 452,1001 612,1038 750,1106 1059,1067

X(388) = isogonal conjugate of X(1036)X(388) = anticomplement of X(958)

X(389) = CENTER OF THE TAYLOR CIRCLE

Trilinears cos A - cos 2A cos(B - C) : cos B - cos 2B cos(C - A) : cos C - cos 2C cos(A - B)Barycentrics a[cos A - cos 2A cos(B - C)] : b[cos B - cos 2B cos(C - A)] : c[cos C - cos 2C cos(A - B)]

If ABC is acute then X(389) is the Spieker center of the orthic triangle. Peter Yff reports (Sept. 19, 2001) that since X(389) is on the Brocard axis, there must exist T for which X(389) is sin(A+T) : sin(B+T) : sin(C+T), and that tan(T) = - cot A cot B cot C.

X(389) lies on these lines:3,6 4,51 24,184 30,143 54,186 115,129 217,232 517,950

X(389) = midpoint between X(I) and X(J) for these (I,J): (3,52), (4,185)X(389) = inverse of X(578) in the Brocard circleX(389) = crosspoint of X(4) and X(54)

X(390) REFLECTION OF GERGONNE POINT ABOUT INCENTER

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc(b + c - a)[3a2 + (b - c)2]Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = (b + c - a)[3a2 + (b - c)2]

X(390) = lies on these lines:1,7 2,11 3,1058 4,495 8,9 30,1056 40,938 144,145 376,999 387,595 496,631 944,971 952,1000

X(390) = midpoint between X(144) and X(145)X(390) = reflection of X(I) about X(J) for these (I,J): (7,1), (8,9)

X(391)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc(3a + b + c)(b + c - a)Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = (3a + b + c)(b + c - a)

X(391) lies on these lines:2,6 8,9 20,573 37,145 75,144 319,344

X(392)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c)(b2 + c2 - a2) + 4abcBarycentrics af(a,b,c): bf(b,c,a): cf(c,a,b)

X(392) lies on these lines:1,6 2,517 8,1000 10,11 21,104 40,474 55,997 63,999 78,1057 210,519 329,1056 354,551 442,946 443,962 452,944 495,908

Let X = X(392) and let V be the vector-sum XA + XB + XC; then V = X(65)X(1) = X(8)X(72).

X(393)

Trilinears bc tan2A : bc tan2B : bc tan2CBarycentrics tan2A : tan2B : tan2C

X(393) lies on these lines:1,836 2,216 4,6 19,208 20,577 24,254 25,1033 27,967 33,42 37,158 69,297 107,111 193,317 230,459 278,1108 342,948 394,837 800,1093

X(393) = cevapoint of X(4) and X(459)X(393) = X(25)-cross conjugate of X(4)

X(394)

Trilinears cos A cot A : cos B cot B : cos C cot CBarycentrics cos2A : cos2B : cos2C

X(394) lies on these lines: 2,6 3,49 20,1032 22,110 25,511 63,77 72,1060 76,275 78,271 287,305 297,315 329,651 393,837 399,541 470,633 471,634 472,622 473,621 611,612 613,614 1062,1069

X(394) = X(69)-Ceva conjugate of X(3)X(394) = crosspoint of X(493) and X(494)

Let W be the vector-sum XA' + XB' + XC', where A'B'C' is the pedal triangle of X(394); then W = X(25)X(394).

X(395) = MIDPOINT OF X(14) AND X(16)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = cos(B - C) + 2 cos(A + π/3)Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,B,A)

X(395) lies on these lines:2,6 3,398 5,13 14,16 15,549 39,618 53,472 61,140 115,530 187,531 202,495 216,465 466,577 532,624 533,619

X(395) = reflection of X(396) about X(230)X(395) = midpoint between X(I) and X(J) for these (I,J): (14,16), (298,385)X(395) = complement of X(299)X(395) = crosspoint of X(2) and X(14)

X(396) = MIDPOINT OF X(13) AND X(15)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = cos(B - C) + 2 cos(A - π/3)Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,B,A)

X(396) lies on these lines:2,6 3,397 5,14 13,15 16,549 39,619 53,473 62,140 115,531 187,530 203,495 216,466 465,577 532,618 533,623

X(396) = midpoint between X(I) and X(J) for these (I,J): (13,15), (299,385)X(396) = reflection of X(395) about X(230)X(396) = anticomplement of X(298)X(396) = crosspoint of X(2) and X(13)

X(397) CROSSPOINT OF ORTHOCENTER AND 1st NAPOLEON POINT

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = cos(B - C) - 2 cos(A + π/3)Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,B,A)

X(397) lies on these lines: 3,396 4,6 5,13 14,546 15,550 16,17 30,61 51,462 141,634 184,463 202,496 524,633 532,635

X(397) = crosspoint of X(4) and X(17)

X(398) CROSSPOINT OF ORTHOCENTER AND 2nd NAPOLEON POINT

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = cos(B - C) - 2 cos(A - π/3)Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,B,A)

X(398) lies on these lines:3,395 4,6 5,14 13,546 15,18 16,550 30,62 51,463 141,633 184,462 203,496 524,634 533,636

X(398) = crosspoint of X(4) and X(18)

X(399) = PARRY REFLECTION POINT

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), wheref(A,B,C) = 5 cos A - 4 cos B cos C - 8 sin B sin C cos2A

Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,B,A)

Let L, M, N be lines through A, B, C, respectively, parallel to the Euler line. Let L' be the reflection of L about sideline BC, let M' be the reflection of M about sideline CA, and let N' be the reflection of N about sideline AB. The lines L', M', N' concur in X(399).

Cyril Parry, Problem 10637, American Mathematical Monthly 105 (1998) 68.

X(399) lies on these lines:3,74 4,195 6,13 30,146 155,382 394,541

X(399) = reflection of X(I) about X(J) for these (I,J): (3,110), (265,113)X(399) = X(I)-Ceva conjugate of X(J) for these (I,J): (30,3), (323,6)

Let W be the vector-sum XA' + XB' + XC', where A'B'C' is the pedal triangle of X(399); then W = X(74)X(399).

X(400) = YFF-MALFATTI POINT

Trilinears csc4(A/4) : csc4(B/4) : csc4(C/4)Barycentrics sin A csc4(A/4) : sin B csc4(B/4) : sin C csc4(C/4)

In 1997, Yff considered the configuration for the 1st Ajima-Malfatti point, X(179). He proved that the same tangencies are possible in another way if the circles are not required to lie inside ABC. With tangency points labeled as before, the lines AA', BB', CC' concur in X(400).

Centers 401- 475, 2- 4, 20- 30, 376, 379, and 381- 384 lie on the Euler line.

X(401) = BAILEY POINT

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), wheref(A,B,C) = [sin 2B sin 2C - sin2(2A)](csc A) Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = sin 2B sin 2C - sin2(2A)

X(401) lies on these lines:2,3 50,338 97,276 248,290 264,577 287,511 323,525

X(401) = anticomplement of X(297)X(401) = X(I)-Ceva conjugate of X(J) for these (I,J): (287,2), (511,385)X(401) = crosspoint of X(276) and X(290)X(401) = X(2)-Hirst inverse of X(3)

X(402) = GOSSARD PERSPECTOR

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), wheref(a,b,c) = p(a,b,c)y(a,b,c)/a, polynomials p and y as given below

Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), whereg(a,b,c) = p(a,b,c)y(a,b,c), polynomials p and y as given below

In A History of Mathematics, Florian Cajori wrote, "H. C. Gossard of the University of Oklahoma showed in 1916 that the three Euler lines of the triangles formed by the Euler line and the sides, taken by twos, of a given triangle, form a triangle . . . perspective with the given triangle and having the same Euler line." Let ABC be the given triangle and A'B'C' the Gossard triangle - that is, the triangle perspective with the given triangle and having the same Euler line. The lines AA', BB', CC' concur in X(402), named the Gosssard perspector by John Conway (1998). Barycentrics for X(402) were received from Paul Yiu (2/20/99); the polynomials p and y referred to above are given as follows:

p(a,b,c) = 2a4 - a2b2 - a2c2 - (b2 - c2)2

y(a,b,c) = a8 - a6(b2 + c2) + a4(2b2 - c2)(2c2 - b2) + [(b2 - c2)2][3a2(b2 + c2) - b4 - c4 - 3b2c2]

X(402) lies on this line: 2,3

X(403) = X(36) OF THE ORTHIC TRIANGLE

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (sec A)(1 + cos 2B + cos 2C)

Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (tan A)(1 + cos 2B + cos 2C)

X(403) lies on these lines: 2,3 112,230 115,232 847,1093

X(403) = midpoint between X(4) and X(186)X(403) = reflection of X(186) about X(468)X(403) = inverse of X(24) in the circumcircleX(403) = inverse of X(4) in the nine-point circleX(403) = inverse of X(378) in the orthocentroidal circleX(403) = X(113)-cross conjugate of X(4)

X(404) = HARMONIC CONJUGATE OF X(21) WRT X(2) AND X(3)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc(a + b + c) - a(b2 + c2 - a2) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = abc(a + b + c) - a2(b2 + c2 - a2)

X(404) lies on these lines:1,88 2,3 8,56 10,36 31,978 46,997 57,78 60,662 63,936 69,1014 81,386 104,355 108,318 145,999 149,496 603,651 612,988 976,982

X(405)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc(a + b + c) + abc cos A Barycentrics b + c + (1 + a)cos A : c + a + (1 + b)cos B : a + b + (1 + c)cos C

X(405) lies on these lines: 1,6 2,3 8,943 10,55 56,226 58,940 63,942 284,965 329,999 756,976 846,986

X(405) = inverse of X(442) in the orthocentroidal circleX(405) = complement of X(377)

X(406)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc(a + b + c) + abc sec A Barycentrics b + c + a(1 + sec A) : c + a + b(1 + sec B) : a + b + c(1 + sec C)

X(406) lies on these lines:2,3 8,1061 10,33 37,158 92,1068 108,388 208,226 261,317

X(406) inverse of X(475) in the orthocentroidal circle

X(407)

Trilinears (v + w)sec A : (w + u)sec B : (u + v)sec C, where u : v : w = X(21); e.g., u = u(A,B,C) = 1/(cos B + cos C)

Barycentrics (v + w)tan A : (w + u)tan B : (u + v)tan C

X(407) lies on these lines: 2,3 12,228 65,225 117,136

X(407) = crosspoint of X(4) and X(225)

X(408)

Trilinears (v + w)cos A : (w + u)cos B : (u + v)cos C, where u : v : w = X(29); e.g., u = u(A,B,C) = (sec A)/(cos B + cos C)

Barycentrics (v + w)sin 2A : (w + u)sin 2B : (u + v)sin 2C

X(408) lies on these lines: 2,3 73,228

X(409)

Trilinears u2 + vw : v2 + wu : w2 + uv, where u : v : w = X(21); e.g., u = u(A,B,C) = 1/(cos B + cos C)

Barycentrics a(u2 + vw) : b(v2 + wu) : c(w2 + uv)

X(409) lies on these lines: 2,3 65,1098

X(410)

Trilinears u2 + vw : v2 + wu : w2 + uv, where u : v : w = X(29); e.g., u = u(A,B,C) = (sec A)/(cos B + cos C)

Barycentrics a(u2 + vw) : b(v2 + wu) : c(w2 + uv)

X(410) lies on this line: 2,3

X(411)

Trilinears (1/v)2 + (1/w)2 - (1/u)2 : (1/w)2 + (1/u)2 - (1/v)2 : (1/u)2 + (1/v)2 - (1/w)2, where u : v : w = X(21); e.g., u = u(A,B,C) = 1/(cos B + cos C)

Barycentrics a[(1/v)2 + (1/w)2 - (1/u)2] : b[(1/w)2 + (1/u)2 - (1/v)2] : c[(1/u)2 + (1/v)2 - (1/w)2]

X(411) lies on these lines: 2,3 35,516 40,78 55,962 81,581 165,936 185,970 243,821 255,651

X(412)

Trilinears (1/v)2 + (1/w)2 - (1/u)2 : (1/w)2 + (1/u)2 - (1/v)2 : (1/u)2 + (1/v)2 - (1/w)2, where u : v : w = X(29); e.g., u = u(A,B,C) = (sec A)/(cos B + cos C)

Barycentrics a[(1/v)2 + (1/w)2 - (1/u)2] : b[(1/w)2 + (1/u)2 - (1/v)2] : c[(1/u)2 + (1/v)2 - (1/w)2]

X(412) lies on these lines: 2,3 40,92 46,158 63,318 65,243 162,580 225,775 278,962

X(413)

Trilinears u3(v2 + w2 - vw) : v3(w2 + u2 - wu) : w3(u2 + v2 - uv), where u : v : w = X(21); e.g., u = u(A,B,C) = 1/(cos B + cos C)

Barycentrics au3(v2 + w2 - vw) : bv3(w2 + u2 - wu) : cw3(u2 + v2 - uv)

X(413) lies on this line: 2,3

X(414)

Trilinears u3(v2 + w2 - vw) : v3(w2 + u2 - wu) : w3(u2 + v2 - uv), where u : v : w = X(29); e.g., u = u(A,B,C) = (sec A)1/(cos B + cos C)

Barycentrics au3(v2 + w2 - vw) : bv3(w2 + u2 - wu) : cw3(u2 + v2 - uv)

X(414) lies on this line: 2,3

X(415)

Trilinears (u2 - vw)sec A : (v2 - wu)sec B : (v2 - uv)sec C, where u : v : w = X(21); e.g., u = u(A,B,C) = 1/(cos B + cos C)

Barycentrics (u2 - vw)tan A : (v2 - wu)tan B : (w2 - uv)tan C

X(415) lies on these lines: 2,3 162,238

X(415) = X(4)-Hirst inverse of X(29)

X(416)

Trilinears (u2 - vw)sec A : (v2 - wu)sec B : (v2 - uv)sec C, where u : v : w = X(29); e.g., u = u(A,B,C) = (sec A)/(cos B + cos C)

Barycentrics (u2 - vw)tan A : (v2 - wu)tan B : (w2 - uv)tan C

X(416) lies on this line: 2,3

X(416) = X(3)-Hirst inverse of X(21)

X(417)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (cos A)(sec2B + sec2C)Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin 2A)(sec2B + sec2C)

X(417) lies on this line: 2,3

X(417) = X(3)-Ceva conjugate of X(185)

X(418)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (cos A)(csc 2B + csc 2C)Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin 2A)(csc 2B + csc 2C)

X(418) lies on these lines: 2,3 51,216 97,110 154,160 157,161 184,577

X(418) = X(I)-Ceva conjugate of X(J) for these (I,J): (3,216), (216,217)

X(419)

Trilinears (u2 - vw)sec A : (v2 - wu)sec B : (v2 - uv)sec C, where u : v : w = X(31); e.g., u = u(A,B,C) = a2

Barycentrics (u2 - vw)tan A : (v2 - wu)tan B : (w2 - uv)tan C

X(419) lies on these lines: 2,3 238,242

X(419) = X(4)-Hirst inverse of X(25)

X(420)

Trilinears (u2 - vw)sec A : (v2 - wu)sec B : (v2 - uv)sec C, where u : v : w = X(38); e.g., u = u(a,b,c) = b2 + c2

Barycentrics (u2 - vw)tan A : (v2 - wu)tan B : (w2 - uv)tan C

X(420) lies on this line: 2,3

X(420) = X(4)-Hirst inverse of X(427)

X(421)

Trilinears (u2 - vw)sec A : (v2 - wu)sec B : (v2 - uv)sec C, where u : v : w = X(47); e.g., u = u(A,B,C) = cos 2A

Barycentrics (u2 - vw)tan A : (v2 - wu)tan B : (w2 - uv)tan C

X(421) lies on this line: 2,3

X(421) = X(4)-Hirst inverse of X(24)

X(422)

Trilinears (u2 - vw)sec A : (v2 - wu)sec B : (v2 - uv)sec C, where u : v : w = X(58); e.g., u = u(a,b,c) = a/(b + c)

Barycentrics (u2 - vw)tan A : (v2 - wu)tan B : (w2 - uv)tan C

X(422) lies on these lines: 2,3 162,242

X(422) = X(4)-Hirst inverse of X(28)

X(423)

Trilinears (u2 - vw)sec A : (v2 - wu)sec B : (v2 - uv)sec C, where u : v : w = X(81); e.g., u = u(a,b,c) = 1/(b + c)

Barycentrics (u2 - vw)tan A : (v2 - wu)tan B : (w2 - uv)tan C

X(423) lies on this line: 2,3

X(423) = X(4)-Hirst inverse of X(27)

X(424)

Trilinears (u2 - vw)sec A : (v2 - wu)sec B : (v2 - uv)sec C, where u : v : w = X(191); e.g., u = u(a,b,c) = (b + c - a)(bc + ca + ab) + b3 + c3 - a3

Barycentrics (u2 - vw)tan A : (v2 - wu)tan B : (w2 - uv)tan C

X(424) lies on this line: 2,3

X(424) = X(4)-Hirst inverse of X(451)

X(425)

Trilinears (u2 - vw)sec A : (v2 - wu)sec B : (v2 - uv)sec C, where u : v : w = X(283); e.g., u = u(A,B,C) = (cos A)/(cos B + cos C)

Barycentrics (u2 - vw)tan A : (v2 - wu)tan B : (w2 - uv)tan C

X(425) lies on this line: 2,3 243,662

X(425) = X(4)-Hirst inverse of X(21)

X(426)

Trilinears (v2 + w2)cos A : (w2 + u2)cos B : (u2 + v2)cos C, where u : v : w = X(19); e.g., u = u(A,B,C) = tan A

Barycentrics (v2 + w2)sin 2A : (w2 + u2)sin 2B : (u2 + v2)sin 2C

X(426) lies on this line: 2,3

X(427)

Trilinears (v + w)sec A : (w + u)sec B : (u + v)sec C, where u : v : w = X(31); e.g., u = u(a,b,c) = a2

Barycentrics (v + w)tan A : (w + u)tan B : (u + v)tan C

X(427) lies on these lines:2,3 6,66 11,33 12,34 51,125 53,232 98,275 112,251 114,136 183,317 230,571 264,305 343,511

X(427) = midpoint between X(4) and X(378)X(427) = inverse of X(468) in the nine-point circleX(427) = inverse of X(25) in the orthocentroidal circle

X(427) = complement of X(22)X(427) = X(112)-Ceva conjugate of X(523)X(427) = X(39)-cross conjugate of X(141)X(427) = crosspoint of X(4) and X(264)X(427) = X(4)-Hirst inverse of X(429)

X(428)

Trilinears (v + w)sec A : (w + u)sec B : (u + v)sec C, where u : v : w = X(38); e.g., u = u(a,b,c) = b2 + c2

Barycentrics (v + w)tan A : (w + u)tan B : (u + v)tan C

X(428) lies on these lines: 2,3 132,137

X(429)

Trilinears (v + w)sec A : (w + u)sec B : (u + v)sec C, where u : v : w = X(58); e.g., u = u(a,b,c) = a/(b + c)

Barycentrics (v + w)tan A : (w + u)tan B : (u + v)tan C

X(429) lies on these lines: 2,3 11,1104 12,37 108,961 119,136 495,1068

X(429) = X(108)-Ceva conjugate of X(523)

X(430)

Trilinears (v + w)sec A : (w + u)sec B : (u + v)sec C, where u : v : w = X(81); e.g., u = u(a,b,c) = 1/(b + c)

Barycentrics (v + w)tan A : (w + u)tan B : (u + v)tan C

X(430) lies on these lines: 2,3 118,136 210,594

X(431)

Trilinears (v + w)sec A : (w + u)sec B : (u + v)sec C, where u : v : w = X(283); e.g., u = u(A,B,C) = (cos A)/(cos B + cos C)

Barycentrics (v + w)tan A : (w + u)tan B : (u + v)tan C

X(431) lies on these lines: 2,3 119,135

X(432)

Trilinears (v2 + w2)sec A : (w2 + u2)sec B : (u2 + v2)sec C, where u : v : w = X(155); e.g., u = u(A,B,C) = (cos A)(cos2B + cos2C - cos2A)

Barycentrics (v2 + w2)tan A : (w2 + u2)tan B : (u2 + v2)tan C

X(432) lies on this line: 2,3

X(433)

Trilinears (v2 + w2)sec A : (w2 + u2)sec B : (u2 + v2)sec C, where u : v : w = X(159)Barycentrics (v2 + w2)tan A : (w2 + u2)tan B : (u2 + v2)tan C

X(433) lies on this line: 2,3

X(434)

Trilinears (v2 + w2)sec A : (w2 + u2)sec B : (u2 + v2)sec C, where u : v : w = X(195)Barycentrics (v2 + w2)tan A : (w2 + u2)tan B : (u2 + v2)tan C

X(434) lies on this line: 2,3

X(435)

Trilinears (v2 + w2)sec A : (w2 + u2)sec B : (u2 + v2)sec C, where u : v : w = X(399)Barycentrics (v2 + w2)tan A : (w2 + u2)tan B : (u2 + v2)tan C

X(435) lies on this line: 2,3

X(436)

Trilinears (u2 + vw)sec A : (v2 + wu)sec B : (w2 + uv)sec C, where u : v : w = X(48); e.g., u(A,B,C) = sin 2A

Barycentrics (u2 + vw)tan A : (v2 + wu)tan B : (w2 + uv)tan C

X(436) lies on these lines: 2,3 51,107 110,324 578,1093

X(437)

Trilinears (u2 + vw)sec A : (v2 + wu)sec B : (w2 + uv)sec C, where u : v : w = X(214)Barycentrics (u2 + vw)tan A : (v2 + wu)tan B : (w2 + uv)tan C

X(437) lies on this line: 2,3

X(438)

Trilinears (u2 + vw)csc A : (v2 + wu)csc B : (w2 + uv)csc C, where u : v : w = X(204); e.g., u(A,B,C) = (tan A)(tan B + tan C - tan A)

Barycentrics u2 + vw : v2 + wu : w2 + uv

X(438) lies on this line: 2,3

X(439)

Trilinears au2 : bv2 : cw2, where u : v : w = X(193); e.g., u(A,B,C) = (csc A)(cot B + cot C - cot A)

Barycentrics (au)2 : (bv)2 : (cw)2

X(439) lies on this line: 2,3

X(439) = X(459)-Ceva conjugate of X(193)

X(440)

Trilinears bc(v + w) : ca(w + u) : ab(u + v), where u : v : w = X(28); e.g., u(a,b,c) = (tan A)/(b + c)

Barycentrics v + w : w + u : u + v

X(440) lies on these lines: 2,3 37,226 72,306 118,122 950,1104

X(440) = complement of X(27)X(440) = X(190)-Ceva conjugate of X(525)X(440) = crosspoint of X(2) and X(306)

X(441)

Trilinears bc(v + w) : ca(w + u) : ab(u + v), where u : v : w = X(240); e.g., u(A,B,C) = sec A cos(A + ω)

Barycentrics v + w : w + u : u + v

X(441) lies on these lines: 2,3 141,577 525,647

X(441) = complement of X(297)

X(442) COMPLEMENT OF SCHIFFLER POINT

Trilinears bc(v + w) : ca(w + u) : ab(u + v), where u : v : w = X(284); e.g., u(A,B,C) = (sin A)/(cos B + cos C)

Barycentrics v + w : w + u : u + v

X(442) lies on these lines: 2,3 8,495 9,46 10,12 11,214 100,943 115,120 119,125 274,325 388,956 392,946

X(442) = midpoint between X(79) and X(191)X(442) = inverse of X(405) in the orthocentroidal circleX(442) = complement of X(21)X(442) = X(100)-Ceva conjugate of X(523)X(442) = crosspoint of X(264) and X(321)

X(443)

Trilinears bc(v + w) : ca(w + u) : ab(u + v), where u : v : w = X(380)

Barycentrics v + w : w + u : u + v

X(443) lies on these lines: 1,142 2,3 7,72 8,942 10,57 69,274 226,936 278,1038 387,940 392,962 579,966

X(443) = complement of X(452)

X(444)

Trilinears (v + w)tan A : (w + u)tan B : (u + v)tan C, where u : v : w = X(256); e.g., u(a,b,c) = 1/(a2 + bc)

Barycentrics (v + w)(sin A tan A) : (w + u)(sin B tan B) : (u + v)(sin C tan C)

X(444) lies on these lines: 2,3 19,232

X(445)

Trilinears (v + w)csc 2A : (w + u)csc 2B : (u + v)csc 2C, where u : v : w = X(79); e.g., u(a,b,c) = 1/(1 + 2 cos A)

Barycentrics (v + w)sec A : (w + u)sec B : (u + v)sec C

X(445) lies on this line: 2,3

X(446)

Trilinears u(v2 + w2) : v(w2 + u2) : w(u2 + v2), where u : v : w = X(98); e.g., u(A,B,C) = sec(A + ω)

Barycentrics au(v2 + w2) : bv(w2 + u2) : cw(u2 + v2)

X(446) lies on this line: 2,3

X(446) = crosspoint of X(98) and X(511)

X(447)

Trilinears bc(u2 - vw) : ca(v2 - wu) : ab(w2 - uv), where u : v : w = X(28); e.g., u(a,b,c) = (tan A)/(b + c)

Barycentrics u2 - vw : v2 - wu : w2 - uv

X(447) lies on this line: 2,3 340,540 350,811 519,648

X(447) = X(2)-Hirst inverse of X(27)

X(448)

Trilinears bc(u2 - vw) : ca(v2 - wu) : ab(w2 - uv), where u : v : w = X(284); e.g., u(A,B,C) = (sin A)/(cos B + cos C)

Barycentrics u2 - vw : v2 - wu : w2 - uv

X(448) lies on this line: 2,3

X(448) = X(2)-Hirst inverse of X(21)

X(449)

Trilinears bc(u2 - vw) : ca(v2 - wu) : ab(w2 - uv), where u : v : w = X(380)Barycentrics u2 - vw : v2 - wu : w2 - uv

X(449) lies on this line: 2,3

X(449) = X(2)-Hirst inverse of X(452)

X(450) X(3)-HIRST INVERSE OF X(4)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (sec A)[cos4A - (cos B cos C)2] = g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (cos A)[sec4A - (sec B sec C)2]

Barycentrics h(A,B,C) : h(B,C,A) : h(C,A,B), where h(A,B,C) = (tan A)(cos4A - (cos B cos C)2]

X(450) lies on these lines: 2,3 107,511 155,1075 1092,1093

X(450) = X(3)-Hirst inverse of X(4)

X(451)

Trilinears u sec A : v sec B : w sec C, where u : v : w = X(191)Barycentrics u tan A : v tan B : w tan C

X(451) lies on these lines: 2,3 12,108 281,1068

X(451) = X(4)-Hirst inverse of X(424)

X(452)

Trilinears u csc A : v csc B : w csc C, where u : v : w = X(380)Barycentrics u : v : w

X(452) lies on these lines: 1,329 2,3 8,9 34,347 63,938 72,145 388,1001 392,944 497,958 956,1058

X(452) = anticomplement of X(443)X(452) = X(2)-Hirst inverse of X(449)

X(453)

Trilinears u2/(cos B + cos C) : v2/(cos C + cos A) : w2/(cos A + cos B), where u : v : w = X(46)Barycentrics (u2sin A)/(cos B + cos C) : (v2sin B)/(cos C + cos A) : (w2sin C)/(cos A + cos B)

X(453) lies on this line: 2,3

X(454)

Trilinears u2sec A : v2sec B : w2sec C, where u : v : w = X(155); e.g., u(A,B,C) = (cos A)[cos2B + cos2C - cos2A]

Barycentrics u2tan A : v2tan B : w2tan C

X(454) lies on this line: 2,3

X(455)

Trilinears u2sec A : v2sec B : w2sec C, where u : v : w = X(159)Barycentrics u2tan A : v2tan B : w2tan C

X(455) lies on this line: 2,3

X(456)

Trilinears u2sec A : v2sec B : w2sec C, where u : v : w = X(195)Barycentrics u2tan A : v2tan B : w2tan C

X(456) lies on this line: 2,3

X(457)

Trilinears u2sec A : v2sec B : w2sec C, where u : v : w = X(399)Barycentrics u2tan A : v2tan B : w2tan C

X(457) lies on this line: 2,3

X(458)

Trilinears u csc 2A : v csc 2B : w csc 2C, where u : v : w = X(182); e.g.; u(A,B,C) = cos(A - ω)

Barycentrics u sec A : v sec B : w sec C

X(458) lies on these lines: 2,3 6,264 76,275 141,317 239,318 273,894 315,343 340,599

X(458) = inverse of X(297) in the orthocentroidal circle

X(459)

Trilinears u tan A : v tan B : w tan C, where u : v : w = X(193); e.g.; u(A,B,C) = (csc A)(cot B + cot C - cot A)

Barycentrics u sin A tan A : v sin B tan B : w sin C tan C

X(459) lies on these lines: 2,3 230,393 232,800 242,278 317,1007

X(459) = cevapoint of X(193) and X(439)X(459) = X(393)-cross conjugate of X(4)X(459) = X(4)-Hirst inverse of X(460)

X(460)

Trilinears u tan A : v tan B : w tan C, where u : v : w = X(230)Barycentrics u sin A tan A : v sin B tan B : w sin C tan C

X(460) lies on this line: 2,3

X(460) = X(4)-Hirst inverse of X(459)

X(461)

Trilinears u tan A : v tan B : w tan C, where u : v : w = X(391); e.g., u(a,b,c) = bc(3a + b + c)(b + c - a)

Barycentrics u sin A tan A : v sin B tan B : w sin C tan C

X(461) lies on these lines: 2,3 33,200

X(462)

Trilinears u tan A : v tan B : w tan C, where u : v : w = X(395); e.g., u(A,B,C) = cos(B - C) + 2 cos(A + π/3)

Barycentrics u sin A tan A : v sin B tan B : w sin C tan C

X(462) lies on these lines: 2,3 51,397 184,398

X(463)

Trilinears u tan A : v tan B : w tan C, where u : v : w = X(396); e.g., u(A,B,C) = cos(B - C) + 2 cos(A - π/3)

Barycentrics u sin A tan A : v sin B tan B : w sin C tan C

X(463) lies on these lines: 2,3 51,398 184,397

X(464)

Trilinears u cot A : v cot B : w cot C, where u : v : w = X(387)Barycentrics u cos A : v cos B : w cos C

X(464) lies on these lines: 2,3 63,69

X(465)

Trilinears u cot A : v cot B : w cot C, where u : v : w = X(397); e.g., u(A,B,C) = cos(B - C) - 2 cos(A + π/3)

Barycentrics u cos A : v cos B : w cos C

X(465) lies on these lines: 2,3 216,395 396,577

X(465) = complement of X(473)

X(466)

Trilinears u cot A : v cot B : w cot C, where u : v : w = X(398); e.g., u(A,B,C) = cos(B - C) - 2 cos(A - π/3)

Barycentrics u cos A : v cos B : w cos C

X(466) lies on these lines: 2,3 216,396 395,577

X(446) = complement of X(472)

X(467)

Trilinears u csc 2A : v csc 2B : w csc 2C, where u : v : w = X(52); e.g., u(A,B,C) = cos 2A cos(B - C)

Barycentrics u sec A : v sec B : w sec C

X(467) lies on these lines: 2,3 53,311

X(467) = X(317)-Ceva conjugate of X(52)

X(468) X(2)-LINE CONJUGATE OF X(3)

Trilinears u csc 2A : v csc 2B : w csc 2C, where u : v : w = X(187); e.g., u(a,b,c) = a(2a2 - b2 - c2)

Barycentrics u sec A : v sec B : w sec C

X(468) lies on these lines: 2,3 98,685 107,842 111,935 230,231 250,325

X(468) = midpoint between X(186) and X(403)X(468) = isogonal conjugate of X(895)X(468) = inverse of X(25) in the circumcircleX(468) = inverse of X(427) in the nine-point circleX(468) = X(187)-cross conjugate of X(524)X(468) = X(2)-line conjugate of X(3)

Let X = X(468) and let V be the vector-sum XA + XB + XC; then V = X(468)X(23).

X(469)

Trilinears u csc 2A : v csc 2B : w csc 2C, where u : v : w = X(386); e.g., u(a,b,c) = a(b2 + c2 + bc + ca + ab)

Barycentrics u sec A : v sec B : w sec C

X(469) lies on these lines: 2,3 92,264 226,273

X(469) = inverse of X(27) in the orthocentroidal circle

X(470)

Trilinears sin(A + π/3) csc 2A : sin(B + π/3) csc 2B : sin(C + π/3) csc 2CBarycentrics sin(A + π/3) sec A : sin(B + π/3) sec B : sin(C + π/3) sec 2C

X(470) lies on these lines: 2,3 18,275 264,301 298,340 302,317 343,634 394,633

X(470) = inverse of X(471) in the orthocentroidal circleX(470) = X(15)-cross conjugate of X(298)X(470) = X(4)-Hirst inverse of X(471)

X(471)

Trilinears sin(A - π/3) csc 2A : sin(B - π/3) csc 2B : sin(C - π/3) csc 2CBarycentrics sin(A - π/3) sec A : sin(B - π/3) sec B : sin(C - π/3) sec 2C

X(471) lies on these lines: 2,3 17,275 264,300 299,340 303,317 343,633 394,634

X(471) = inverse of X(470) in the orthocentroidal circleX(471) = X(16)-cross conjugate of X(299) for these (I,J): (71,10), (72,307), (440,2)X(471) = X(4)-Hirst inverse of X(470)

X(472)

Trilinears cos(A + π/3) csc 2A : cos(B + π/3) csc 2B : cos(C + π/3) csc 2CBarycentrics cos(A + π/3) sec A : cos(B + π/3) sec B : cos(C + π/3) sec 2C

X(472) lies on these lines: 2,3 13,275 53,395 264,298 299,317 343,621 394,622

X(472) = inverse of X(473) in the orthocentroidal circleX(472) = anticomplement of X(466)X(472) = X(62)-cross conjugate of X(303)

X(473)

Trilinears cos(A - π/3) csc 2A : cos(B - π/3) csc 2B : cos(C - π/3) csc 2CBarycentrics cos(A - π/3) sec A : cos(B - π/3) sec B : cos(C - π/3) sec 2C

X(473) lies on these lines: 2,3 14,275 53,396 264,299 298,317 343,622 394,621

X(473) = inverse of X(472) in the orthocentroidal circleX(473) = anticomplement of X(465)X(473) = X(61)-cross conjugate of X(302)

X(474)

Trilinears cos A - (a + b + c)/a : cos B - (a + b + c)/b : cos C - (a + b + c)/cBarycentrics a cos A - (a + b + c) : b cos B - (a + b + c) : c cos C - (a + b + c)

X(474) lies on these lines: 2,3 8,999 10,56 35,1001 36,958 40,392 46,960 57,72 65,997 78,942 142,954 171,978 183,274 244,976 283,582 386,940 579,965 986,1054

X(475)

Trilinears sec A - (a + b + c)/a : sec B - (a + b + c)/b : sec C - (a + b + c)/cBarycentrics a sec A - (a + b + c) : b sec B - (a + b + c) : c sec C - (a + b + c)

X(475) lies on these lines: 2,3 8,1063 10,34 264,274 318,1068

X(475) = inverse of X(406) in the orthocentroidal circle

X(476) = TIXIER POINT

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = 1/[(1 + 2 cos 2A)(sin(B - C)]Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

The reflection of X(110) about the Euler line; X(476) lies on the circumcircle. (Michel Tixier, 5/9/98). Also, X(476) is the center of the polar conic of X(30) with respect to the Neuberg cubic; this conic is a rectangular hyperbola passing through the incenter, the excenters, and X(30). (Peter Yff, 5/23/99)

X(476) lies on these lines: 2,842 3,477 23,94 30,74 99,850 110,523 111,230 250,933 376,841

X(476) = reflection of X(477) about X(3)X(476) = isogonal conjugate of X(526)X(476) = cevapoint of X(30) and X(523)

X(477) = TIXIER ANTIPODE

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = 1/[4 cos A + cos(A - 2B) + cos(A - 2C) - 3 cos(B - C)]Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

The reflection of X(476) about X(3), on the circumcircle. (Michel Tixier, 5/16/98)

X(477) lies on these lines: 3,476 30,110 50,112 74,523 107,186 376,691 378,935

X(477) = reflection of X(476) about X(3)

X(478) = CENTER OF YIU CONIC

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a[a4 - 2abc(b + c - a) - (b2 - c2)2]/(b + c - a)Barycentrics h(a,b,c) : h(b,c,a) : h(c,a,b), where h(a,b,c) = af(a,b,c)

Center of the Yiu conic, which passes through the points outside the circumcircle at which the excircles of ABC are tangent to the sidelines of ABC. (PaulYiu, "The Clawson point and excircles," 1999)

X(478) lies on these lines: 6,19 9,1038 69,651 109,573 198,577 222,226

X(479)

Trilinears (tan A/2 sec A/2)2 : (tan B/2 sec B/2)2 : (tan C/2 sec C/2)2

Barycentrics tan3(A/2) : tan3(B/2) : tan3(C/2)= 1/(b + c - a)3 : 1/(c + a - b)3 : 1/(a + b - c)3

Let A' be the point in which the incircle is tangent to a circle that passes through vertices B and C, and determine B' and C' cyclically. The lines AA', BB', CC' concur in X(479)

Clark Kimberling and Peter Yff, Problem 10678, American Mathematical Monthly 105 (1998) 666.

X(479) lies on these lines: 7,354 57,279 269,614

X(479) = isogonal conjugate of X(480)X(479) = X(269)-cross conjugate of X(279)

X(480)

Trilinears (cot A/2 cos A/2)2 : (cot B/2 cos B/2)2 : (cot C/2 cos C/2)2

Barycentrics (sin A)(cot A/2 cos A/2)2 : (sin B)(cot B/2 cos B/2)2 : (sin C)(cot C/2 cos C/2)2

The radical center of the three circles used to construct X(479). (Peter Yff, 5/6/98)

X(480) lies on these lines: 8,344 9,55 10,954 56,78 100,144

X(480) = isogonal conjugate of X(479)X(480) = X(200)-Ceva conjugate of X(220)

X(481) = 1st EPPSTEIN POINT

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = 1 - 2 sec A/2 cos B/2 cos C/2= 1 - 4(area)/[a(b + c - a)] : 1 - 4(area)/[b(c + a - b)] : 1 - 4(area)/[c(a + b - c) [E. Brisse, 3/20/01]Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B).

Let S be the inner Soddy circle and Sa, Sb, Sc the Soddy circles tangent to S. Let Ia = S^Sa, Ea = Sb^Sc, and determine Ib, Ic, Eb, Ec cyclically. Then X(481) is the point of concurrence of lines Ia-to-Ea, Ib-to-Eb, Ic-to-Ec.

David Eppstein, "Tangent spheres and triangle centers," American Mathematical Monthly, 108 (2001) 63-66.

X(481) lies on these lines: 1,7 226,485

X(481) = X(79)-Ceva conjugate of X(482)

X(482) = 2nd EPPSTEIN POINT

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = 1 + 2 sec A/2 cos B/2 cos C/2= 1 + 4(area)/[a(b + c - a)] : 1 + 4(area)/[b(c + a - b)] : 1 + 4(area)/[c(a + b - c) [E. Brisse, 3/20/01]Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B).

Let S' be the outer Soddy circle and Sa, Sb, Sc the Soddy circles tangent to S. Let Ja = S'^Sa, Ea = Sb^Sc, and determine Jb, Jc, Eb, Ec cyclically. Then X(482) is the point of concurrence of lines Ja-to-Ea, Jb-to-Eb, Jc-to-Ec.

David Eppstein, "Tangent spheres and triangle centers," American Mathematical Monthly, 108 (2001) 63-66.

X(482) lies on these lines: 1,7 226,486

X(482) = X(79)-Ceva conjugate of X(481)

X(483) = RADICAL CENTER OF AJIMA-MALFATTI CIRCLES

Trilinears sec2A/4 : sec2B/4 : sec2C/4= 1/(1 + cos A/2) : 1/(1 + cos B/2) : 1/(1 + cos C/2)

Barycentrics sin A sec2A/4 : sin B sec2B/4 : sin C sec2C/4

The Ajima-Malfatti circles are described at X(179). (Peter Yff, 6/1/98)

X(483) lies on these lines: 8,178 173,180 174,175

X(484) = EVANS PERSPECTOR

Trilinears 1 + 2(cos A - cos B - cos C) : 1 + 2(cos B - cos C - cos A) : 1 + 2(cos C + cos A - cos B)Barycentrics a[1 + 2(cos A - cos B - cos C)] : b[1 + 2(cos B - cos C - cos A)] : c[1 + 2(cos C + cos A - cos B)]

X(484) is the perspector of the extriangle and the triangle A'B'C', where A' is the reflection of vertex A about sideline BC and B', C' are determined cyclically. (Lawrence Evans, 10/22/98)

X(484) lies on these lines: 1,3 10,191 12,79 30,80 63,535 100,758 499,962 759,901 1046,1048

X(484) = reflection of X(1) about X(36)X(484) = inverse of X(35) in the circumcircleX(484) = X(80)-Ceva conjugate of X(1)

Centers 485- 495, 371, and 372: Vierkanten in een driehoek - triangle centers associated with squares.

X(485) = VECTEN POINT

Trilinears sec(A - π/4) : sec(B - π/4) : sec(C - π/4)Barycentrics sin A sec(A - π/4) : sin B sec(B - π/4) : sin C sec(C - π/4)

X(485) is the perspector of triangles associated with squares that circumscribe ABC. For details, visit Floor van Lamoen's site, Vierkanten in een driehoek: 1. Omgeschreven vierkanten (van Lamoen, 4/26/98) and his article "Friendship Among Triangle Centers," Forum Geometricorum, 1 (2001) 1-6.

X(485) lies on these lines: 2,372 3,590 4,371 5,6 69,639 76,491 226,481 489,671

X(485) = isogonal conjugate of X(371)X(485) = isotomic conjugate of X(492)X(485) = complement of X(488)X(485) = anticomplement of X(641)X(485) = X(3)-cross conjugate of X(486)

X(486) = INNER VECTEN POINT

Trilinears sec(A + π/4) : sec(B + π/4) : sec(C + π/4)Barycentrics sin A sec(A + π/4) : sin B sec(B + π/4) : sin C sec(C + π/4)

X(486) is a perspector of triangles associated with squares that circumscribe ABC. For details and references, see X(485). (Floor van Lamoen, 4/26/98)

X(486) lies on these lines: 2,371 3,615 4,372 5,6 76,492 141,591 226,482 490,671

X(486) = isogonal conjugate of X(372)X(486) = isotomic conjugate of X(491)X(486) = complement of X(487)X(485) = anticomplement of X(642)X(486) = X(3)-cross conjugate of X(485)

X(487)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (cos A)(1 - csc A sin B sin C)Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(487) is a perspector of triangles associated with squares that circumscribe ABC. (Floor van Lamoen, 4/29/98)

X(487) lies on these lines: 2,371 3,69 4,489 20,638 193,372 376,490 492,631

X(487) = anticomplement of X(486)X(487) = X(I)-Ceva conjugate of X(J) for these (I,J): (4,488), (489,20), (491,2)

X(488)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (cos A)(1 + csc A sin B sin C)Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(488) is a perspector of triangles associated with squares that circumscribe ABC. For details, see Vierkanten in een driehoek: 2. Meer punten uit omgeschreven vierkanten (Floor van Lamoen, 4/29/98)

X(488) lies on these lines: 2,372 3,69 4,490 193,371 376,489 491,631

X(488) = anticomplement of X(485)X(488) = X(I)-Ceva conjugate of X(J), for these (I,J): (4,487), (490,20), (492,2)

X(489)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (cos A)(1 - csc A sin B sin C) - cos B cos CBarycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(489) is a perspector of triangles associated with squares that circumscribe ABC. For details and reference, see X(488). (Floor van Lamoen, 4/29/98)

X(489) lies on these lines: 3,492 4,487 20,64 30,638 176,664 376,488 485,671

X(489) = cevapoint of X(20) and X(487)

X(490)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (cos A)(1 + csc A sin B sin C) - cos B cos CBarycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(490) is a perspector of triangles associated with squares that circumscribe ABC. For details and reference, see X(488). (Floor van Lamoen, 4/29/98)

X(490) lies on these lines: 3,491 4,488 20,64 30,637 175,664 376,487 486,671

X(490) = cevapoint of X(20) and X(488)

X(491)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (cos A)(1 - csc A sin B sin C) + cos B cos CBarycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(491) is a pole associated with squares that circumscribe ABC. For details and reference, see X(488). (Floor van Lamoen, 4/26/98)

X(491) lies on these lines: 2,6 3,490 4,487 5,637 76,485 315,371 372,642 488,631

X(491) = isotomic conjugate of X(486)X(491) = anticomplement of X(615)X(491) = X(264)-Ceva conjugate of X(492)X(491) = cevapoint of X(2) and X(487)

X(492)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (cos A)(1 + csc A sin B sin C) + cos B cos CBarycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(492) is a pole associated with squares that circumscribe ABC. For details and reference, see X(488). (Floor van Lamoen, 4/27/98)

X(492) lies on these lines: 2,6 3,489 4,488 5,638 76,486 315,372 371,641 487,631

X(492) = isotomic conjugate of X(485)X(492) = anticomplement of X(590)X(492) = X(264)-Ceva conjugate of X(491)X(492) = cevapoint of X(2) and X(488)

X(493)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = 1/(sin A + sin B sin C)Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(493) is a homothetic center of triangles associated with squares that circumscribe ABC. For details, see Vierkanten in een driehoek: 4. Ingeschreven vierkanten (Floor van Lamoen, 4/27/98)

X(493) lies on these lines: 25,371 39,494

X(493) = X(394)-cross conjugate of X(494)

X(494)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = 1/(sin A - sin B sin C)Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(494) is a homothetic center of triangles associated with squares that circumscribe ABC. For details and reference, see X(393). (Floor van Lamoen, 4/27/98)

X(494) lies on these lines: 25,372 39,493

X(494) = X(394)-cross conjugate of X(493)

X(495) = JOHNSON MIDPOINT

Trilinears 2 + cos(B - C) : 2 + cos(C - A) : 2 + cos(A - B)Barycentrics (sin A)[2 + cos(B - C)] : (sin B)[2 + cos(C - A)] : (sin C)[2 + cos(A - B)]

X(495) is the midpoint of segments C1-to-P1, C2-to-P2, C3-to-P3 in the Johnson four-circle configuration.

Roger A. Johnson, Advanced Euclidean Geometry, Dover, New York, 1960, page 75.

Peter Yff, "Three concurrent congruent circles 'inscribed' in a triangle," manuscript, 1998; X(495) is the point R on page 5.

X(495) lies on these lines:1,5 2,956 3,388 4,390 8,442 10,141 30,55 35,550 36,549 56,140 202,395 203,396 226,517 381,497 392,908 429,1068 529,993 612,1060

X(495) = complement of X(956)

X(496) = HARMONIC CONJUGATE OF X(495) WRT X(1) AND X(5)

Trilinears 2 - cos(B - C) : 2 - cos(C - A) : 2 - cos(A - B)Barycentrics (sin A)[2 - cos(B - C)] : (sin B)[2 - cos(C - A)] : (sin C)[2 - cos(A - B)]

Peter Yff, "Three concurrent congruent circles 'inscribed' in a triangle," manuscript, 1998; X(496) is the point R' on page 5.

X(496) lies on these lines: 1,5 2,1058 3,497 4,999 30,56 35,549 36,550 55,140 149,404 202,397 203,398 381,388 390,631 613,1069 614,1062 942,946

X(497) CROSSPOINT OF GERGONNE POINT AND NAGEL POINT

Trilinears 1 - cos B cos C : 1 - cos C cos A : 1 - cos A cos BBarycentrics (sin A)(1 - cos B cos C) : (sin B)(1 - cos C cos A) : (sin C)(1 - cos A cos B)

X(497) is the harmonic conjugate of X(388) with respect to X(1) and X(4)

Peter Yff, "Three concurrent congruent circles 'inscribed' in a triangle," manuscript, 1998; X(497) is the point C' on page 5.

X(497) lies on these lines:1,4 2,11 3,496 7,354 8,210 20,56 29,1036 30,999 35,499 36,376 57,516 65,938 69,350 80,1000 212,238 329,518 381,495 452,958 614,1040 1057,1065

X(497) = isogonal conjugate of X(1037)X(497) = crosspoint of X(I) and X(J) for these (I,J): (7,8), (29,314)

X(498)

Trilinears 1 + 2 sin B sin C : 1 + 2 sin C sin A : 1 + 2 sin A sin BBarycentrics (sin A)(1 + 2 sin B sin C) : (sin B)(1 + 2 sin C sin A) : (sin C)(1 + 2 sin A sin B)

X(498) and X(499) are harmonic conjugate points with respect to X(1) and X(2), in analogy with such pairs with respect to X(1), X(4) and with respect to X(1), X(5).

Peter Yff, "Three concurrent congruent circles 'inscribed' in a triangle," manuscript, 1998; X(498) is the point S on page 6.

X(498) lies on these lines: 1,2 3,12 4,35 5,55 9,920 36,388 37,91 46,226 47,171 56,140 141,611 191,329 255,750 345,1089

X(499)

Trilinears 1 - 2 sin B sin C : 1 - 2 sin C sin A : 1 - 2 sin A sin BBarycentrics (sin A)(1 - 2 sin B sin C) : (sin B)(1 - 2 sin C sin A) : (sin C)(1 - 2 sin A sin B)

X(499) is the harmonic conjugate of X(498) with respect to X(1) and X(2).

Peter Yff, "Three concurrent congruent circles 'inscribed' in a triangle," manuscript, 1998; X(498) is the point S' on page 6.

X(499) lies on these lines: 1,2 3,11 4,36 5,56 12,999 17,202 18,203 35,497 46,946 47,238 55,140 57,920 80,944 141,613 255,748 348,1111 484,962

X(500) = ORTHOCENTER OF THE INCENTRAL TRIANGLE

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), wheref(a,b,c) = a(b2 +c2 - a2 + bc)[2abc + (b + c)(a2 - (b - c)2)]

Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(500) lies on these lines: 1,30 3,6 651,943

X(500) = inverse of X(582) in the Brocard circleX(500) = crosspoint of X(1) and X(35)

X(501) = MIQUEL ASSOCIATE OF INCENTER

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), wheref(a,b,c) = a/[(b + c)u], u = u(a,b,c) = a3 - b3 - c3 + ba2 - ab2 + ca2 - ac2 - cb2 - bc2 - abc

Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)

Miquel's theorem states that if A', B', C' are points (other than A, B, C) on sidelines BC, CA, AB, respectively, then the circles AB'C', BC'A', CA'B' meet at a point. Suppose P is a point and A' = P^BC, B' = P^CA, C' = P^AB; the point in which the three circles is the Miquel associate of P. (Paul Yiu, 7/6/99)

X(501) lies on this line: 1,229

X(501) = isogonal conjugate of X(502)X(501) = X(267)-Ceva conjugate of X(58)

X(502) = CYCLOCEVIAN CONJUGATE OF X(1)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), wheref(a,b,c) = (b + c)u/a, u = u(a,b,c) = a3 - b3 - c3 + ba2 - ab2 + ca2 - ac2 - cb2 - bc2 - abc

Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)

If A', B', C' are points (other than A, B, C) on sidelines BC, CA, AB, respectively, then the circle passing through A', B', C' passes through through points A" on BC, B" on CA, C" on AB. Suppose P is a point and A' = P^BC, B' = P^CA, C' = P^AB. Then lines AA", BB", CC" concur in the cyclocevian conjugate of P, as defined in TCCT p.226. (Paul Yiu, 7/6/99)

X(502) lies on these lines: 1,2 261,319

X(502) = isogonal conjugate of X(501)X(502) = cyclocevian conjugate of X(1)>BR> X(502) = X(191)-cross conjugate of X(10)

Centers 503- 510, 173, 174, 258, and 351- 364 are associated with isoscelizers. A line L perpendicularto the line that bisects angle A is an A-isoscelizer. Let E = L∩CA and F = L∩AB;

then |AE| = |AF|. Centers defined by Peter Yff as points X of concurrence of A-, B-, and C-

isoscelizers depend on these notations:t(X,A) = |AE|, h(X,A) = A-altitude of AEF, v(X,A) = |EF|.

X(503)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = sec B/2 + sec C/2 - sec A/2Barycentrics af(A,B,C) : bf(B,C,A) : cf(C,A,B)

The isoscelizer equations ah(X,A) = bh(X,B) = ch(X,C) have solution X = X(503). (Peter Yff, 4/9/99)

X(503) lies on these lines: 1,167 164,361 173,844

X(503) = X(259)-Ceva conjugate of X(1)

X(504)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = b sin B/2 + c sin C/2 - a sin A/2Barycentrics af(A,B,C) : bf(B,C,A) : cf(C,A,B)

The isoscelizer equations [h(X,A)]/a = [h(X,B)]/b = [h(X,C)]/c have solution X = X(504). (Peter Yff, 4/9/99)

X(504) lies on this line: 164,173

X(505)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = 1/(sin B/2 + sin C/2 - sin A/2)Barycentrics af(A,B,C) : bf(B,C,A) : cf(C,A,B)

The isoscelizer equations h(X,A)v(X,A) = h(X,B)v(X,B) = h(X,C)v(X,C) have solution X = X(505). (Peter Yff, 4/9/99)

X(505) lies on this line: 40, 164

X(505) = isogonal conjugate of X(164)X(505) = X(266)-cross conjugate of X(1)

X(506)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (cos A/2)-2/3

Barycentrics af(A,B,C) : bf(B,C,A) : cf(C,A,B)

The isoscelizer equations

[v(X,A)][area of t(X,A)] = [v(X,B)][area of t(X,B)] = [v(X,C)][area of t(X,C)]

have solution X = X(506). (Peter Yff, 4/9/99)

X(507)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (cos A/2)-1/2

Barycentrics af(A,B,C) : bf(B,C,A) : cf(C,A,B)

The isoscelizer equations

[area of T(X,A)]/v(X,A) = [area of T(X,B)]/v(X,B) = [area of T(X,C)]/v(X,C)

have solution X = X(507). (Peter Yff, 4/9/99)

X(508)

Trilinears a-1/2sec(A/2) : b-1/2sec(B/2) : c-1/2sec(C/2)Barycentrics a1/2sec(A/2) : b1/2sec(B/2) : c1/2sec(C/2)

The isoscelizer equations

a[area of T(X,A)] = b[area of T(X,B)] = c[area of T(X,C)]

have solution X = X(508). (Peter Yff, 4/9/99)

X(509)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (tan A/2)1/2

Barycentrics af(A,B,C) : bf(B,C,A) : cf(C,A,B)

The isoscelizer equations

[area of T(X,A)]/a = [area of T(X,B)]/b = [area of T(X,C)]/c

have solution X = X(509). (Peter Yff, 4/9/99)

X(510)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = b3/2 + c3/2 - a3/2

Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)

The isoscelizer equations

t(X,A)[area of T(X,A)]/a = t(X,B)[area of T(X,B)]/b = t(X,C)[area of T(X,C)]/c

have solution X = X(510). (Peter Yff, 4/9/99)

Centers 511- 526 and 30 lie on the line at infinity.

Thus, each collection of collinearities comprises a family of parallel lines.

X(511) = ISOGONAL CONJUGATE OF X(98)

Trilinears cos(A + ω) : cos(B + ω) : cos(C + ω)Barycentrics sin A cos(A + ω) : cos B cos(B + ω) : cos C cos(C + ω)

As the isogonal conjugate of a point on the circumcircle, X(511) lies on the line at infinity.

X(511) lies on these lines:1,256 2,51 3,6 4,69 5,141 20,185 22,184 23,110 24,1092 25,394 26,206 30,512 40,1045 55,611 56,613 66,68 67,265 74,691 98,385 107,450 111,352 114,325 125,858 140,143 155,159 171,181 186,249 287,401 298,1080 299,383 343,427 381,599 549,597

X(511) = isogonal conjugate of X(98)X(511) = isotomic conjugate of X(290)X(511) = cevapoint of X(385) and X(401)X(511) = X(I)-cross conjugate of X(J) for these (I,J): (4,114), (290,2), (297,232)X(511) = crosspoint of X(I) and X(J) for these (I,J): (2,290), (297,325)X(511) = X(3)-Hirst inverse of X(6)X(511) = X(I)-line conjugate of X(J) for these (I,J): (3,6), (30,523)

X(512) = ISOGONAL CONJUGATE OF X(99)

Trilinears a(b2 - c2) : b(c2 - a2) : c(a2 - b2)Barycentrics a2(b2 - c2) : b2(c2 - a2) : c2(a2 - b2)

X(512) is the point in which the line of the 1st and 2nd Brocard points meets the line at infinity.

X(512) lies on these lines: 1,875 4,879 30,511 32,878 39,881 74,842 99,805 110,249 111,843 187,237 316,850 660,1016 670,886

X(512) = isogonal conjugate of X(99)X(512) = isotomic conjugate of X(670)X(512) = X(I)-Ceva conjugate of X(J) for these (I,J): (4,115), (66,125), (99,39), (110,6), (112,32)X(512) = crosspoint of X(I) and X(J) for these (I,J): (4,112), (6,110), (83,99)X(512) = X(112)-line conjugate of X(30)

X(513) = ISOGONAL CONJUGATE OF X(100)

Trilinears b - c : c - a : a - bBarycentrics ab - ac : bc - ba : ca - cb

As the isogonal conjugate of a point on the circumcircle, X(513) lies on the line at infinity.

X(513) lies on these lines: 1,764 6,1024 7,885 30,511 36,238 37,876 44,649 59,651 100,765 104,953 105,840 190,660 320,350 663,855 668,889 1052,1054

X(513) = X(244)-cross conjugate of X(1) X(513) = X(I)-Ceva conjugate of X(J) for these (I,J): (1,244), (4,11), (100,1), (101,354), (108,56), (109,65), (190,37)

X(513) = isogonal conjugate of X(100)X(513) = isotomic conjugate of X(668)X(513) = crosspoint of X(I) and X(J) for these (I,J): (1,100), (4,108), (58,109), (86,190)X(513) = X(I)-line conjugate of X(J) for these (I,J): (30,518), (36,238)

X(514) = ISOGONAL CONJUGATE OF X(101)

Trilinears (b - c)/a : (c - a)/b : (a - b)/cBarycentrics b - c : c - a : a - b

As the isogonal conjugate of a point on the circumcircle, X(514) lies on the line at infinity.

X(514) lies on this line: 1,663 2,1022 10,764 30,511 101,664 109,929 190,1016 239,649 241,650 551,676 651,655 659,667 661,693

X(514) = isogonal conjugate of X(101)X(514) = isotomic conjugate of X(190)X(514) = X(I)-Ceva conjugate of X(J) for these (I,J): (4,116), (7,11), (75,244), (100,142), (190,2)X(514) = X(I)-cross conjugate of X(J) for these (I,J): (11,7), (244,75)X(514) = crosspoint of X(2) and X(190)

X(515) = ISOGONAL CONJUGATE OF X(102)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c)sec A - b sec B - c sec CBarycentrics af(a,b,c) : bf(b,c,a) cf(c,a,b)

As the isogonal conjugate of a point on the circumcircle, X(515) lies on the line at infinity.

X(515) lies on these lines: 1,4 3,10 8,20 29,947 30,511 36,80 55,1012 103,929 119,214 153,908 165,376 281,610 284,1065 381,551

X(515) = isogonal conjugate of X(102)X(515) = X(4)-Ceva conjugate of X(117)

X(516) = ISOGONAL CONJUGATE OF X(103)

Trilinears 1/f(a,b,c) : 1/f(b,c,a) : 1/f(c,a,b), where f(a,b,c) : f(b,c,a) : f(c,a,b) = X(103)Barycentrics a/f(a,b,c) : b/f(b,c,a) c/f(c,a,b)

As the isogonal conjugate of a point on the circumcircle, X(516) lies on the line at infinity.

X(516) lies on these lines:1,7 2,165 3,142 4,9 8,144 30,511 35,411 55,226 57,497 65,950 80,655 100,908 102,929 103,927 118,910 200,329 238,673 354,553 355,382 376,551 993,1012

X(516) = isogonal conjugate of X(103)X(516) = X(4)-Ceva conjugate of X(118)

X(517) = ISOGONAL CONJUGATE OF X(104)

Trilinears -1 + cos B + cos C : -1 + cos C + cos A : -1 + cos A + cos BBarycentrics (sin A)(-1 + cos B + cos C) : (sin B)(-1 + cos C + cos A) : (sin C)(-1 + cos A + cos B)

As the isogonal conjugate of a point on the circumcircle, X(517) lies on the line at infinity.

X(517) lies on these lines:1,3 2,392 4,8 5,10 6,998 7,1000 9,374 19,219 20,145 30,511 37,573 42,1064 63,956 78,945 100,953 101,910 104,901 119,908 169,220 210,381 226,495 238,1052 389,950 549,551 572,1100 580,595 582,602 938,1058 1042,1066

X(517) = isogonal conjugate of X(104)X(517) = X(4)-Ceva conjugate of X(119)X(517) = crosspoint of X(I) and X(J) for these (I,J): (1,80), (7,88)

X(518) = ISOGONAL CONJUGATE OF X(105)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = ab + ac - b2 - c2

Barycentrics af(a,b,c) : bf(b,c,a) cf(c,a,b)

As the isogonal conjugate of a point on the circumcircle, X(518) lies on the line at infinity.

X(518) lies on these lines:1,6 2,210 7,8 10,141 11,908 30,511 38,42 43,982 55,63 56,78 57,200 59,765 144,145 209,306 239,335 244,899 329,497 551,597 583,1009 612,940 896,902 997,999

X(518) = isogonal conjugate of X(105)X(518) = X(I)-Ceva conjugate of X(J) for these (I,J): (4,120), (335,37)X(518) = crosspoint of X(1) and X(291)X(518) = X(1)-Hirst inverse of X(9)X(518) = X(I)-line conjugate of X(J) for these (I,J): (1,6), (30,513)

X(519) = ISOGONAL CONJUGATE OF X(106)

Trilinears (2a - b - c)/a : (2b - c - a)/b : (2c - a - b)/cBarycentrics 2a - b - c : 2b - c - a : 2c - a - b

As the isogonal conjugate of a point on the circumcircle, X(519) lies on the line at infinity.

X(519) lies on these lines: 1,2 6,996 9,1000 30,511 36,100 40,376 55,956 58,1043 65,553 72,950 80,908 210,392 238,765 320,668 355,381 447,648 594,1100 672,1018 751,984

X(519) = isogonal conjugate of X(106)X(519) = isotomic conjugate of X(903)X(519) = X(I)-Ceva conjugate of X(J) for these (I,J): (4,121), (80,10)

X(520) = ISOGONAL CONJUGATE OF X(107)

Trilinears (cos A)(sin 2B - sin 2C) : (cos B)(sin 2C - sin 2A) : (cos C)(sin 2A - sin 2B)Barycentrics (sin 2A)(sin 2B - sin 2C) : (sin 2B)(sin 2C - sin 2A) : (sin 2C)(sin 2A - sin 2B)

As the isogonal conjugate of a point on the circumcircle, X(520) lies on the line at infinity.

X(520) lies on these lines: 30,511 69,879 110,250 340,850 647,652

X(520) = isogonal conjugate of X(107)X(520) = X(I)-Ceva conjugate of X(J) for these (I,J): (4,122), (68,125), (110,3)X(520) = crosspoint of X(3) and X(110)

X(521) = ISOGONAL CONJUGATE OF X(108)

Trilinears (sec B - sec C)(csc A) : (sec C - sec A)(csc B) : (sec A - sec B)(csc C)Barycentrics sec B - sec C : sec C - sec A : sec A - sec B

As the isogonal conjugate of a point on the circumcircle, X(521) lies on the line at infinity.

X(521) lies on these lines: 30,511 59,100 650,1021 656,810

X(521) = isogonal conjugate of X(108)X(521) = X(I)-Ceva conjugate of X(J) for these (I,J): (4,123), (100,3)X(521) = crosspoint of X(8) and X(100)

X(522) = ISOGONAL CONJUGATE OF X(109)

Trilinears (cos B - cos C)(csc A) : (cos C - cos A)(csc B) : (cos A - cos B)(csc C)Barycentrics cos B - cos C : cos C - cos A : cos A - cos B

As the isogonal conjugate of a point on the circumcircle, X(522) lies on the line at infinity.

X(522) lies on this line: 9,657 30,511 100,655 101,929 190,666 240,656 243,652

X(522) = isogonal conjugate of X(109)X(522) = isotomic conjugate of X(664)X(522) = X(I)-Ceva conjugate of X(J) for these (I,J): (4,124), (8,11), (100,10), (190,9)X(522) = X(11)-cross conjugate of X(8)X(522) = crosspoint of X(I) and X(J) for these (I,J): (21,100), (75,190)

X(523) = ISOGONAL CONJUGATE OF X(110)

Trilinears sin(B - C) : sin(C - A) : sin(A - B)Barycentrics b2 - c2 : c2 - a2 : a2 - b2

As the isogonal conjugate of a point on the circumcircle, X(523) lies on the line at infinity.

X(523) lies on these lines: 6,879 11,1090 23,385 30,511 69,655 74,477 75,876 98,842 99,691 110,476 112,935 141,882 230,231 250,648 325,684

X(523) = X(I)-Ceva conjugate of X(J) for these (I,J): (1,11), (2,115), (4,125), (99,2), (100,442), (107,4), (108,429), (108,429), (110,5), (112,427), (254,136), (476,30)

X(523) = isogonal conjugate of X(110)X(523) = isotomic conjugate of X(99)X(523) = cevapoint of X(2) and X(148)X(523) = X(I)-cross conjugate of X(J) for these (I,J): (115,2), (125,4)X(523) = crosspoint of X(I) and X(J) for these (I,J): (2,99), (4,107), (54,110), (112,251)X(523) = X(30)-line conjugate of X(511)

X(524) = ISOGONAL CONJUGATE OF X(111)

Trilinears (2a2 - b2 - c2)/a : (2b2 - c2 - a2)/b : (2c2 - a2 - b2)/cBarycentrics 2a2 - b2 - c2 : 2b2 - c2 - a2 : 2c2 - a2 - b2

As the isogonal conjugate of a point on the circumcircle, X(524) lies on the line at infinity.

X(524) lies on these lines: 2,6 5,576 30,511 53,317 67,858 76,598 99,843 140,575 182,549 239,320 297,340 316,594 319,594 397,633 398,634

X(524) = isogonal conjugate of X(111)X(524) = isotomic conjugate of X(671)X(524) = X(I)-Ceva conjugate of X(J) for these (I,J): (4,126), (67,141)X(524) = X(187)-cross conjugate of X(468)X(524) = X(I)-line conjugate of X(J) for these (I,J): (2,6), (30,512)

X(525) = ISOGONAL CONJUGATE OF X(112)

Trilinears (b cos B - c cos C)/a : (c cos C - a cos A)/b : (a cos A - b cos B)/cBarycentrics b cos B - c cos C : c cos C - a cos A : a cos A - b cos B

As the isogonal conjugate of a point on the circumcircle, X(525) lies on the line at infinity.

X(525) lies on these lines: 3,878 30,511 99,249 110,935 297,850 323,401 441,647

X(525) = X(I)-Ceva conjugate of X(J) for these (I,J): (4,127), (69,125), (76,339), (99,3), (110,141), (190,440), (253, 122)

X(525) = isogonal conjugate of X(112)X(525) = isotomic conjugate of X(648)X(525) = X(I)-cross conjugate of X(J) for these (I,J): (115,68), (122,253), (125,69)X(525) = crosspoint of X(76) and X(99)

X(526) = ISOGONAL CONJUGATE OF X(476)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (1 + 2 cos 2A)sin(B - C)Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

As the isogonal conjugate of a point on the circumcircle, X(526) lies on the line at infinity.

X(526) = isogonal conjugate of X(476)X(526) lies on these lines: 30,511 67,879 110,351X(526) = crosspoint of X(74) and X(110)

Centers 527- 565 were added to ETC on 1/1/01.

X(527) = DIRECTION OF VECTOR AX + BX + CX, where X = X(7)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), wheref(a,b,c) = (2ax - by - cz)/a, x = x(a,b,c) = 1/[a(b + c - a)], y = x(b,c,a), z = x(c,a,b)

Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(527) lies on the line at infinity.

X(527) lies on these lines: 2,7 30,511 44,1086 190,320 551,993 666,673

X(528) = DIRECTION OF VECTOR AX + BX + CX, where X = X(11)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), wheref(a,b,c) = (2ax - by - cz)/a, x = x(A,B,C) = 1 - cos(B-C), y = x(B,C,A), z = x(C,A,B)

Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(528) lies on the line at infinity.

X(528) lies on these lines: 1,1086 2,11 7,664 8,190 9,80 30,511 104,376 119,381 142,214

X(528) = isogonal conjugate of X(840)

X(529) = DIRECTION OF VECTOR AX + BX + CX, where X = X(12)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), wheref(a,b,c) = (2ax - by - cz)/a, x = x(A,B,C) = 1 + cos(B-C), y = x(B,C,A), z = x(C,A,B)

Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(529) lies on the line at infinity.

X(529) lies on these lines: 2,12 30,511 495,993 1001,1056

X(530) = DIRECTION OF VECTOR AX + BX + CX, where X = X(13)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), wheref(a,b,c) = (2ax - by - cz)/a, x = x(A,B,C) = csc(A + π/3), y = x(B,C,A), z = x(C,A,B)

Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(530) lies on the line at infinity.

X(530) lies on these lines: 2,13 14,671 30,511 99,299 115,395 187,396 298,316

X(531) = DIRECTION OF VECTOR AX + BX + CX, where X = X(14)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), wheref(a,b,c) = (2ax - by - cz)/a, x = x(A,B,C) = csc(A - π/3), y = x(B,C,A), z = x(C,A,B)

Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(531) lies on the line at infinity.

X(531) lies on these lines: 2,14 13,671 30,511 99,298 115,396 187,395 299,316

X(532) = DIRECTION OF VECTOR AX + BX + CX, where X = X(17)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), wheref(a,b,c) = (2ax - by - cz)/a, x = x(A,B,C) = csc(A + π/6), y = x(B,C,A), z = x(C,A,B)

Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(532) lies on the line at infinity.

X(532) lies on these lines: 2,17 13,298 14,622 15,616 16,299 30,511 395,624 396,618 397,635

X(533) = DIRECTION OF VECTOR AX + BX + CX, where X = X(18)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), wheref(a,b,c) = (2ax - by - cz)/a, x = x(A,B,C) = csc(A - π/6), y = x(B,C,A), z = x(C,A,B)

Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(533) lies on the line at infinity.

X(533) lies on these lines: 2,18 13,621 14,299 15,298 16,617 30,511 395,619 396,623 398,636

X(534) = DIRECTION OF VECTOR AX + BX + CX, where X = X(19)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), wheref(a,b,c) = (2ax - by - cz)/a, x = x(A,B,C) = tan A, y = x(B,C,A), z = x(C,A,B)

Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(534) lies on the line at infinity.

X(534) lies on these lines: 2,19 30,511

X(535) = DIRECTION OF VECTOR AX + BX + CX, where X = X(36)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), wheref(a,b,c) = (2ax - by - cz)/a, x = x(A,B,C) = 1 - 2 cos A, y = x(B,C,A), z = x(C,A,B)

Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(535) lies on the line at infinity.

X(535) lies on these lines: 2,36 30,511 63,484 214,908 226,551

X(536) = DIRECTION OF VECTOR AX + BX + CX, where X = X(37)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), wheref(a,b,c) = (2ax - by - cz)/a, x = x(a,b,c) = b + c, y = x(b,c,a), z = x(c,a,b)

Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(536) lies on the line at infinity.

X(536) lies on these lines: 2,37 30,511 44,190 335,903 894,1100

X(536) = isogonal conjugate of X(739)

X(537) = DIRECTION OF VECTOR AX + BX + CX, where X = X(38)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), wheref(a,b,c) = (2ax - by - cz)/a, x = x(a,b,c) = b2 + c2, y = x(b,c,a), z = x(c,a,b)

Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(537) lies on the line at infinity.

X(537) lies on these lines: 1,190 2,38 10,1086 30,511 37,551 75,668

X(538) = DIRECTION OF VECTOR AX + BX + CX, where X = X(39)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), wheref(a,b,c) = (2ax - by - cz)/a, x = x(a,b,c) = a(b2 + c2), y = x(b,c,a), z = x(c,a,b)

Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(538) lies on the line at infinity.

X(538) lies on these lines: 2,39 30,511 32,1003 99,187 115,325 148,316 183,574 230,620 350,1015

X(538) = isogonal conjugate of X(729)

X(539) = DIRECTION OF VECTOR AX + BX + CX, where X = X(54)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), wheref(a,b,c) = (2ax - by - cz)/a, x = x(A,B,C) = 1/cos(B-C), y = x(B,C,A), z = x(C,A,B)

Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(539) lies on the line at infinity.

X(539) lies on these lines: 2,54 30,511 155,195

X(540) = DIRECTION OF VECTOR AX + BX + CX, where X = X(58)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), wheref(a,b,c) = (2ax - by - cz)/a, x = x(a,b,c) = a/(b + c), y = x(b,c,a), z = x(c,a,b)

Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(540) lies on the line at infinity.

X(540) lies on these lines: 2,58 30,511 340,447

X(541) = DIRECTION OF VECTOR AX + BX + CX, where X = X(74)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), wheref(a,b,c) = (2ax - by - cz)/a, x = x(A,B,C) = 1/(cos A - 2 cos B cos C), y = x(B,C,A), z = x(C,A,B)

Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(541) lies on the line at infinity.

X(541) lies on these lines: 2,74 30,511 110,376 125,381 394,399

X(541) = isogonal conjugate of X(841)

X(542) = DIRECTION OF VECTOR AX + BX + CX, where X = X(98)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), wheref(a,b,c) = (2ax - by - cz)/a, x = x(A,B,C) = sec(A + ω), y = x(B,C,A), z = x(C,A,B)

Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(542) lies on the line at infinity.

X(542) lies on these lines: 2,98 3,67 4,576 5,575 6,13 30,511 69,74 141,549 146,148

X(542) = isogonal conjugate of X(842)

X(543) = DIRECTION OF VECTOR AX + BX + CX, where X = X(99)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), wheref(a,b,c) = (2ax - by - cz)/a, x = x(a,b,c) = bc/(b2 - c2), y = x(b,c,a), z = x(c,a,b)

Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(543) lies on the line at infinity.

X(543) lies on these lines: 2,99 30,511 98,376 114,381

X(543) = isogonal conjugate of X(843)

X(544) = DIRECTION OF VECTOR AX + BX + CX, where X = X(101)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), wheref(a,b,c) = (2ax - by - cz)/a, x = x(a,b,c) = a/(b - c), y = x(b,c,a), z = x(c,a,b)

Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(544) lies on the line at infinity.

X(544) lies on these lines: 2,101 30,511 63,1018 103,376 118,381

X(545) = DIRECTION OF VECTOR AX + BX + CX, where X = X(190)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), wheref(a,b,c) = (2ax - by - cz)/a, x = x(a,b,c) = bc/(b - c), y = x(b,c,a), z = x(c,a,b)

Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(545) lies on the line at infinity.

X(545) lies on these lines: 2,45 30,511

X(546) = NINE-POINT CENTER OF X(4)-EULER MIDWAY TRIANGLE

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), wheref(A,B,C) = 3 cos(B - C) - 2 cos A

Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(546) lies on the Euler line. (Antreas Hatzipolakis, 1/20/00, Hyacinthos #201)

X(546) lies on these lines: 2,3 13,398 14,397 113,137 156,578 946,952

X(546) = inverse of X(382) in the orthocentroidal circleX(546) = complement of X(550)

X(547) = CENTROID OF X(5)-EULER MIDWAY TRIANGLE

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), wheref(A,B,C) = 5 cos(B - C) + 2 cos A

Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(547) lies on the Euler line. (Antreas Hatzipolakis, 1/20/00, Hyacinthos #201)

X(547) lies on these lines: 2,3 551,952

X(547) = complement of X(549)

X(548) = DE LONGCHAMPS POINT OF X(5)-EULER MIDWAY TRIANGLE

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), wheref(A,B,C) = - cos(B - C) + 6 cos A

Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(548) lies on the Euler line. (Antreas Hatzipolakis, 1/20/00, Hyacinthos #201)

X(549) = ORTHOCENTER OF X(2)-EULER MIDWAY TRIANGLE

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), wheref(A,B,C) = cos(B - C) + 4 cos A

Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(549) lies on the Euler line. (Antreas Hatzipolakis, 1/20/00, Hyacinthos #201)

X(549) lies on these lines: 2,3 15,395 16,396 35,496 36,495 141,542 182,524 230,574 302,617 303,616 511,597 517,551

X(549) = complement of X(381)X(549) = anticomplement of X(547)

X(550) = ORTHOCENTER OF X(20)-EULER MIDWAY TRIANGLE

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), wheref(A,B,C) = - cos(B - C) + 4 cos A

Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(550) lies on the Euler line. (Antreas Hatzipolakis, 1/20/00, Hyacinthos #201)

X(550) lies on these lines: 2,3 15,397 16,398 35,495 36,496 40,952 74,930 156,1092 165,355

X(550) = complement of X(382) X(550) = anticomplement of X(546)

X(551) = INCENTER OF X(1)-EULER MIDWAY TRIANGLE

Trilinears (4a + b + c)/a : (4b + c + a)/b : (4c + a + b)/c Barycentrics 4a + b + c : 4b + c + a : 4c + a + b

X(551) lies on these lines: 1,2 30,946 37,537 56,553 86,99 142,214 226,535 354,392 376,516 381,515 514,676 517,549 518,597 527,993 547,952

(Antreas Hatzipolakis, 1/24/00, Hyacinthos #223)

X(552)

Trilinears 1/[a(b + c - a)(b + c)2] : 1/[b(c +a - b)(c + a)2] : 1/[c(a + b - c)(a + b)2] Barycentrics 1/[(b+c-a)(b+c)2] : 1/[(c+a-b)(c+a)2] : 1/[(a+b-c)(a+b)2]

X(552) lies on this line: 261,873

(Floor van Lamoen, 1/30/00, Hyacinthos #255)

X(553)

Trilinears bc(2a + b + c)/(b + c - a) : ca(2b + c + a)/(c + a - b) : ab/(2c + a + b)/(a + b - c) Barycentrics (2a + b + c)/(b + c - a) : (2b + c + a)/(c + a - b) : /(2c + a + b)/(a + b - c)

X(553) lies on these lines: 1,376 2,7 30,942 56,551 65,519 354,516

(Floor van Lamoen, 1/30/00, Hyacinthos #257)

X(554)

Trilinears sec(A/2) csc(A/2 + π/3) : sec(B/2) csc(B/2 + π/3) : sec(C/2) csc(C/2 + π/3) Barycentrics sin A sec(A/2) csc(A/2 + π/3) : sin B sec(B/2) csc(B/2 + π/3) : sin C sec(C/2) csc(C/2 + π/3)

X(554) lies on these lines: 1,30 7,1082 14,226 75,299

Clark Kimberling, "Major Centers of Triangles," Amer. Math. Monthly 104 (1997) 431-438; in (9) on page 435, take Y = X(13).

X(555)

Trilinears sec3(A/2) : sec3(B/2) : sec3(C/2) Barycentrics sin A sec3(A/2) : sin B sec3(B/2) : sin C sec3(C/2)

X(555) lies on these lines: 7,177 234,1088

Clark Kimberling, "Major Centers of Triangles," Amer. Math. Monthly 104 (1997) 431-438; in (9) on page 435, take Y = X(7).

X(556)

Trilinears csc A csc A/2 : csc B csc B/2 : csc C csc C/2 Barycentrics csc A/2 : csc B/2 : csc C/2

X(556) lies on these lines: 8,177 74,234

X(556) = isotomic conjugate of X(174)

Clark Kimberling, "Major Centers of Triangles," Amer. Math. Monthly 104 (1997) 431-438; in (9) on page 435, take Y = X(8).

X(557)

Trilinears sec A/2 cot A/4 : sec B/2 cot B/4 : sec C/2 cot C/4 Barycentrics cos A/2 cot A/4 : cos B/2 cot B/4 : cos C/2 cot C/4

X(557) lies on this line: 2,178

Clark Kimberling, "Major Centers of Triangles," Amer. Math. Monthly 104 (1997) 431-438; in (9) on page 435, take Y = X(9).

X(558)

Trilinears sec A/2 tan A/4 : sec B/2 tan B/4 : sec C/2 tan C/4 Barycentrics sin2(A/4) : sin2(B/4) : sin2(C/4)

X(558) lies on this line: 2,178

Clark Kimberling, "Major Centers of Triangles," Amer. Math. Monthly 104 (1997) 431-438; in (9) on page 435, take Y = X(57).

X(559)

Trilinears (sec A/2) sin(A/2 + π/3) : (sec B/2) sin(B/2 + π/3) : (sec C/2) sin(C/2 + π/3) Barycentrics (sin A/2) sin(A/2 + π/3) : (sin B/2) sin(B/2 + π/3) : (sin C/2) sin(C/2 + π/3)

X(559) lies on these lines: 1,3 14,226 299,319

Clark Kimberling, "Major Centers of Triangles," Amer. Math. Monthly 104 (1997) 431-438; in (9) on page 435, take Y = X(15).

X(560) 4th POWER POINT

Trilinears a4 : b4 : c4

Barycentrics a5 : b5 : c5

X(560) lies on these lines: 1,82 31,48 41,872 42,584 100,697 101,713 110,715 717,825 719,827

X(560) = isogonal conjugate of X(561)

X(561) ISOGONAL CONJUGATE OF 4th POWER POINT

Trilinears a - 4 : b - 4 : c - 4

Barycentrics a - 3 : b - 3 : c - 3

X(561) lies on these lines: 1,718 2,716 6,720 31,722 32,724 38,75 63,799 76,321 92,304 313,696

X(561) = isogonal conjugate of X(560)X(561) = isotomic conjugate of X(31)

X(562)

Trilinears csc A tan 3A : csc B tan 3B : csc C tan 3CBarycentrics tan 3A : tan 3B : tan 3C

X(562) lies on this line: 4,93

Clark Kimberling and Cyril Parry, "Products, Square Roots, and Layers in Triangle Geometry," Mathematics and Informatics Quarterly 10 (2000) 9-22; on page 21, section 8.

X(563)

Trilinears sin 4A : sin 4B : sin 4CBarycentrics sin A sin 4A : sin B sin 4B : sin C sin 4C

X(563) lies on these lines: 19,163 48,255

Clark Kimberling and Cyril Parry, "Products, Square Roots, and Layers in Triangle Geometry," Mathematics and Informatics Quarterly 10 (2000) 9-22; on page 21, section 8.

X(564)

Trilinears cos(2B - 2C) : cos(2C - 2A) : cos(2A - 2B)Barycentrics sin A cos(2B - 2C) : sin B cos(2C - 2A) : sin C cos(2A - 2B)

X(564) lies on these lines: 1,1048 47,91

Clark Kimberling and Cyril Parry, "Products, Square Roots, and Layers in Triangle Geometry," Mathematics and Informatics Quarterly 10 (2000) 9-22; on page 21, section 8.

X(565)

Trilinears cos(3B - 3C) : cos(3C - 3A) : cos(3A - 3B)Barycentrics sin A cos(3B - 3C) : sin B cos(3C - 3A) : sin C cos(3A - 3B)

X(565) lies on these lines: 49,93 143,324

Clark Kimberling and Cyril Parry, "Products, Square Roots, and Layers in Triangle Geometry," Mathematics and Informatics Quarterly 10 (2000) 9-22; on page 21, section 8.

Centers 566- 584 are on the Brocard axis, L(3,6). Each is the center X of a circlemeeting the sides of triangle ABC with three equal angles at X.

Let AB, AC, BC, BA, CA, CB denote the meeting-points; e.g., AB and CB are on side CA. The equal angles are given by

D = angle(ABXAC) = angle(BCXBA) = angle(CAXCB)

Then trilinears for X are given by

X = sin A + cot D/2 cos A : sin B + cot D/2 cos B : sin C + cot D/2 cos C.

Definitions: Y is the orthogonal of X if D(X) + D(Y) = π/2;Y is the harmonic of X if X and Y are harmonic conjugates with respect to X(3) and

X(6);Y is the orthoharmonic if Y is the harmonic of the orthogonal of X.

For a figure and details, see Edward Brisse, "Some properties of points on the axis of Brocard."

X(566) = HARMONIC OF X(50)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = sin A + cot D/2 cos A,cot D/2 = 4*area/(a2 + b2 + c2 - 6r2), where r = abc/(4*area)

Barycentrics (sin A)f(A,B,C): (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(566) lies on these lines: 2,94 3,6

X(566) = inverse of X(50) in the Brocard circle

Edward Brisse, as cited above X(566), page 7.

X(567) = ORTHOGONAL OF X(50)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = sin A + cot D/2 cos A,cot D/2 = (a2 + b2 + c2 - 6r2)/(4*area), where r = abc/(4*area)

Barycentrics (sin A)f(A,B,C): (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(567) lies on these lines: 3,6 5,49 184,381

X(567) = inverse of X(568) in the Brocard circle

Edward Brisse, as cited above X(566), page 7.

X(568) = ORTHOHARMONIC OF X(50)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = sin A + cot D/2 cos A,cot D/2 = (6r2 - a2 - b2 - c2)/(4*area), where r = abc/(4*area)

Barycentrics (sin A)f(A,B,C): (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(568) lies on these lines: 3,6 4,94 24,49 51,381 68,973 185,382

X(568) = inverse of X(567) in the Brocard circle

Edward Brisse, as cited above X(566), page 7.

X(569) = HARMONIC OF X(52)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = sin A + cot D/2 cos A,cot D/2 = (2e3 + e2 - e1)/[64*(area)3], wheree1 = a6 + b6 + c6 e2 = a4(b2 + c2) + b4(c2 + a2) + c4(a2 + b2) e3 = a2b2c2

Barycentrics (sin A)f(A,B,C): (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(569) lies on these lines: 2,54 3,6 5,156 26,51 140,343

X(569) = inverse of X(52) in the Brocard circle

Edward Brisse, as cited above X(566), page 7.

X(570) = ORTHOGONAL OF X(52)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = sin A + cot D/2 cos A,cot D/2 = [64*(area)3]/(2e3 + e2 - e1), wheree1, e2, e3 are as for X(569)

Barycentrics (sin A)f(A,B,C): (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(570) lies on these lines: 2,311 3,6 53,232 115,128 140,231 157,184

X(570) = inverse of X(571) in the Brocard circleX(570) = complement of X(311)

Edward Brisse, as cited above X(566), page 7.

X(571) = ORTHOHARMONIC OF X(52)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = sin A + cot D/2 cos A,cot D/2 = [64*(area)3]/(e1 - 2e3 - e2), wheree1, e2, e3 are as for X(569)

Barycentrics (sin A)f(A,B,C): (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(571) lies on these lines: 3,6 4,96 66,248 112,393 160,184 206,237 230,427 608,913

X(571) = inverse of X(570) in the Brocard circle

Edward Brisse, as cited above X(566), page 7.

X(572) = ORTHOGONAL OF X(58)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = sin A + cot D/2 cos A,cot D/2 = (a + b + c)2/(4*area)

Barycentrics (sin A)f(A,B,C): (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(572) lies on these lines: 1,604 3,6 9,48 51,199 54,71 103,825 165,1051 169,610 184,1011 219,947 261,662 517,1100 594,952 631,966

X(572) = inverse of X(573) in the Brocard circle

Edward Brisse, as cited above X(566), page 7.

X(573) = ORTHOHARMONIC OF X(58)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = sin A + cot D/2 cos A,cot D/2 = - (a + b + c)2/(4*area)

Barycentrics (sin A)f(A,B,C): (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(573) lies on these lines: 1,941 3,6 4,9 20,391 36,604 37,517 43,165 51,1011 55,181 101,102 109,478 184,199 256,981 346,1018 347,1020

X(573) = inverse of X(572) in the Brocard circle

Edward Brisse, as cited above X(566), page 7.

X(574) = HARMONIC OF X(187)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = sin A + cot D/2 cos A,cot D/2 = 12*area/(a2 + b2 + c2)

Barycentrics (sin A)f(A,B,C): (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(574) lies on these lines: 2,99 3,6 55,1015 110,353 183,538 230,549 232,378 805,843

X(574) = isogonal conjugate of X(598)X(574) = inverse of X(187) in the Brocard circle

Edward Brisse, as cited above X(566), page 7.

X(575) = ORTHOGONAL OF X(187)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = sin A + cot D/2 cos A,cot D/2 = (a2 + b2 + c2)/(12*area)

Barycentrics (sin A)f(A,B,C): (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(575) lies on these lines: 3,6 4,598 5,542 23,51 54,895 110,373 140,524 141,629

X(575) = inverse of X(576) in the Brocard circle

Edward Brisse, as cited above X(566), page 7.

X(576) = ORTHOHARMONIC OF X(187)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = sin A + cot D/2 cos A,cot D/2 = - (a2 + b2 + c2)/(12*area)

Barycentrics (sin A)f(A,B,C): (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(576) lies on these lines: 3,6 4,542 5,524 23,184 140,597 262,385

X(576) = inverse of X(575) in the Brocard circle

Edward Brisse, as cited above X(566), page 7.

X(577) = HARMONIC OF X(216)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = sin A + cot D/2 cos A,cot D/2 = - 8*(area)3/(a2b2c2 cos A cos B cos C)

Barycentrics (sin A)f(A,B,C): (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(577) lies on these lines: 2,95 3,6 20,393 22,232 30,53 48,603 69,248 112,376 141,441 160,206 172,1038 184,418 198,478 219,906 220,268 264,401 395,466 396,465

X(577) = inverse of X(216) in the Brocard circleX(577) = complement of X(317)

Edward Brisse, as cited above X(566), page 7.

X(578) = ORTHOHARMONIC OF X(216)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = sin A + cot D/2 cos A,cot D/2 = (a2b2c2 cos A cos B cos C)/[8*(area)3]

Barycentrics (sin A)f(A,B,C): (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(578) lies on these lines: 2,1092 3,6 4,54 24,51 49,381 156,546 185,378 436,1093

X(578) = inverse of X(389) in the Brocard circle

Edward Brisse, as cited above X(566), page 7.

X(579) = HARMONIC OF X(284)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = sin A + cot D/2 cos A,cot D/2 = 4*area*(a + b + c)/(e1 - e2 - 2abc), wheree1 = a3 + b3 + c3 e2 = a2(b + c) + b2(c + a) + c2(a + b)

Barycentrics (sin A)f(A,B,C): (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(579) lies on these lines: 1,71 2,7 3,6 19,46 36,48 37,942 40,387 56,219 109,608 165,380 198,218 443,966 474,965 517,1108

X(579) = inverse of X(284) in the Brocard circle

Edward Brisse, as cited above X(566), page 7.

X(580) = ORTHOGONAL OF X(284)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = sin A + cot D/2 cos A,cot D/2 = (e1 - e2 - 2abc)/[4*area*(a + b + c)], where e1, e2 are as for X(579)

Barycentrics (sin A)f(A,B,C): (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(580) lies on these lines: 1,201 2,283 3,6 31,40 34,46 36,54 57,255 162,412 165,601 223,603 238,946 517,595

X(580) = inverse of X(581) in the Brocard circle

Edward Brisse, as cited above X(566), page 7.

X(581) = ORTHOHARMONIC OF X(284)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = sin A + cot D/2 cos A,cot D/2 = (2abc - e1 + e2)/[4*area*(a + b + c)], where e1, e2 are as for X(579)

Barycentrics (sin A)f(A,B,C): (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(581) lies on these lines: 1,4 3,6 35,47 40,42 81,411 84,941 222,1035 936,966 947,1036 995,1104

X(581) = inverse of X(580) in the Brocard circle

Edward Brisse, as cited above X(566), page 7.

X(582) = HARMONIC OF X(500)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = sin A + cot D/2 cos A,cot D/2 = (e1 - e2 - 4abc)/[4*area*(a + b + c)], where

e1 = a3 + b3 + c3 e2 = a2(b + c) + b2(c + a) + c2(a + b)

Barycentrics (sin A)f(A,B,C): (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(582) lies on these lines: 3,6 212,942 283,474 517,602

X(582) = inverse of X(500) in the Brocard circle

Edward Brisse, as cited above X(566), page 7.

X(583) = ORTHOGONAL OF X(500)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = sin A + cot D/2 cos A,cot D/2 = 4*area*(a + b + c)/(e1 - e2 - 4abc), where e1, e2 are as for X(582)

Barycentrics (sin A)f(A,B,C): (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(583) lies on these lines: 3,6 37,38 44,992 71,1100 518,1009

X(583) = inverse of X(584) in the Brocard circle

Edward Brisse, as cited above X(566), page 7.

X(584) = ORTHOHARMONIC OF X(500)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = sin A + cot D/2 cos A,cot D/2 = 4*area*(a + b + c)/(4abc - e1 + e2), where e1, e2 are as for X(582)

Barycentrics (sin A)f(A,B,C): (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(584) lies on these lines: 3,6 37,41 42,560 48,354

X(584) = inverse of X(583) in the Brocard circle

Edward Brisse, as cited above X(566), page 7.

X(585) = 1st CONGRUENT SHRUNK INSQUARES POINT

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = [2*area*(1/b + 1/c - 1/a) + b + c - a]/aBarycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where f(a,b,c) = 2*area*(1/b + 1/c - 1/a) + b + c - a

X(585) lies on this line: 8,192

For a discussion, see

Floor van Lamoen, "Vierkanten in een driehoek: 5. Gekrompen ingeschreven vierkanten"

X(586) = 2nd CONGRUENT SHRUNK INSQUARES POINT

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = [2*area*(1/b + 1/c - 1/a) + a - b - c]/aBarycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = 2*area*(1/b + 1/c - 1/a) + a - b - c

X(586) lies on this line: 8,192

For a discussion, see

Floor van Lamoen, "Vierkanten in een driehoek: 5. Gekrompen ingeschreven vierkanten"

X(587) = REFLECTION OF X(20) ABOUT X(1)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = [2*(a + b + c) + (a - b - c) tan A]/aBarycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = 2*(a + b + c) + (a - b - c) tan A

X(587) is the radical center of the circles centered at A, B, C, with respectiveradii |CA| + |AB|, |AB| + |BC|, |BC| + |CA|.

X(587) lies on this line: 2,92

Floor van Lamoen, Problem 10734, American Mathematical Monthly 107 (2000) 658-659.

X(588)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a/(a2 + 4*area)Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(588) is the perspector of triangles ABC and A'B'C', where A' is the circumcenter of X(371) and the points where the squares in the Kenmotu configuration with center X(371) meet sideline BC, and B' and C' are defined cyclically. A discussion will appear in "Some concurrences from Tucker hexagons," forthcoming in Forum Geometricorum. (Floor van Lamoen, 1/4/01)

X(588) lies on this line: 39,589

X(588) = isogonal conjugate of X(590)

X(589)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a/(a2 - 4*area)Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(589) is the perspector of triangles ABC and A'B'C', where A' is the circumcenter of X(372) and the points where the squares in the Kenmotu configuration with center X(372) meet sideline BC, and B' and C' are defined cyclically. A discussion will appear in "Some concurrences from Tucker hexagons," forthcoming in Forum Geometricorum. (Floor van Lamoen, 1/4/01)

X(589) lies on this line: 39,588

X(589) = isogonal conjugate of X(615)

X(590) = ISOGONAL CONJUGATE OF X(588)

f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (a2 + 4*area)/aBarycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a2 + 4*area

X(590) lies on these lines: 2,6 3,485 5,371 32,640 39,642 140,372 605,748 606,750

X(590) = isogonal conjugate of X(588)X(590) = complement of X(492)

X(591)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (csc A)(2 + 2 cot A - cot B - cot C + 2 cot B cot C)Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = 2 + 2 cot A - cot B - cot C + 2 cot B cot C

Erect squares outwardly on the sides of triangle ABC. Two edges emanate from A; let P and Q be their endpoints. Let a' be the perpendicular bisector of PQ, and define b' and c' cyclically. Then a', b', c' concur in X(591). (Floor van Lamoen, 1/4/01, Hyacinthos #2123)

X(591) lies on these lines: 5,524 141,486 615,637

X(592)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = 1/(cos A + 2 cos(B - ω) cos(C - ω)Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A)f(A,B,C)

X(592) is the perspector of triangles ABC and A'B'C', where A' is the circumcenter of the triangles PQX,

where X = X(39), and P and Q are the points where the parallels through X(6) to lines CA and AB meet line BC,and B' and C' are defined cyclically. (Floor van Lamoen, 1/4/01)

X(593)

Trilinears a/(b + c)2 : b/(c + a)2 : c/(a + b)2

Barycentrics [a/(b + c)]2 : [b/(c + a)]2 : [c/(a + b)]2

Let O(A) be the circle tangent to line BC and to the circumcircle of triangle ABC at vertex A. Let AB and AC be where O(A) meets lines AB and AC, respectively. Let L(A) be the line joining AB and AC, and define L(B) and L(C) cyclically. Let A' be where L(B) and L(C) meet, and define B' and C' cyclically. Then triangle A'B'C' is homothetic to triangle ABC, and the center of homothety is X(593).

X(593) lies on these lines: 2,261 31,110 36,58 81,757 115,1029 229,1104

X(593) = isogonal conjugate of X(594)

(Antreas Hatzipolakis, Paul Yiu, Hyacinthos #2070)

X(594) = ISOGONAL CONJUGATE OF X(593)

Trilinears (b + c)2/a : (c + a)2/b : (a + b)2/cBarycentrics (b + c)2 : (c + a)2 : (a + b)2

X(594) lies on these lines: 6,8 7,599 9,80 10,37 45,346 53,318 75,141 100,1030 210,430 220,281 313,321 319,524 519,1100 572,952 762,1089

X(594) = isogonal conjugate of X(593)

X(595)

Trilinears a(a2 + ab + ac - bc) : b(b2 + bc + ba - ca) : c(c2 + ca + cb - ab)Barycentrics a2(a2 + ab + ac - bc) : b2(b2 + bc + ba - ca) : c2(c2 + ca + cb - ab)

Let O(A) be the circle tangent to line BC and to the circumcircle of triangle ABC at vertex A, and define O(B) and O(C) cyclically. X(595) is the radical center of O(A), O(B), O(C).

X(595) lies on these lines: 1,21 3,995 10,82 32,101 35,902 40,602 46,614 55,386 56,106 110,849 387,390 517,580

X(595) = isogonal conjugate of X(596)

(Antreas Hatzipolakis, Paul Yiu, Hyacinthos #2070)

X(596) = ISOGONAL CONJUGATE OF X(595)

Trilinears 1/[a(a2 + ab + ac - bc)] : 1/[b(b2 + bc + ba - ca)] : 1/[c(c2 + ca + cb - ab)]Barycentrics 1/(a2 + ab + ac - bc) : 1/(b2 + bc + ba - ca) : 1/(c2 + ca + cb - ab)

X(596) lies on these lines: 10,38 37,39 58,82 65,519 244,1089

X(596) = isogonal conjugate of X(595)

X(597) = MIDPOINT OF X(2) AND X(6)

Trilinears (4a2 + b2 + c2)/a : (4b2 + c2 + a2)/b : (4c2 + a2 + b2)/cBarycentrics 4a2 + b2 + c2 : 4b2 + c2 + a2 : 4c2 + a2 + b2

X(597) lies on these lines: 2,6 5,542 30,182 39,1084 83,671 140,576 511,549 518,551

X(597) = complement of X(599)

(Bernard Gibert, 1/5/01, Hyacinthos #2334)

X(598)

Trilinears bc/(a2 - 2b2 - 2c2) : ca/(b2 - 2c2 - 2a2) : ab/(c2 - 2a2 - 2b2)Barycentrics 1/(a2 - 2b2 - 2c2) : 1/(b2 - 2c2 - 2a2) : 1/(c2 - 2a2 - 2b2)

The Lemoine ellipse is the ellipse inscribed in triangle ABC having X(2) and X(6) as foci. Let A' be where this ellipse meets sideline BC, and define B' and C' cyclically. Then triangles ABC and A'B'C' are perspective, and their perspector is X(598). (Bernard Gibert, 1/5/01, Hyacinthos #2334)

X(598) lies on these lines: 2,187 4,575 6,671 30,262 76,524 98,381

X(598) = isogonal conjugate of X(574)X(598) = isotomic conjugate of X(599)

X(599) = ISOTOMIC CONJUGATE OF X(598)

Trilinears bc(a2 - 2b2 - 2c2) : ca(b2 - 2c2 - 2a2) : ab(c2 - 2a2 - 2b2)Barycentrics a2 - 2b2 - 2c2 : b2 - 2c2 - 2a2 : c2 - 2a2 - 2b2

X(599) lies on these lines: 2,6 3,67 7,594 8,1086 76,338 340,458 381,511

X(599) = isotomic conjugate of X(598) X(599) = anticomplement of X(597)

(Bernard Gibert, 1/5/01, Hyacinthos #2334)

X(600)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where= a(a2b2 + a2c2 - b2c2)/[bc + 2(a2 - b2 - c2)]

Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)

(Antreas Hatzipolakis, Paul Yiu, 1/5/01, Hyacinthos #2344, #2346-8)

X(601)

Trilinears sin2A + cos A : sin2B + cos B : sin2C + cos C Barycentrics (sin A)(sin2A + cos A) : (sin B)(sin2B + cos B) : (sin C)(sin2C + cos C)

X(601) lies on these lines: 1,104 3,31 4,171 5,750 35,47 40,58 41,906 55,255 140,748 165,580 201,920 371,606 372,605 774,1060 912,976 999,1106

X(602)

Trilinears sin2A - cos A : sin2B - cos B : sin2C - cos C Barycentrics (sin A)(sin2A - cos A) : (sin B)(sin2B - cos B) : (sin C)(sin2C - cos C)

X(602) lies on these lines: 1,201 3,31 4,238 5,748 36,47 40,595 56,255 140,750 171,631 371,605 372,606 517,582 774,1062

X(603)

Trilinears cos2A - cos A : cos2B - cos B : cos2C - cos C Barycentrics (sin A)(cos2A - cos A) : (cos B)(cos2B - cos B) : (cos C)(cos2C - cos C)

X(603) lies on these lines: 1,104 3,73 6,1035 12,750 28,34 31,56 33,84 36,47 41,911 48,577 63,201 77,283 171,388 223,580 404,651

X(603) = isogonal conjugate of X(318)

X(604)

Trilinears a(1 - cos A) : b(1 - cos B) : c(1 - cos c) Barycentrics a2(1 - cos A) : b2(1 - cos B) : c2(1 - cos C)

X(604) lies on these lines: 1,572 6,41 19,909 31,184 32,1106 36,573 57,77 65,1100 109,739 219,672 608,1042

X(604) = isogonal conjugate of X(312)

X(605)

Trilinears a(1 + sin A) : b(1 + sin B) : c(1 + sin c) Barycentrics a2(1 + sin A) : b2(1 + sin B) : c2(1 + sin C)

X(605) lies on these lines: 6,31 371,602 372,601 590,748 615,750

X(606)

Trilinears a(1 - sin A) : b(1 - sin B) : c(1 - sin c) Barycentrics a2(1 - sin A) : b2(1 - sin B) : c2(1 - sin C)

X(606) lies on these lines: 6,31 371,601 372,602 590,750 615,748

X(607)

Trilinears a(1 + sec A) : b(1 + sec B) : c(1 + sec c) Barycentrics a2(1 + sec A) : b2(1 + sec B) : c2(1 + sec C)

X(607) lies on these lines: 1,949 4,218 6,19 8,29 9,1039 25,41 28,1002 33,210 56,911 92,239 213,1096 227,910 240,611

X(607) = isogonal conjugate of X(348)

X(608)

Trilinears a(1 - sec A) : b(1 - sec B) : c(1 - sec c) Barycentrics a2(1 - sec A) : b2(1 - sec B) : c2(1 - sec C)

X(608) lies on these lines: 6,19 7,27 9,1041 25,31 28,959 92,894 108,739 109,579 193,651 223,380 240,613 571,913 604,1042

X(608) = isogonal conjugate of X(345)

X(609)

Trilinears area + a2 sin A : area + b2 sin B : area + c2 sin C Barycentrics a(area + a2 sin A) : b(area + b2 sin B) : c(area + c2 sin C)

X(609) lies on these lines: 1,32 6,36 31,101 33,112 41,58 251,614 995,1055

X(610)

Trilinears area - a2 cot A : area - b2 cot B : area - c2 cot C Barycentrics a(area - a2 cot A) : b(area - b2 cot B) : c(area - c2 cot C)

X(610) lies on these lines: 1,19 3,9 6,57 40,219 71,165 159,197 169,572 281,515 326,662

X(611)

Trilinears W + sin A : W + sin B : W + sin C, where W = (a2 + b2 + c2)/(4*area) Barycentrics a(W + sin A) : b(W + sin B) : c(W + sin C)

X(611) lies on these lines: 1,6 55,511 56,182 141,498 240,607 394,612

X(612)

Trilinears W + csc A : W + csc B : W + csc C, where W = (a2 + b2 + c2)/(4*area) Barycentrics a(W + csc A) : b(W + csc B) : c(W + csc C)

X(612) lies on these lines: 1,2 6,210 9,31 12,34 19,25 21,989 22,35 38,57 63,171 165,990 394,611 404,988 495,1060 518,940

X(613)

Trilinears W - sin A : W - sin B : W - sin C, where W = (a2 + b2 + c2)/(4*area) Barycentrics a(W - sin A) : b(W - sin B) : c(W - sin C)

X(613) lies on these lines: 1,6 55,182 56,511 141,499 222,982 240,608 394,614 496,1069

X(614)

Trilinears W - csc A : W - csc B : W - csc C, where W = (a2 + b2 + c2)/(4*area) Barycentrics a(W - csc A) : b(W - csc B) : c(W - csc C)

X(614) lies on these lines: 1,2 6,354 9,38 11,33 21,988 22,36 25,34 31,57 46,595 63,238 106,998 165,902 251,609 269,479 278,1096 305,350 394,613 496,1062 497,1040 968,1001

X(615) = ISOGONAL CONJUGATE OF X(589)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (a2 - 4*area)/a Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a2 - 4*area

X(615) lies on these lines: 2,6 3,486 5,372 32,639 39,641 140,371 591,637 605,750 606,748

X(615) = isogonal conjugate of X(589)X(615) = complement of X(491)

Centers 616- 642 were contributed by Bernard Gibert, March 2, 2001. Notation:

SA = (b2 + c2 - a2)/2 SB = (c2 + a2 - b2)/2 SC = (a2 + b2 - c2)/2

X(616) = ANTICOMPLEMENT OF X(13)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = [SB*SC - 2*SA*(a2 + sqr(3)*area]/a Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

The midpoint of X(616) and X(617) is the Steiner point, X(99).

X(616) lies on these lines: 2,13 3,299 4,627 14,148 15,532 20,633 30,298 69,74 302,381 303,549

X(616) = anticomplement of X(13)

X(617) = ANTICOMPLEMENT OF X(14)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = [SB*SC - 2*SA*(a2 - sqr(3)*area]/a

Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(617) lies on these lines: 2,14 3,298 4,628 13,148 16,533 20,634 30,299 69,74 302,549 303,381

X(617) = anticomplement of X(14)

X(618) = COMPLEMENT OF X(13)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), wheref(a,b,c) = [2*SB*SC + 5*SA*a2 + 2*sqr(3)*(b2 + c2)*area]/a Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(618) lies on these lines: 2,13 3,635 5,629 14,99 15,298 30,623 39,395 61,627 140,630 141,542 396,532

X(619) = COMPLEMENT OF X(14)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), wheref(a,b,c) = [2*SB*SC + 5*SA*a2 - 2*sqr(3)*(b2 + c2)*area]/a Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(619) lies on these lines:2,14 3,636 5,630 13,99 16,299 30,624 39,396 62,628 140,629 141,542 395,533

X(620) = MIDPOINT OF X(618) AND X(619)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = [4* SA*a2 - (b4 + c4)]/a Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(620) lies on these lines:2,99 3,114 30,625 98,631 141,542 187,325 230,538

X(620) = complement of X(115)

X(621) = ANTICOMPLEMENT OF X(15)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = [sqr(3)*SB*SC + 2*SA*area]/a Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(621) lies on these lines: 2,14 3,302 4,69 5,303 13,533 20,627 30,298 183,383 265,300 299,381 325,1080 343,472 394,473

X(621) = anticomplement of X(15)

X(622) = ANTICOMPLEMENT OF X(16)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = [sqr(3)*SB*SC - 2*SA*area]/a Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(622) lies on these lines: 2,13 3,303 4,69 5,302 14,532 20,628 30,299 183,1080 265,301 298,381 325,383 343,473 394,472

X(622) = anticomplement of X(16)

X(623) = COMPLEMENT OF X(15)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), wheref(a,b,c) = [2*(b2 + c2)*area + sqr(3)*(SA*a2 + 2*SB*SC)]/a Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(623) lies on these lines: 2,14 3,629 5,141 13,298 16,302 17,633 18,83 30,618 396,533

X(623) = inverse of X(624) in the nine-point circleX(623) = complement of X(15)

X(624) = COMPLEMENT OF X(16)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), wheref(a,b,c) = [2*(b2 + c2)*area - sqr(3)*(SA*a2 + 2*SB*SC)]/a Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(624) lies on these lines: 2,13 3,630 5,141 14,299 15,303 17,83 30,619 395,532

X(624) = inverse of X(623) in the nine-point circleX(624) = complement of X(16)

X(625) = MIDPOINT OF X(623) AND X(624)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = [2*(b4 + c4 - b2c2) - a2(b2 + c2)]/a Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(625) lies on these lines: 2,187 5,141 30,620 115,325 126,858 230,754

X(625) = inverse of X(141) in the nine-point circleX(625) = complement of X(187)

X(626) = COMPLEMENT OF X(32)

Trilinears (b4 + c4)/a : (c4 + a4)/a : (a4 + b4)/a Barycentrics b4 + c4 : c4 + a4 : a4 + b4

X(626) lies on these lines: 2,32 3,114 5,141 10,760 37,746 39,325 76,115 316,384

X(626) = complement of X(32)

X(627) = ANTICOMPLEMENT OF X(17)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = [SB*SC + 2*SA*(a2 + sqr(3)*area)]/a Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(627) lies on these lines: 2,17 3,298 4,616 5,302 16,635 20,621 54,69 61,618 140,299

X(627) = anticomplement of X(17)

X(628) = ANTICOMPLEMENT OF X(18)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = [SB*SC + 2*SA*(a2 - sqr(3)*area)]/a Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(628) lies on these lines: 2,18 3,299 4,617 5,303 15,636 20,622 54,69 62,619 140,298

X(628) = anticomplement of X(18)

X(629) = COMPLEMENT OF X(17)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), wheref(a,b,c) = [6*SB*SC + 7*SA*a2 + 2*sqr(3)*(b2 + c2)*area]/a Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(629) lies on these lines: 2,17 3,623 5,618 61,302 140,619 141,575

X(629) = complement of X(17)

X(630) = COMPLEMENT OF X(18)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), wheref(a,b,c) = [6*SB*SC + 7*SA*a2 - 2*sqr(3)*(b2 + c2)*area]/a Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(630) lies on these lines: 2,18 3,624 5,619 62,303 140,618 141,575

X(630) = complement of X(18)

X(631) = (3/5)OG

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (SB*SC + 2*SA*a2)/a Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(631) lies on these lines: 1,1000 2,3 10,944 35,497 36,388 54,69 55,1058 56,1056 98,620 104,958 171,602 216,1075 238,601 315,1007 390,496 487,492 488,491 572,966 978,1064

X(632) = (9/10)OG

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (6*SB*SC + 7*SA*a2)/a Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(632) lies on these lines: 2,3 141,575

X(633) = ANTICOMPLEMENT OF X(61)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (SB*SC + 2*sqr(3)*SA*area)/a Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(633) lies on these lines: 2,18 3,298 4,69 5,299 14,636 17,623 20,616 140,302 141,398 343,471 394,470 397,524

X(633) = anticomplement of X(61)

X(634) = ANTICOMPLEMENT OF X(62)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (SB*SC - 2*sqr(3)*SA*area)/a Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(634) lies on these lines: 2,17 3,299 4,69 5,298 13,635 18,624 20,617 140,303 141,397 343,470 394,471 398,524

X(634) = anticomplement of X(62)

X(635) = COMPLEMENT OF X(61)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = [2*SB*SC + SA*a2 + 2*sqr(3)*(b2 + c2)*area]/a Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(635) lies on these lines: 2,18 3,618 5,141 13,634 16,627 17,299 62,298 140,619 397,532

X(635) = complement of X(61)

X(636) = COMPLEMENT OF X(62)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = [2*SB*SC + SA*a2 - 2*sqr(3)*(b2 + c2)*area]/a Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(636) lies on these lines: 2,17 3,619 5,141 14,633 15,628 18,298 61,299 140,618 398,533

X(636) = complement of X(62)

X(637) = ANTICOMPLEMENT OF X(371)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (SB*SC + 2*SA*area)/a Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(637) lies on these lines: 2,371 3,489 4,69 5,491 20,488 30,490 591,615

X(637) = anticomplement of X(371)

X(638) = ANTICOMPLEMENT OF X(372)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (SB*SC - 2*SA*area)/a Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(638) lies on these lines: 2,372 3,490 4,69 5,492 20,487 30,489

X(638) = anticomplement of X(372)

X(639) = COMPLEMENT OF X(371)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = [2*SB*SC + SA*a2 + 2*(b2 + c2)*area]/a Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(639) lies on these lines: 2,371 3,641 5,141 32,615 69,485 315,372

X(639) = complement of X(371)

X(640) = COMPLEMENT OF X(372)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = [2*SB*SC + SA*a2 - 2*(b2 + c2)*area]/a Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(640) lies on these lines: 2,372 3,642 5,141 69,486 315,371

X(640) = complement of X(372)

X(641) = COMPLEMENT OF X(485)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = [2*SB*SC + 3*SA*a2 + 2*(b2 + c2)*area]/a Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(641) lies on these lines: 2,372 3,639 39,615 140,141 371,492

X(641) = complement of X(485)

X(642) = COMPLEMENT OF X(486)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = [2*SB*SC + 3*SA*a2 - 2*(b2 + c2)*area]/a Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(642) lies on these lines: 2,371 3,640 140,141 372,491

X(642) = complement of X(486)

X(643) = TRILINEAR MULTIPLIER FOR KIEPERT PARABOLA

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c - a)/(b2 - c2) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(643) satisfies the equation X*(incircle) = Kiepert parabola, where * denotes trilinear multiplication, defined by(u : v : w) * (x : y : z) = ux : vy : wz.

X(643) lies on these lines: 8,1098 99,109 100,110 101,931 162,190 163,1018 212,312 283,1043

X(644) = TRILINEAR MULTIPLIER FOR YFF PARABOLA

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c - a)/(b - c) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(644) satisfies the equation X*(incircle) = Yff parabola, where * denotes trilinear multiplication, defined by(u : v : w) * (x : y : z) = ux : vy : wz.

X(644) lies on these lines: 8,220 78,728 100,101 105,1083 145,218 190,651 219,346 645,646 666,668 813,932 934,1025

X(645) = BARYCENTRIC MULTIPLIER FOR KIEPERT PARABOLA

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c - a)/[a(b2 - c2)] Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = (b + c - a)/(b2 - c2)

X(645) satisfies the equation X*(incircle) = Kiepert parabola, where * denotes barycentric multiplication, defined by(u : v : w) * (x : y : z) = ux : vy : wz (barycentric coordinates; see note at X(2)).

X(645) lies on these lines: 9,261 99,101 100,931 294,314 644,646 648,668 651,799 666,670

X(646) = BARYCENTRIC MULTIPLIER FOR YFF PARABOLA

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c - a)/[a2(b - c)] Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = (b + c - a)/[a(b - c)]

X(646) satisfies the equation X*(incircle) = Yff parabola, where * denotes barycentric multiplication, defined by(u : v : w) * (x : y : z) = ux : vy : wz (barycentric coordinates, see note at X(2)).

X(646) lies on these lines: 190,668,646 644,646

X(647) = ISOGONAL CONJUGATE OF TRILINEAR POLE OF EULER LINE

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b2 - c2)(b2 + c2 - a2)= u(A,B,C) : u(B,C,A) : u(C,A,B), where u(A,B,C) = sin 2A sin(B - C) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(647) is the point whose trilinears are coefficients for the Euler line.

X(647) lies on these lines: 1,1021 2,850 50,654 111,842 184,878 187,237 230,231 441,525 520,652

X(647) = isogonal conjugate of X(648) X(647) = complement of X(850)

X(648) = TRILINEAR POLE OF EULER LINE

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[a(b2 - c2)(b2 + c2 - a2)]= u(A,B,C) : u(B,C,A) : u(C,A,B), where u(A,B,C) = csc 2A csc(B - C) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(648) is constructed as the pole of the Euler line L as follows: let A", B", C" be the points where L meets the sidelines BC, CA, AB of the reference triangle ABC. Let A', B', C' be the harmonic conjugates of A", B", C" with respect to {B,C}, {C,A}, {A,B}, respectively, The lines AA', BB', CC' concur in X(648).

X(648) lies on these lines:4,452 6,264 27,903 94,275 95,216 99,112 107,110 108,931 132,147 155,1093 162,190 185,1105 193,317 232,385 249,687 250,523 297,340 447,519 645,668 651,823 653,662 925,933 1020,1021 1075,1092

X(648) = isogonal conjugate of X(647)X(648) = isotomic conjugate of X(525)

X(649) = ISOGONAL CONJUGATE OF TRILINEAR POLE OF LINE X(1)X(2)

Trilinears a(b - c) : b(c - a) : c(a - b)

Barycentrics a2(b - c) : b2(c - a) : c2(a - b)

X(649) lies on these lines:31,884 42,788 44,513 57,1024 89,1022 100,660 101,901 109,919 187,237 190,889 239,514 693,812

X(649) = isogonal conjugate of X(650)

X(650) = ISOGONAL CONJUGATE OF TRILINEAR POLE OF LINE X(1)X(3)

Trilinears cos B - cos C : cos C - cos A : cos A - cos B= (b - c)(b + c - a) : (c - a)(c + a - b) : (a - b)(a + b - c) Barycentrics sin A (cos B - cos C) : sin B (cos C - cos A) : sin C(cos A - cos B)= a(b - c)(b + c - a) : b(c - a)(c + a - b) : c(a - b)(a + b - c)

X(650) lies on these lines:2,693 44,513 55,884 100,919 230,231 241,514 521,1021 663,861

X(650) = isogonal conjugate of X(651)X(650) = complement of X(693)

X(651) = TRILINEAR POLE OF LINE X(1)X(3)

Trilinears 1/(cos B - cos C) : 1/(cos C - cos A) : 1/(cos A - cos B)= 1/[(b - c)(b + c - a)] : 1/[(c - a)(c + a - b)] : 1[(a - b)(a + b - c)] Barycentrics (sin A)/(cos B - cos C) : (sin B)/(cos C - cos A) : (sin C)/(cos A - cos B)= a/[(b - c)(b + c - a)] : b/[(c - a)(c + a - b)] : c/[(a - b)(a + b - c)]

X(651) lies on these lines:2,222 6,7 8,221 9,77 21,73 44,241 57,88 59,513 63,223 65,895 69,478 81,226 100,109 101,934 108,110 144,219 155,1068 190,644 193,608 218,279 255,411 287,894 329,394 404,603 500,943 514,655 645,799 648,823 978,1106

X(651) = isogonal conjugate of X(650)

X(652) = ISOGONAL CONJUGATE OF TRILINEAR POLE OF LINE X(1)X(4)

Trilinears sec B - sec C : sec C - sec A : sec A - sec B Barycentrics sin A (sec B - sec C) : sin B (sec C - sec A) : sin C(sec A - sec B)

X(652) lies on these lines: 44,513 243,522 520,647

X(652) = isogonal conjugate of X(653)

X(653) = TRILINEAR POLE OF LINE X(1)X(4)

Trilinears 1/(sec B - sec C) : 1/(sec C - sec A) : 1/(sec A - sec B)Barycentrics (sin A)/(sec B - sec C) : (sin B)/(sec C - sec A) : (sin C)/(sec A - sec B)

X(653) lies on these lines:2,196 7,281 9,342 19,273 29,65 46,158 57,92 78,207 88,278 100,108 107,109 208,318 225,897 648,662

X(653) = isogonal conjugate of X(652)

X(654) = ISOGONAL CONJUGATE OF TRILINEAR POLE OF LINE X(1)X(5)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = cos(A - B) - cos(C - A) Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = sin A f(A,B,C)

X(654) lies on these lines: 44,513 50,647 55,926 63,918 101,109

X(654) = isogonal conjugate of X(655)

X(655) = TRILINEAR POLE OF LINE X(1)X(5)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = 1/[cos(A - B) - cos(C - A)] Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = sin A f(A,B,C)

X(655) lies on these lines: 59,523 80,516 100,522 514,651

X(655) = isogonal conjugate of X(654)

X(656) = ISOGONAL CONJUGATE OF TRILINEAR POLE OF LINE X(1)X(19)

Trilinears tan B - tan C : tan C - tan A : tan A - tan B Barycentrics sin A (tan B - tan C) : sin B (tan C - tan A) : sin C (tan A - tan B)

X(656) lies on these lines: 44,513 240,522 521,810 662,1101 667,832

X(656) = isogonal conjugate of X(162)X(656) = isotomic conjugate of X(811)

X(657) = ISOGONAL CONJUGATE OF TRILINEAR POLE OF LINE X(1)X(30)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (1 + cos A)(cos B - cos C) Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = sin A f(A,B,C)

X(657) lies on these lines: 9,522 44,513 59,101 663,853

X(657) = isogonal conjugate of X(658)

X(658) = TRILINEAR POLE OF LINE X(1)X(30)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = 1/[(1 + cos A)(cos B - cos C)] Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = sin A f(A,B,C)

X(658) lies on these lines: 7,11 57,673 88,279 100,664 109,927 190,1020

X(658) = isogonal conjugate of X(657)

X(659) = ISOGONAL CONJUGATE OF TRILINEAR POLE OF LINE X(1)X(39)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (a2 - bc)(b - c) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(659) lies on these lines: 1,891 23,385 44,513 100,190 105,884 291,875 292,665 514,667

X(659) = isogonal conjugate of X(660)

X(660) = TRILINEAR POLE OF LINE X(1)X(39)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[(a2 - bc)(b - c)] Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(660) lies on these lines: 44,292 88,291 100,649 190,513 239,335 320,334 512,1016 662,765

X(660) = isogonal conjugate of X(659)

X(661) = ISOGONAL CONJUGATE OF TRILINEAR POLE OF LINE X(1)X(63)

Trilinears cot B - cot C : cot C - cot A : cot A - cot B Barycentrics sin A (cot B - cot C) : sin B (cot C - cot A) : sin C (cot A - cot B)

X(661) lies on these lines: 44,513 514,693 663,810

X(661) = isogonal conjugate of X(662)X(661) = isotomic conjugate of X(799)

X(662) = TRILINEAR POLE OF LINE X(1)X(63)

Trilinears 1/(cot B - cot C) : 1/(cot C - cot A) : 1/(cot A - cot B) Barycentrics (sin A)/(cot B - cot C) : (sin B)/(cot C - cot A) : (sin C)/(cot A - cot B)

X(662) lies on these lines:1,897 3,1098 6,757 27,913 48,75 60,404 81,88 86,142 99,101 100,110 109,931 214,759 243,425 261,572 326,610 333,909 648,653 656,1101 660,765 689,787 775,820 811,823 827,831

X(662) = isogonal conjugate of X(661)

X(663) = ISOGONAL CONJUGATE OF TRILINEAR POLE OF LINE X(2)X(7)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b - c)(b + c - a) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(663) lies on these lines: 1,514 41,884 101,919 106,840 187,237 513,855 650,861 657,853 661,810

X(663) = isogonal conjugate of X(664)

X(664) = TRILINEAR POLE OF LINE X(2)X(7)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[a(b - c)(b + c - a)] Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(664) lies on these lines:1,85 7,528 8,348 69,347 73,290 75,77 99,109 100,658 101,514 145,279 150,952 175,490 176,489 190,644 223,312 226,671 239,241 307,319 322,326 648,653 668,1026 1018,1025

X(664) = isogonal conjugate of X(663)X(664) = isotomic conjugate of X(522)

X(665) = ISOGONAL CONJUGATE OF TRILINEAR POLE OF LINE X(2)X(11)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a[(a - b)2(a + b - c) - (c - a)2(c + a - b)] Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(665) lies on these lines: 37,900 101,109 187,237 241,514 244,866 292,659 743,761

X(665) = isogonal conjugate of X(666)

X(666) = TRILINEAR POLE OF LINE X(2)X(11)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc/[a - b)2(a + b - c) - (c - a)2(c + a - b)] Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(666) lies on these lines: 99,919 101,514 105,898 190,522 239,294 527,673 644,668 645,670 1026,1027

X(666) = isogonal conjugate of X(665)X(666) = isotomic conjugate of X(918)

X(667) = ISOGONAL CONJUGATE OF TRILINEAR POLE OF LINE X(2)X(37)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(b - c) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(667) lies on these lines: 3,1083 36,238 56,764 100,898 101,813 187,237 213,875 514,659 656,832 668,932 692,1110 788,798

X(667) = isogonal conjugate of X(668)X(667) = inverse of X(1083) in the circumcircle

X(668) = TRILINEAR POLE OF LINE X(2)X(37)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[a2(b - c)] Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(668) lies on these lines:2,1015 8,76 10,274 69,150 72,290 75,537 80,313 99,100 101,789 110,839 190,646 304,341 321,671 350,519 513,889 644,666 645,648 664,1026 667,932

X(668) = isogonal conjugate of X(667)X(668) = isotomic conjugate of X(513)X(668) = anticomplement of X(1015)

X(669) = ISOGONAL CONJUGATE OF TRILINEAR POLE OF LINE X(2)X(39)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a3(b2 - c2) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(669) lies on these lines: 23,385 25,878 31,875 99,886 110,805 187,237 684,924 688,864 804,850

X(669) = isogonal conjugate of X(670)

X(670) = TRILINEAR POLE OF LINE X(2)X(39)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[a3(b2 - c2)] Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(670) lies on these lines: 2,1084 69,290 76,338 99,804 110,689 141,308 190,799 310,903 512,886 645,666 850,892

X(670) = isogonal conjugate of X(669)X(670) = isotomic conjugate of X(512)X(670) = anticomplement of X(1084)

X(671) = TRILINEAR POLE OF LINE X(2)X(99)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[a(2a2 - b2 - c2)] Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(671) lies on these lines:2,99 4,542 6,598 10,190 13,531 14,530 30,98 76,338 83,597 226,664 262,381 316,524 321,668 485,489 486,490

X(671) = isogonal conjugate of X(187)X(671) = isotomic conjugate of X(524)

X(672) = ISOGONAL CONJUGATE OF TRILINEAR POLE OF LINE X(2)X(100)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a[b2 + c2 - a(b + c)] Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(672) lies on these lines:1,1002 2,7 3,41 6,31 36,101 37,38 39,213 43,165 44,513 46,169 56,220 72,1009 103,919 105,238 190,350 219,604 519,1018

X(672) = isogonal conjugate of X(673)

X(673) = TRILINEAR POLE OF LINE X(2)X(100)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc/[b2 + c2 - a(b + c)] Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(673) lies on these lines:2,11 6,7 9,75 19,273 27,162 57,658 86,142 238,516 239,335 310,333 527,666 675,919 812,1024 885,900

X(673) = isogonal conjugate of X(672)

X(674) = ISOGONAL CONJUGATE OF TRILINEAR POLE OF LINE X(2)X(101)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a[b3 + c3 - a(b2 + c2)] Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

As the isogonal conjugate of a point on the circumcircle, X(674) lies on the line at infinity.

X(674) lies on these lines: 6,31 30,511 51,210

X(674) = isogonal conjugate of X(675)

X(675) = TRILINEAR POLE OF LINE X(2)X(101)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc/[b3 + c3 - a(b2 + c2)] Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(675) lies on the circumcircle.

X(675) lies on these lines:2,101 7,109 27,112 75,100 86,110 99,310 108,273 335,813 673,919 789,871 901,903 934,1088

X(675) = isogonal conjugate of X(674)

X(676) = ISOGONAL CONJUGATE OF TRILINEAR POLE OF LINE X(3)X(101)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc(b - c)[b3 + c3 - a3 + (b + c)(a2 - bc)] Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(676) lies on this line: 514,551

X(676) = isogonal conjugate of X(677)

X(677) = TRILINEAR POLE OF LINE X(3)X(101)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a/{(b - c)[b3 + c3 - a3 + (b + c)(a2 - bc)]} Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(677) = isogonal conjugate of X(676)

X(678)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c - 2a)2 Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(678) lies on these lines: 1,88 44,902 45,55

X(678) = isogonal conjugate of X(679)

X(679)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/(b + c - 2a)2 Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(679) lies on these lines: 44,88 320,519

X(679) = isogonal conjugate of X(678)

X(680) = ISOGONAL CONJUGATE OF TRILINEAR POLE OF LINE X(4)X(9)

Trilinears sin B sec2C - sin C sec2B : sin C sec2A - sin A sec2C : sin A sec2B - sin B sec2ABarycentrics csc C sec2C - csc B sec2B : csc A sec2A - csc C sec2C : csc B sec2B - csc A sec2A

As the isogonal conjugate of a point on the circumcircle, X(680) lies on the line at infinity.

X(680) lies on this line: 30,511

X(680) = isogonal conjugate of X(681)

X(681) = TRILINEAR POLE OF LINE X(4)X(9)

Trilinears f(A,B,C): f(B,C,A) : f(C,A,B), where f(A,B,C) = cos2A (sin B cos2B - sin C cos2C) Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = sin A f(A,B,C)

X(681) lies on the circumcircle.

X(681) lies on this line: 110,823

X(681) = isogonal conjugate of X(680)

X(682) = ISOGONAL CONJUGATE OF TRILINEAR POLE OF LINE X(4)X(76)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = sec B csc3C + sec C csc3B Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = sin A f(A,B,C)

X(682) lies on these lines: 3,69 154,237 248,695

X(682) = isogonal conjugate of X(683)

X(683) = TRILINEAR POLE OF LINE X(4)X(76)

Trilinears f(A,B,C): f(B,C,A) : f(C,A,B), where f(A,B,C) = 1/[sec B csc3C + sec C csc3B]

Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = sin A f(A,B,C)

X(683) lies on this line: 25,305

X(683) = isogonal conjugate of X(682)

X(684) = ISOGONAL CONJUGATE OF TRILINEAR POLE OF LINE X(4)X(98)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = sec B sin3 C - sec C sin3 B Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = sin A f(A,B,C)

X(684) lies on these lines: 110,351 114,132 122,125 147,804 325,523 520,647 669,924

X(684) = isogonal conjugate of X(685)

X(685) = TRILINEAR POLE OF LINE X(4)X(98)

Trilinears f(A,B,C): f(B,C,A) : f(C,A,B), where f(A,B,C) = 1/(sec B sin3 C - sec C sin3 B)

Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = sin A f(A,B,C)

X(685) lies on these lines: 98,468 110,850 250,523 287,297

X(685) = isogonal conjugate of X(684)

X(686) = ISOGONAL CONJUGATE OF TRILINEAR POLE OF LINE X(4)X(110)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = sec B csc(A - B) + sec C csc(A - C) Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = sin A f(A,B,C)

X(686) lies on these lines: 115,125 184,351 520,647

X(686) = isogonal conjugate of X(687)

X(687) = TRILINEAR POLE OF LINE X(4)X(110)

Trilinears f(A,B,C): f(B,C,A) : f(C,A,B), where f(A,B,C) = 1/[sec B csc(A - B) + sec C csc(A - C)] Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = sin A f(A,B,C)

X(687) lies on these lines: 107,250 249,648

X(687) = isogonal conjugate of X(686)

X(688) = ISOGONAL CONJUGATE OF TRILINEAR POLE OF LINE X(6)X(99)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a3(b4 - c4) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

As the isogonal conjugate of a point on the circumcircle, X(688) lies on the line at infinity.

X(688) lies on this line: 6,882 30,511 669,864 798,872

X(688) = isogonal conjugate of X(689)

X(689) = TRILINEAR POLE OF LINE X(6)X(99)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[a3(b4 - c4)] Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(689) lies on the circumcircle.

X(689) lies on these lines:1,719 2,733 6,703 75,745 76,755 82,715 83,729 110,670 111,308 251,699 662,787 741,873 799,813

X(689) = isogonal conjugate of X(688)

X(690) = ISOGONAL CONJUGATE OF TRILINEAR POLE OF LINE X(6)X(110)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc(b2 - c2)(2a2 - b2 - c2) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

As the isogonal conjugate of a point on the circumcircle, X(690) lies on the line at infinity.

X(690) lies on these lines: 30,511 74,98 99,110 113,114 115,125 146,147

X(690) = isogonal conjugate of X(691)X(690) = isotomic conjugate of X(892)

X(691) = TRILINEAR POLE OF LINE X(6)X(110)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a/[(b2 - c2)(2a2 - b2 - c2)] Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(691) lies on the circumcircle.

X(691) lies on these lines: 3,842 6,843 23,111 30,98 74,511 99,523 110,249 112,250 316,858 376,477 741,923 759,897 805,882

X(691) = isogonal conjugate of X(690)

X(692)

Trilinears a2/(b - c) : b2/(c - a) c2/(a - b)Barycentrics a3/(b - c) : b3/(c - a) c3/(a - b)

X(692) lies on these lines: 25,913 48,911 55,184 59,513 99,785 100,110 101,926 154,197 163,906 182,1001 206,219 213,923 667,1110 813,825

X(692) = isogonal conjugate of X(693)

X(693)

Trilinears (b - c)/a2 : (c - a)/b2 : (a - b)/c2

Barycentrics (b - c)/a : (c - a)/b : (a - b)/c

X(693) lies on these lines: 2,650 76,764 100,927 320,350 321,824 325,523 514,661 649,812

X(693) = isogonal conjugate of X(692)X(693) = isotomic conjugate of X(100)X(693) = anticomplement of X(650)

X(694) = ISOGONAL CONJUGATE OF X(385)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a/(a4 - b2c2) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(694) lies on these lines: 6,1084 37,256 42,893 110,251 111,805 141,308 172,904 257,335 351,881 384,695 882,888

X(694) = isogonal conjugate of X(385)

X(695) = ISOGONAL CONJUGATE OF X(384)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a/(a4 + b2c2) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(695) lies on these lines: 69,194 99,711 248,682 384,694

X(695) = isogonal conjugate of X(384)

X(696) = EVEN (- 4, - 3) INFINITY POINT

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a-5(b-3 + c-3) - a-4(b-4 + c-4) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

As the isogonal conjugate of a point on the circumcircle, X(696) lies on the line at infinity. The first trilinear coordinate has the form

am-1(bn + cn) - an-1(bm + cm).

If m and n are distinct integers, this form fits the definition of even polynomial center as in Clark Kimberling, "Functional equations associated with triangle geometry," Aequationes Mathematicae 45 (1993) 127-152. This form, perhaps appearing initially here (July 7, 2001) defines a triangle center for arbitrary distinct real numbers m and n. Selected even infinity and circumcircle points begin at X(696); odd ones begin at X(768).

Certain points of this type occur prior to this section. They are as follows:

X(538) = even (- 2, 0) infinity pointX(536) = even (- 1, 0) infinity pointX(519) = even (0, 1) infinity pointX(106) = even (0, 1) circumcircle pointX(524) = even (0, 2) infinity pointX(111) = even (0, 2) circumcircle pointX(518) = even (1, 2) infinity pointX(105) = even (1, 2) circumcircle pointX(674) = even (2, 3) infinity pointX(675) = even (2, 3) circumcircle pointX(511) = even (2, 4) infinity pointX(98) = even (2, 4) circumcircle point

X(696) lies on these lines: 30,511 313,561

X(696) = isogonal conjugate of X(697)

X(697) = EVEN (- 4, - 3) CIRCUMCIRCLE POINT

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[a-5(b-3 + c-3) - a-4(b-4 + c-4)] Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(697) lies on the circumcircle. This is one of several points of the form given by first trilinear

1/[am-1(bn + cn) - an-1(bm + cm)],

hence the name "(m, n)-circumcircle point".

X(697) lies on this line: 100,560

X(697) = isogonal conjugate of X(696)

X(698) = EVEN (- 4, - 2) INFINITY POINT

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a-5(b-2 + c-2) - a-3(b-4 + c-4) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

As the isogonal conjugate of a point on the circumcircle, X(698) lies on the line at infinity.

X(698) lies on these lines: 6,194 30,511 75,257 76,141

X(698) = isogonal conjugate of X(699)

X(699) = EVEN (- 4, - 2) CIRCUMCIRCLE POINT

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[a-5(b-2 + c-2) - a-3(b-4 + c-4)] Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(699) lies on the circumcircle.

X(699) lies on these lines: 32,99 172,932 251,689

X(699) = isogonal conjugate of X(698)

X(700) = EVEN (- 4, - 1) INFINITY POINT

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a-5(b-1 + c-1) - a-2(b-4 + c-4) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

As the isogonal conjugate of a point on the circumcircle, X(700) lies on the line at infinity.

X(700) lies on this line: 30,511 75,871

X(700) = isogonal conjugate of X(701)

X(701) = EVEN (- 4, - 1) CIRCUMCIRCLE POINT

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[a-5(b-1 + c-1) - a-2(b-4 + c-4)] Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(701) lies on the circumcircle.

X(701) lies on this line: 31,789

X(701) = isogonal conjugate of X(700)

X(702) = EVEN (- 4, 0) INFINITY POINT

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a-5(b0 + c0) - a-1(b-4 + c-4) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

As the isogonal conjugate of a point on the circumcircle, X(702) lies on the line at infinity.

X(702) lies on these lines: 2,308 30,511

X(702) = isogonal conjugate of X(703)

X(703) = EVEN (- 4, 0) CIRCUMCIRCLE POINT

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[a-5(b0 + c0) - a-1(b-4 + c-4)] Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(703) lies on the circumcircle.

X(703) lies on this line: 6,689

X(703) = isogonal conjugate of X(702)

X(704) = EVEN (- 4, 1) INFINITY POINT

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a-5(b1 + c1) - a0(b-4 + c-4) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

As the isogonal conjugate of a point on the circumcircle, X(704) lies on the line at infinity.

X(704) lies on this line: 30,511

X(704) = isogonal conjugate of X(705)

X(705) = EVEN (- 4, 1) CIRCUMCIRCLE POINT

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[a-5(b1 + c1) - a0(b-4 + c-4)] Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(705) lies on the circumcircle.

X(705) = isogonal conjugate of X(704)

X(706) = EVEN (- 4, 2) INFINITY POINT

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a-5(b2 + c2) - a1(b-4 + c-4) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

As the isogonal conjugate of a point on the circumcircle, X(706) lies on the line at infinity.

X(706) lies on this line: 30,511

X(706) = isogonal conjugate of X(707)

X(707) = EVEN (- 4, 2) CIRCUMCIRCLE POINT

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[a-5(b2 + c2) - a1(b-4 + c-4)] Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(707) lies on the circumcircle.

X(707) = isogonal conjugate of X(706)

X(708) = EVEN (- 4, 3) INFINITY POINT

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a-5(b3 + c3) - a2(b-4 + c-4) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

As the isogonal conjugate of a point on the circumcircle, X(708) lies on the line at infinity.

X(708) lies on this line: 30,511

X(708) = isogonal conjugate of X(709)

X(709) = EVEN (- 4, 3) CIRCUMCIRCLE POINT

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[a-5(b3 + c3) - a2(b-4 + c-4)] Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(709) lies on the circumcircle.

X(709) = isogonal conjugate of X(708)

X(710) = EVEN (- 4, 4) INFINITY POINT

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a-5(b4 + c4) - a3(b-4 + c-4) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

As the isogonal conjugate of a point on the circumcircle, X(710) lies on the line at infinity.

X(710) lies on this line: 30,511

X(710) = isogonal conjugate of X(711)

X(711) = EVEN (- 4, 4) CIRCUMCIRCLE POINT

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[a-5(b4 + c4) - a3(b-4 + c-4)] Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(711) lies on the circumcircle.

X(711) lies on this line: 99,695

X(711) = isogonal conjugate of X(710)

X(712) = EVEN (- 3, - 2) INFINITY POINT

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a-4(b-2 + c-2) - a-3(b-3 + c-3) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

As the isogonal conjugate of a point on the circumcircle, X(712) lies on the line at infinity.

X(712) lies on these lines: 30,511 76,321

X(712) = isogonal conjugate of X(713)

X(713) = EVEN (- 3, - 2) CIRCUMCIRCLE POINT

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[a-4(b-2 + c-2) - a-3(b-3 + c-3)] Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(713) lies on the circumcircle.

X(713) lies on these lines: 32,100 101,560

X(713) = isogonal conjugate of X(712)

X(714) = EVEN (- 3, - 1) INFINITY POINT

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a-4(b-1 + c-1) - a-2(b-3 + c-3) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

As the isogonal conjugate of a point on the circumcircle, X(714) lies on the line at infinity.

X(714) lies on these lines: 30,511 38,75

X(714) = isogonal conjugate of X(715)

X(715) = EVEN (- 3, - 1) CIRCUMCIRCLE POINT

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[a-4(b-1 + c-1) - a-2(b-3 + c-3)] Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(715) lies on the circumcircle.

X(715) lies on these lines: 31,99 81,932 82,689 110,560

X(715) = isogonal conjugate of X(714)

X(716) = EVEN (- 3, 0) INFINITY POINT

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a-4(b0 + c0) - a-1(b-3 + c-3) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

As the isogonal conjugate of a point on the circumcircle, X(716) lies on the line at infinity.

X(716) lies on these lines: 2,561 30,511

X(716) = isogonal conjugate of X(717)

X(717) = EVEN (- 3, 0) CIRCUMCIRCLE POINT

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[a-4(b0 + c0) - a-1(b-3 + c-3)] Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(717) lies on the circumcircle.

X(717) lies on thse lines: 6,789 560,825

X(717) = isogonal conjugate of X(716)

X(718) = EVEN (- 3, 1) INFINITY POINT

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a-4(b1 + c1) - a0(b-3 + c-3) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

As the isogonal conjugate of a point on the circumcircle, X(718) lies on the line at infinity.

X(718) lies on these lines: 1,561 30,511

X(718) = isogonal conjugate of X(717)

X(719) = EVEN (- 3, 1) CIRCUMCIRCLE POINT

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[a-4(b1 + c1) - a0(b-3 + c-3)] Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(719) lies on the circumcircle.

X(719) lies on this line: 1,689 560,827

X(719) = isogonal conjugate of X(718)

X(720) = EVEN (- 3, 2) INFINITY POINT

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a-4(b2 + c2) - a1(b-3 + c-3) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

As the isogonal conjugate of a point on the circumcircle, X(720) lies on the line at infinity.

X(720) lies on these lines: 6,561 30,511

X(720) = isogonal conjugate of X(721)

X(721) = EVEN (- 3, 0) CIRCUMCIRCLE POINT

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[a-4(b2 + c2) - a1(b-3 + c-3)] Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(721) lies on the circumcircle.

X(721) = isogonal conjugate of X(720)

X(722) = EVEN (- 3, 3) INFINITY POINT

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a-4(b3 + c3) - a2(b-3 + c-3) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

As the isogonal conjugate of a point on the circumcircle, X(722) lies on the line at infinity.

X(722) lies on this line: 30,511

X(722) = isogonal conjugate of X(723)

X(723) = EVEN (- 3, 3) CIRCUMCIRCLE POINT

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[a-4(b3 + c3) - a2(b-3 + c-3)] Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(723) lies on the circumcircle.

X(723) = isogonal conjugate of X(722)

X(724) = EVEN (- 3, 4) INFINITY POINT

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a-4(b4 + c4) - a3(b-3 + c-3) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

As the isogonal conjugate of a point on the circumcircle, X(724) lies on the line at infinity.

X(724) lies on this line: 30,511

X(724) = isogonal conjugate of X(725)

X(725) = EVEN (- 3, 4) CIRCUMCIRCLE POINT

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[a-4(b4 + c4) - a3(b-3 + c-3)] Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(725) lies on the circumcircle.

X(725) = isogonal conjugate of X(724)

X(726) = EVEN (- 2, -1) INFINITY POINT

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a-3(b-1 + c-1) - a-2(b-2 + c-2) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

As the isogonal conjugate of a point on the circumcircle, X(726) lies on the line at infinity.

X(726) lies on these lines: 1,87 10,75 30,511 37,39 38,321 190,238 291,350 312,982

X(726) = isogonal conjugate of X(727)

X(727) = EVEN (- 2, -1) CIRCUMCIRCLE POINT

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[a-3(b-1 + c-1) - a-2(b-2 + c-2)] Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(727) lies on the circumcircle.

X(727) lies on these lines: 1,932 31,43 32,101 58,99 789,985 934,1106

X(727) = isogonal conjugate of X(728)

X(728)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c - a)3 Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(728) lies on these lines: 8,9 40,1018 57,345 78,644 200,220

X(728) = isogonal conjugate of X(738)

X(729) = EVEN (- 2, 0) CIRCUMCIRCLE POINT

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[a-3(b0 + c0) - a-1(b-2 + c-2)] Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(729) lies on the circumcircle.

X(729) lies on these lines: 6,99 32,110 83,689 100,213 187,805

X(729) = isogonal conjugate of X(538)

X(730) = EVEN (- 2, 1) INFINITY POINT

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a-3(b1 + c1) - a0(b-2 + c-2) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

As the isogonal conjugate of a point on the circumcircle, X(730) lies on the line at infinity.

X(730) lies on these lines: 1,76 8,194 10,39 30,511

X(730) = isogonal conjugate of X(727)

X(731) = EVEN (- 2, 1) CIRCUMCIRCLE POINT

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[a-3(b1 + c1) - a0(b-2 + c-2)] Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(731) lies on the circumcircle.

X(731) lies on these lines: 1,789 32,825 100,869

X(731) = isogonal conjugate of X(730)

X(732) = EVEN (- 2, 2) INFINITY POINT

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a-3(b2 + c2) - a1(b-2 + c-2) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

As the isogonal conjugate of a point on the circumcircle, X(732) lies on the line at infinity.

X(732) lies on these lines: 6,76 30,511 39,141 69,194

X(732) = isogonal conjugate of X(733)

X(733) = EVEN (- 2, 2) CIRCUMCIRCLE POINT

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[a-3(b2 + c2) - a1(b-2 + c-2)] Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(733) lies on the circumcircle.

X(733) lies on these lines: 2,689 32,827 39,83 39,141 100,893 101,904 110,251 755,882

X(733) = isogonal conjugate of X(732)

X(734) = EVEN (- 2, 3) INFINITY POINT

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a-3(b3 + c3) - a2(b-2 + c-2) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

As the isogonal conjugate of a point on the circumcircle, X(733) lies on the line at infinity.

X(734) lies on these lines: 30,511 31,76

X(734) = isogonal conjugate of X(735)

X(735) = EVEN (- 2, 3) CIRCUMCIRCLE POINT

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[a-3(b3 + c3) - a2(b-2 + c-2)] Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(735) lies on the circumcircle.

X(735) = isogonal conjugate of X(734)

X(736) = EVEN (- 2, 4) INFINITY POINT

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a-3(b4 + c4) - a3(b-2 + c-2) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

As the isogonal conjugate of a point on the circumcircle, X(736) lies on the line at infinity.

X(736) lies on these lines: 30,511 32,76 39,325 194,315

X(736) = isogonal conjugate of X(737)

X(737) = EVEN (- 2, 4) CIRCUMCIRCLE POINT

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[a-3(b4 + c4) - a3(b-2 + c-2)] Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(737) lies on the circumcircle.

X(737) = isogonal conjugate of X(736)

X(738)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c - a)-3 Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(738) lies on these lines: 9,348 56,269 57,279 77,951

X(738) = isogonal conjugate of X(728)

X(739) = EVEN (- 1, 0) CIRCUMCIRCLE POINT

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[a-2(b0 + c0) - a-1(b-1 + c-1)] Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(739) lies on the circumcircle.

X(739) lies on these lines: 6,100 31,101 81,99 108,608 109,604 813,902

X(739) = isogonal conjugate of X(737)

X(740) = EVEN (- 1, 1) INFINITY POINT

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a-2(b1 + c1) - a0(b-1 + c-1) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

As the isogonal conjugate of a point on the circumcircle, X(740) lies on the line at infinity.

X(740) lies on these lines: 1,75 8,192 10,37 30,511 42,321 43,312 238,239 872,1089

X(740) = isogonal conjugate of X(741)

X(741) = EVEN (- 1, 1) CIRCUMCIRCLE POINT

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a/[a-2(b1 + c1) - a0(b-1 + c-1)] Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(741) lies on the circumcircle.

X(741) lies on these lines: 1,99 21,932 31,110 42,81 58,101 86,789 107,1096 334,839 335,835 689,873 691,923 759,876 827,849 934,1042

X(741) = isogonal conjugate of X(740)

X(742) = EVEN (- 1, 2) INFINITY POINT

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a-2(b2 + c2) - a1(b-1 + c-1) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

As the isogonal conjugate of a point on the circumcircle, X(742) lies on the line at infinity.

X(742) lies on these lines: 6,75 30,511 37,141 69,192 320,335

X(742) = isogonal conjugate of X(743)

X(743) = EVEN (- 1, 2) CIRCUMCIRCLE POINT

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a/[a-2(b2 + c2) - a1(b-1 + c-1)] Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(743) lies on the circumcircle.

X(743) lies on these lines: 2,789 31,825 101,869 665,761

X(743) = isogonal conjugate of X(742)

X(744) = EVEN (- 1, 3) INFINITY POINT

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a-2(b3 + c3) - a2(b-1 + c-1) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

As the isogonal conjugate of a point on the circumcircle, X(744) lies on the line at infinity.

X(744) lies on these lines: 30,511 31,75

X(744) = isogonal conjugate of X(745)

X(745) = EVEN (- 1, 3) CIRCUMCIRCLE POINT

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a/[a-2(b3 + c3) - a2(b-1 + c-1)] Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(745) lies on the circumcircle.

X(745) lies on these lines: 31,827 38,99 75,689

X(745) = isogonal conjugate of X(744)

X(746) = EVEN (- 1, 4) INFINITY POINT

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a-2(b4 + c4) - a3(b-1 + c-1) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

As the isogonal conjugate of a point on the circumcircle, X(746) lies on the line at infinity.

X(746) lies on these lines: 30,511 32,75 37,626 192,315

X(746) = isogonal conjugate of X(747)

X(747) = EVEN (- 1, 4) CIRCUMCIRCLE POINT

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a/[a-2(b4 + c4) - a3(b-1 + c-1)] Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(747) lies on the circumcircle.

X(747) = isogonal conjugate of X(746)

X(748)

Trilinears a2 - 2bc : b2 - 2ca : c2 - 2ab Barycentrics a3 - 2abc : b3 - 2abc : c3 - 2abc

X(748) lies on these lines: 1,756 2,31 5,602 9,38 11,212 21,978 42,1001 44,354 55,899 63,244 140,601 181,373 255,499 590,605 606,615

X(748) = isogonal conjugate of X(749)

X(749)

Trilinears 1/(a2 - 2bc) : 1/(b2 - 2ca) : 1/(c2 - 2ab) Barycentrics a/(a2 - 2bc) : b/(b2 - 2ca) : c/(c2 - 2ab)

X(749) = isogonal conjugate of X(748)

X(750)

Trilinears a2 + 2bc : b2 + 2ca : c2 + 2ab Barycentrics a3 + 2abc : b3 + 2abc : c3 + 2abc

X(750) lies on these lines:1,88 2,31 5,601 6,899 9,896 12,603 38,57 42,940 43,81 46,975 63,756 140,602 165,968 255,498 388,1106 590,606 605,615 902,1001 942,976

X(750) = isogonal conjugate of X(751)

X(751)

Trilinears 1/(a2 + 2bc) : 1/(b2 + 2ca) : 1/(c2 + 2ab) Barycentrics a/(a2 + 2bc) : b/(b2 + 2ca) : c/(c2 + 2ab)

X(751) = isogonal conjugate of X(750)

X(751) lies on this line: 519,984

X(752) = EVEN (0, 3) INFINITY POINT

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a-1(b3 + c3) - a2(b0 + c0) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

As the isogonal conjugate of a point on the circumcircle, X(752) lies on the line at infinity.

X(752) lies on these lines: 1,320 2,31 10,44 30,511

X(752) = isogonal conjugate of X(753)

X(753) = EVEN (0, 3) CIRCUMCIRCLE POINT

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a/[a-1(b3 + c3) - a2(b0 + c0)] Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(753) lies on the circumcircle.

X(753) lies on these lines: 6,825 75,789 100,984

X(753) = isogonal conjugate of X(752)

X(754) = EVEN (0, 4) INFINITY POINT

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a-1(b4 + c4) - a3(b0 + c0) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

As the isogonal conjugate of a point on the circumcircle, X(754) lies on the line at infinity.

X(754) lies on these lines: 2,32 30,511 115,316 187,325 230,625

X(754) = isogonal conjugate of X(755)

X(755) = EVEN (0, 4) CIRCUMCIRCLE POINT

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a/[a-1(b4 + c4) - a3(b0 + c0)] Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(755) lies on the circumcircle.

X(755) lies on these lines: 6,827 39,110 76,689 99,141 733,882

X(755) = isogonal conjugate of X(754)

X(756)

Trilinears (b + c)2 : (c + a)2 : (a + b)2 Barycentrics a(b + c)2 : b(c + a)2 : c(a + b)2

X(756) lies on these lines: 1,748 2,38 9,31 10,321 12,201 37,42 45,55 63,750 100,846 171,896 200,968 405,976

X(756) = isogonal conjugate of X(757)X(756) = isotomic conjugate of X(873)

X(757)

Trilinears (b + c)-2 : (c + a)-2 : (a + b)-2 Barycentrics a(b + c)-2 : b(c + a)-2 : c(a + b)-2

X(757) lies on these lines: 6,662 58,86 60,1014 81,593 171,319 763,849

X(757) = isogonal conjugate of X(756)X(757) = isotomic conjugate of X(1089)

X(758) = EVEN (1, 3) INFINITY POINT

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a0(b3 + c3) - a2(b1 + c1) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

As the isogonal conjugate of a point on the circumcircle, X(758) lies on the line at infinity.

X(758) lies on these lines: 1,21 8,79 10,12 30,511 36,214 46,78 57,997 100,484 354,392 386,986 942,960 982,995

X(758) = isogonal conjugate of X(757)

X(759) = EVEN (1, 3) CIRCUMCIRCLE POINT

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a/[a0(b3 + c3) - a2(b1 + c1)] Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(759) lies on the circumcircle.

X(759) lies on these lines:1,60 10,21 19,112 28,108 31,994 37,101 58,65 75,99 82,827 91,925 107,158 214,662 270,933 484,901 691,897 741,876 833,1010 840,1019 934,1014

X(759) = isogonal conjugate of X(754)

X(760) = EVEN (1, 4) INFINITY POINT

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a0(b4 + c4) - a3(b1 + c1) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

As the isogonal conjugate of a point on the circumcircle, X(761) lies on the line at infinity.

X(760) lies on these lines: 1,32 8,315 10,626 30,511

X(760) = isogonal conjugate of X(761)

X(761) = EVEN (1, 4) CIRCUMCIRCLE POINT

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a/[a0(b4 + c4) - a3(b1 + c1)] Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(761) lies on the circumcircle.

X(761) lies on this line: 1,825 76,789 101,984 665,743

X(761) = isogonal conjugate of X(762)

X(762)

Trilinears (b + c)3 : (c + a)3 : (a + b)3 Barycentrics a(b + c)3 : b(c + a)3 : c(a + b)3

X(762) lies on this line: 210,213 594,1089

X(762) = isogonal conjugate of X(763)

X(763)

Trilinears (b + c)-3 : (c + a)-3 : (a + b)-3 Barycentrics a(b + c)-3 : b(c + a)-3 : c(a + b)-3

X(763) lies on line 757,849

X(763) = isogonal conjugate of X(762)

X(764)

Trilinears (b - c)3 : (c - a)3 : (a - b)3 Barycentrics a(b - c)3 : b(c - a)3 : c(a - b)3

X(764) lies on these lines: 1,513 10,514 56,667 76,693

X(765)

Trilinears (b - c)-2 : (c - a)-2 : (a - b)-2 Barycentrics a(b - c)-2 : b(c - a)-2 : c(a - b)-2

X(765) lies on these lines: 1,1052 59,518 100,513 101,898 109,522 238,519 660,662 798,813

X(765) = isogonal conjugate of X(244)X(765) = isotomic conjugate of X(1111)

X(766) = EVEN (3, 4) INFINITY POINT

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(b4 + c4) - a3(b3 + c3) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

As the isogonal conjugate of a point on the circumcircle, X(766) lies on the line at infinity.

X(766) lies on these lines: 30,511 31,32

X(766) = isogonal conjugate of X(767)

X(767) = EVEN (3, 4) CIRCUMCIRCLE POINT

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a/[a2(b4 + c4) - a3(b3 + c3)] Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(767) lies on the circumcircle.

X(767) lies on these lines: 75,101 76,100 85,109 108,331 110,274 112,286 334,813 825,870

X(767) = isogonal conjugate of X(766)

X(768) = ODD (- 4, - 3) INFINITY POINT

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a-5(b-3 - c-3) + a-4(b-4 - c-4) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

As the isogonal conjugate of a point on the circumcircle, X(768) lies on the line at infinity. The first trilinear coordinate has the form

am-1(bn - cn) + an-1(bm - cm),

corresponding to an odd polynomial center in case m and n are distinct integers. See the note accompanying X(696), where even (m,n) infinity points and even (m,n) circumcircle points are introduced. [For nonzero n, "odd (m,n) circumcircle point" is would be a misnomer (as the point is an even polynomial center); consequently, the prefix o- is used to distinguish this point from "even (m,n) circumcircle point" defined at X(696).] Certain points of these classes occur prior to this section. They are as follows:

X(523) = odd (- 4, - 2) infinity pointX(688) = odd (- 4, 0) infinity pointX(689) = o-(- 4, 0) circumcircle pointX(514) = odd (- 2, - 1) infinity pointX(101) = o-(- 2, - 1) circumcircle pointX(512) = odd (- 2, 0) infinity pointX(99) = o-(- 2, 0) circumcircle pointX(513) = odd (- 1, 0) infinity pointX(100) = o-(- 1, 0) circumcircle pointX(514) = odd (0, 1) infinity pointX(101) = o-(0, 1) circumcircle pointX(523) = odd (0, 2) infinity point

X(110) = o-(0, 2) circumcircle pointX(513) = odd (1, 2) infinity pointX(100) = o-(1, 2) circumcircle pointX(512) = odd (2, 4) infinity pointX(99) = o-(2, 4) circumcircle point

X(768) lies on this line: 30,511

X(768) = isogonal conjugate of X(769)

X(769) = o-(- 4, - 3) CIRCUMCIRCLE POINT

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[a-5(b-3 - c-3) + a-4(b-4 - c-4)] Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(769) lies on the circumcircle. This is one of several points of the form given by first trilinear

1/[am-1(bn - cn) + an-1(bm - cm)],

hence the name "(m, n)-circumcircle point".

X(769) = isogonal conjugate of X(768)

X(770)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = cos3B - cos3C Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A)f(A,B,C)

X(770) lies on this line: 44,513

X(770) = isogonal conjugate of X(771)

X(771)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = 1/(cos3B - cos3C) Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A)f(A,B,C)

X(771) = isogonal conjugate of X(770)

X(772) = ODD (- 4, - 1) INFINITY POINT

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a-5(b-1 - c-1) + a-2(b-4 - c-4) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

As the isogonal conjugate of a point on the circumcircle, X(772) lies on the line at infinity.

X(772) lies on this line: 30,511

X(772) = isogonal conjugate of X(773)

X(773) = o-(- 4, - 1) CIRCUMCIRCLE POINT

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[a-5(b-1 - c-1) + a-2(b-4 - c-4)] Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(773) lies on the circumcircle.

X(773) = isogonal conjugate of X(772)

X(774)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = cos2B + cos2C Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A)f(A,B,C)

X(774) lies on these lines: 1,21 55,201 601,1060 602,1062 821,823 912,1066 938,986

X(774) = isogonal conjugate of X(775)

X(775)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = 1/[cos2B + cos2C] Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A)f(A,B,C)

X(775) lies on these lines: 10,801 31,1097 158,255 225,412 662,820

X(775) = isogonal conjugate of X(774)

X(776) = ODD (- 4, 1) INFINITY POINT

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a-5(b1 - c1) + a0(b-4 - c-4) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

As the isogonal conjugate of a point on the circumcircle, X(776) lies on the line at infinity.

X(776) lies on this line: 30,511

X(776) = isogonal conjugate of X(773)

X(777) = o-(- 4, 1) CIRCUMCIRCLE POINT

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[a-5(b1 - c1) + a0(b-4 - c-4)] Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(777) lies on the circumcircle.

X(777) = isogonal conjugate of X(776)

X(778) = ODD (- 4, 2) INFINITY POINT

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a-5(b2 - c2) + a1(b-4 - c-4) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

As the isogonal conjugate of a point on the circumcircle, X(778) lies on the line at infinity.

X(778) lies on this line: 30,511

X(778) = isogonal conjugate of X(779)

X(779) = o-(- 4, 2) CIRCUMCIRCLE POINT

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[a-5(b2 - c2) + a1(b-4 - c-4)] Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(779) lies on the circumcircle.

X(779) = isogonal conjugate of X(778)

X(780) = ODD (- 4, 3) INFINITY POINT

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a-5(b3 - c3) + a2(b-4 - c-4) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

As the isogonal conjugate of a point on the circumcircle, X(780) lies on the line at infinity.

X(780) lies on this line: 30,511

X(780) = isogonal conjugate of X(781)

X(781) = o-(- 4, 3) CIRCUMCIRCLE POINT

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[a-5(b3 - c3) + a2(b-4 - c-4)] Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(781) lies on the circumcircle.

X(781) = isogonal conjugate of X(780)

X(782) = ODD (- 4, 4) INFINITY POINT

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a-5(b4 - c4) + a3(b-4 - c-4) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

As the isogonal conjugate of a point on the circumcircle, X(782) lies on the line at infinity.

X(782) lies on this line: 30,511

X(782) = isogonal conjugate of X(783)

X(783) = o-(- 4, 4) CIRCUMCIRCLE POINT

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[a-5(b4 - c4) + a3(b-4 - c-4)] Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(783) lies on the circumcircle.

X(783) = isogonal conjugate of X(782)

X(784) = ODD (- 3, - 2) INFINITY POINT

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a-4(b-2 - c-2) + a-3(b-3 - c-3) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

As the isogonal conjugate of a point on the circumcircle, X(784) lies on the line at infinity.

X(784) lies on this line: 30,511

X(784) = isogonal conjugate of X(785)

X(785) = o-(- 3, - 2) CIRCUMCIRCLE POINT

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[a-4(b-2 - c-2) + a-3(b-3 - c-3)] Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(785) lies on the circumcircle.

X(785) lies on this line: 99,692

X(785) = isogonal conjugate of X(782)

X(786) = ODD (- 3, - 1) INFINITY POINT

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a-4(b-1 - c-1) + a-2(b-3 - c-3) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

As the isogonal conjugate of a point on the circumcircle, X(786) lies on the line at infinity.

X(786) lies on this line: 30,511

X(786) = isogonal conjugate of X(787)

X(787) = o-(- 3, - 1) CIRCUMCIRCLE POINT

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[a-4(b-1 - c-1) + a-2(b-3 - c-3)] Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(787) lies on the circumcircle.

X(787) lies on this line: 662,689

X(787) = isogonal conjugate of X(786)

X(788) = ODD (- 3, 0) INFINITY POINT

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a-4(b0 - c0) + a-1(b-3 - c-3) =(b3 - c3)/a Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

As the isogonal conjugate of a point on the circumcircle, X(788) lies on the line at infinity.

X(788) lies on these lines: 30,511 42,649 667,798

X(788) = isogonal conjugate of X(789)

X(789) = o-(- 3, 0) CIRCUMCIRCLE POINT

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[a-4(b0 - c0) + a-1(b-3 - c-3)] =a/(b3 - c3) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(789) lies on the circumcircle.

X(789) lies on these lines:1,731 2,743 6,717 31,701 75,753 76,761 86,741 100,874 101,668 106,870 110,799 112,811 190,813 675,871 727,985

X(789) = isogonal conjugate of X(788)

X(790) = ODD (- 3, 1) INFINITY POINT

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a-4(b1 - c1) + a0(b-3 - c-3) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

As the isogonal conjugate of a point on the circumcircle, X(790) lies on the line at infinity.

X(790) lies on this line: 30,511

X(790) = isogonal conjugate of X(791)

X(791) = o-(- 3, 1) CIRCUMCIRCLE POINT

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[a-4(b1 - c1) + a0(b-3 - c-3)] Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(791) lies on the circumcircle.

X(791) = isogonal conjugate of X(790)

X(792) = ODD (- 3, 2) INFINITY POINT

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a-4(b2 - c2) + a1(b-3 - c-3) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

As the isogonal conjugate of a point on the circumcircle, X(792) lies on the line at infinity.

X(792) lies on this line: 30,511

X(792) = isogonal conjugate of X(793)

X(793) = o-(- 3, 2) CIRCUMCIRCLE POINT

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[a-4(b2 - c2) + a1(b-3 - c-3)] Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(793) lies on the circumcircle.

X(793) = isogonal conjugate of X(792)

X(794) = ODD (- 3, 3) INFINITY POINT

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a-4(b3 - c3) + a2(b-3 - c-3) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

As the isogonal conjugate of a point on the circumcircle, X(794) lies on the line at infinity.

X(794) lies on this line: 30,511

X(794) = isogonal conjugate of X(795)

X(795) = o-(- 3, 3) CIRCUMCIRCLE POINT

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[a-4(b3 - c3) + a2(b-3 - c-3)] Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(795) lies on the circumcircle.

X(795) = isogonal conjugate of X(794)

X(796) = ODD (- 3, 4) INFINITY POINT

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a-4(b4 - c4) + a3(b-3 - c-3) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

As the isogonal conjugate of a point on the circumcircle, X(794) lies on the line at infinity.

X(796) lies on this line: 30,511

X(796) = isogonal conjugate of X(797)

X(797) = o-(- 3, 4) CIRCUMCIRCLE POINT

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[a-4(b4 - c4) + a3(b-3 - c-3)] Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(797) lies on the circumcircle.

X(797) = isogonal conjugate of X(796)

X(798)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = sin2A (cos2B - cos2C) Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = sin3A (cos2B - cos2C)

X(798) lies on these lines: 44,513 163,1101 667,788 688,872 765,813

X(798) = isogonal conjugate of X(799)

X(799)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = csc2A/(cos2B - cos2C) Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (csc A)/(cos2B - cos2C)

X(799) lies on these lines:2,873 63,561 75,897 88,274 99,100 110,789 162,811 190,670 310,333 645,651 689,813

X(799) = isogonal conjugate of X(798)

X(800)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = sin A (cos2B + cos2C) Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = sin2A (cos2B + cos2C)

X(800) lies on these lines: 3,6 53,115 232,459 393,1093

X(800) = isogonal conjugate of X(801)

X(801)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (csc A)/(cos2B + cos2C) Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (csc A)/(cos2B + cos2C)

X(801) lies on these lines: 4,1092 10,775

X(801) = isogonal conjugate of X(800)

X(802) = ODD (- 2, 1) INFINITY POINT

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a-3(b1 - c1) + a0(b-2 - c-2) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

As the isogonal conjugate of a point on the circumcircle, X(802) lies on the line at infinity.

X(802) lies on this line: 30,511

X(802) = isogonal conjugate of X(803)

X(803) = o-(- 2, 1) CIRCUMCIRCLE POINT

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[a-3(b1 - c1) + a0(b-2 - c-2)] Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(803) lies on the circumcircle.

X(803) = isogonal conjugate of X(802)

X(804) = ODD (- 2, 2) INFINITY POINT

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a-3(b2 - c2) + a1(b-2 - c-2) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

As the isogonal conjugate of a point on the circumcircle, X(804) lies on the line at infinity.

X(804) lies on these lines: 2,351 30,511 98,878 99,670 115,1084 147,684 669,850

X(804) = isogonal conjugate of X(805)

X(805) = o-(- 2, 2) CIRCUMCIRCLE POINT

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[a-3(b2 - c2) + a1(b-2 - c-2)] Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(805) lies on the circumcircle.

X(805) lies on these lines: 98,385 99,512 110,669 111,694 187,729 249,827 574,843 691,882 888,892

X(805) = isogonal conjugate of X(804)

X(806) = ODD (- 2, 3) INFINITY POINT

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a-3(b3 - c3) + a2(b-2 - c-2) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

As the isogonal conjugate of a point on the circumcircle, X(806) lies on the line at infinity.

X(806) lies on this line: 30,511

X(806) = isogonal conjugate of X(807)

X(807) = o-(- 2, 3) CIRCUMCIRCLE POINT

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[a-3(b3 - c3) + a2(b-2 - c-2)] Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(807) lies on the circumcircle.

X(807) = isogonal conjugate of X(806)

X(808) = ODD (- 2, 4) INFINITY POINT

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a-3(b4 - c4) + a3(b-2 - c-2) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

As the isogonal conjugate of a point on the circumcircle, X(808) lies on the line at infinity.

X(808) lies on this line: 30,511

X(808) = isogonal conjugate of X(809)

X(809) = o-(- 2, 4) CIRCUMCIRCLE POINT

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[a-3(b4 - c4) + a3(b-2 - c-2)] Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(809) lies on the circumcircle.

X(809) = isogonal conjugate of X(808)

X(810)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = sin 2A (cos2B - cos2C) Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = sin2A cos A (cos2B - cos2C)

X(810) lies on these lines: 521,656 661,663 667,788

X(810) = isogonal conjugate of X(811)

X(811)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (csc 2A)/(cos2B - cos2C) Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sec A)/(cos2B - cos2C)

X(811) lies on these lines: 1,336 75,1099 99,108 112,789 162,799 350,447 645,648 662,823

X(811) = isogonal conjugate of X(810)

X(812) = ODD (- 1, 1) INFINITY POINT

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a-2(b1 - c1) + a0(b-1 - c-1) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

As the isogonal conjugate of a point on the circumcircle, X(812) lies on the line at infinity.

X(812) lies on these lines: 30,511 190,646 649,693 673,1024 903,1022 1015,1086

X(812) = isogonal conjugate of X(813)

X(813) = o-(- 1, 1) CIRCUMCIRCLE POINT

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[a-2(b1 - c1) + a0(b-1 - c-1)] Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(813) lies on the circumcircle.

X(813) lies on these lines:99,1016 100,649 101,667 103,295 105,238 106,292 163,827 190,789 334,767 335,675 644,932 689,799 692,825 739,902 765,798 898,1023 927,1025

X(813) = isogonal conjugate of X(812)

X(814) = ODD (- 1, 2) INFINITY POINT

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a-2(b2 - c2) + a1(b-1 - c-1) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

As the isogonal conjugate of a point on the circumcircle, X(814) lies on the line at infinity.

X(814) lies on this line: 30,511

X(814) = isogonal conjugate of X(815)

X(815) = o-(- 1, 2) CIRCUMCIRCLE POINT

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[a-2(b2 - c2) + a1(b-1 - c-1)] Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(815) lies on the circumcircle.

X(815) = isogonal conjugate of X(814)

X(816) = ODD (- 1, 3) INFINITY POINT

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a-2(b3 - c3) + a2(b-1 - c-1) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

As the isogonal conjugate of a point on the circumcircle, X(816) lies on the line at infinity.

X(816) lies on this line: 30,511

X(816) = isogonal conjugate of X(817)

X(817) = o-(- 1, 3) CIRCUMCIRCLE POINT

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[a-2(b3 - c3) + a2(b-1 - c-1)] Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(817) lies on the circumcircle.

X(817) = isogonal conjugate of X(816)

X(818) = ODD (- 1, 4) INFINITY POINT

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a-2(b4 - c4) + a3(b-1 - c-1) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

As the isogonal conjugate of a point on the circumcircle, X(818) lies on the line at infinity.

X(818) lies on this line: 30,511

X(818) = isogonal conjugate of X(819)

X(819) = o-(- 1, 4) CIRCUMCIRCLE POINT

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[a-2(b4 - c4) + a3(b-1 - c-1)] Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(819) lies on the circumcircle.

X(819) = isogonal conjugate of X(818)

X(820)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = cos2A (cos2B + cos2C) Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = cos A cos2A (cos2B + cos2C)

X(820) lies on these lines: 1,29 3,296 662,775 836,1100

X(820) = isogonal conjugate of X(821)

X(821)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = sec2A /(cos2B + cos2C)Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A)f(A,B,C)

X(821) lies on these lines: 158,255 243,411 774,823

X(821) = isogonal conjugate of X(820)

X(822)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = sec2B - sec2C Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = sin A (sec2B - sec2C)

X(822) lies on this line: 44,513

X(822) = isogonal conjugate of X(823)

X(823)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = 1/(sec2B - sec2C) Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A)f(A,B,C)

X(823) lies on these lines: 100,107 110,681 158,897 264,379 648,651 662,811 774,821

X(823) = isogonal conjugate of X(822)

X(824) = ODD (0, 3) INFINITY POINT

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a-1(b3 - c3) + a2(b0 - c0) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

As the isogonal conjugate of a point on the circumcircle, X(824) lies on the line at infinity.

X(824) lies on these lines: 30,511 321,693

X(824) = isogonal conjugate of X(825)

X(825) = o-(0, 3) CIRCUMCIRCLE POINT

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[a-1(b3 - c3) + a2(b0 - c0)] Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(825) lies on the circumcircle.

X(825) lies on these lines:1,761 6,753 31,743 32,731 99,163 103,572 105,985 560,717 692,813 767,870

X(825) = isogonal conjugate of X(824)

X(826) = ODD (0, 4) INFINITY POINT

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a-1(b4 - c4) + a3(b0 - c0) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

As the isogonal conjugate of a point on the circumcircle, X(826) lies on the line at infinity.

X(826) lies on this line: 30,511 54,879 76,882

X(826) = isogonal conjugate of X(827)

X(827) = o-(0, 4) CIRCUMCIRCLE POINT

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[a-1(b4 - c4) + a3(b0 - c0)] Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(827) lies on the circumcircle.

X(827) lies on these lines:5,83 6,755 31,745 32,733 82,759 111,251 163,813 249,805 250,935 560,719 662,831 741,849

X(827) = isogonal conjugate of X(826)

X(828)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = sin C sec2B + sin B sec2C Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = sin A f(A,B,C)

X(828) = isogonal conjugate of X(829)

X(829)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = 1/(sin C sec2B + sin B sec2C)

Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A)f(A,B,C)

X(829) = isogonal conjugate of X(828)

X(830) = ODD (1, 3) INFINITY POINT

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a0(b3 - c3) + a2(b1 - c1) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

As the isogonal conjugate of a point on the circumcircle, X(830) lies on the line at infinity.

X(830) lies on this line: 30,511

X(830) = isogonal conjugate of X(831)

X(831) = o-(1, 3) CIRCUMCIRCLE POINT

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[a0(b3 - c3) + a2(b1 - c1)] Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(831) lies on the circumcircle.

X(831) lies on this line: 662,827

X(831) = isogonal conjugate of X(830)

X(832) = ODD (1, 4) INFINITY POINT

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a0(b4 - c4) + a3(b1 - c1) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

As the isogonal conjugate of a point on the circumcircle, X(832) lies on the line at infinity.

X(832) lies on these lines: 30,511 656,667

X(832) = isogonal conjugate of X(833)

X(833) = o-(1, 4) CIRCUMCIRCLE POINT

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[a0(b4 - c4) + a3(b1 - c1)] Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(833) lies on the circumcircle.

X(833) lies on these lines: 106,977 759,1010

X(833) = isogonal conjugate of X(832)

X(834) = ODD (2, 3) INFINITY POINT

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a1(b3 - c3) + a2(b2 - c2) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

As the isogonal conjugate of a point on the circumcircle, X(834) lies on the line at infinity.

X(834) lies on this line: 30,511

X(834) = isogonal conjugate of X(835)

X(835) = o-(2, 3) CIRCUMCIRCLE POINT

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[a1(b3 - c3) + a2(b2 - c2)] Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(835) lies on the circumcircle.

X(835) lies on these lines: 110,190 335,741

X(835) = isogonal conjugate of X(834)

X(836)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = sin B sec2B + sin C sec2C Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = sin A f(A,B,C)

X(836) lies on these lines: 1,393 37,73 820,1100

X(836) = isogonal conjugate of X(837)

X(837)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = 1/(sin B sec2B + sin C sec2C)

Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A)f(A,B,C)

X(837) lies on this line: 393,394

X(837) = isogonal conjugate of X(836)

X(838) = ODD (3, 4) INFINITY POINT

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(b4 - c4) + a3(b3 - c3) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

As the isogonal conjugate of a point on the circumcircle, X(838) lies on the line at infinity.

X(838) lies on this line: 30,511

X(838) = isogonal conjugate of X(839)

X(839) = o-(3, 4) CIRCUMCIRCLE POINT

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[a2(b4 - c4) + a3(b3 - c3)] Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(839) lies on the circumcircle.

X(839) lies on these lines: 110,668 334,741

X(839) = isogonal conjugate of X(838)

X(840) = ISOGONAL CONJUGATE of X(528)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a/(2ax - by - cz), x = x(A,B,C) = 1 - cos(B - C) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(840) lies on the circumcircle.

X(840) lies on these lines: 6,919 7,927 36,101 55,901 100,518 105,513 106,663 109,902 759,1019 898,1083

X(840) = isogonal conjugate of X(528)

X(841) = ISOGONAL CONJUGATE of X(541)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a/(2ax - by - cz), x = x(A,B,C) = 1/(cos A - 2 cos B cos C) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(841) lies on the circumcircle.

X(841) lies on this line: 376,476

X(841) = isogonal conjugate of X(541)

X(842) = ISOGONAL CONJUGATE of X(542)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a/(2ax - by - cz), x = x(A,B,C) = sec(A + ω) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(842) lies on the circumcircle.

X(842) lies on these lines: 2,476 3,691 4,935 23,110 30,99 74,512 98,523 107,468 111,647 112,186 858,925

X(842) = isogonal conjugate of X(542)

X(843) = ISOGONAL CONJUGATE of X(543)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a/(2ax - by - cz), x = x(a,b,c) = bc/(b2 - c2) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(843) lies on the circumcircle.

X(843) lies on these lines: 6,691 99,525 110,187 111,512 574,805

X(843) = isogonal conjugate of X(543)

X(844)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = - x + y + z, x = x(A,B,C) = sin(A/2) sec2(A/2) Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = sin A f(A,B,C)

This point solves a problem posed by Antreas Hatzipolakis in Hyacinthos #2768, May 3, 2001.

X(844) lies on these lines: 166,167 173,503

X(845)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = - x + y + z, x = x(A,B,C) = sin2(A/2) sec(A/2) Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = sin A f(A,B,C)

This point solves a problem posed by Antreas Hatzipolakis in Hyacinthos #2768, May 3, 2001.

X(845) lies on these lines: 164,362 165,166 173,503

X(846)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = - a2 + b2 + c2 + bc + ca + abBarycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a f(a,b,c)

This point solves a problem posed by Antreas Hatzipolakis in Hyacinthos #3001, June 11, 2001.

X(846) lies on these lines: 1,21 2,1054 6,1051 9,43 35,228 37,171 55,984 100,756 333,740 405,986 982,1001

X(847)

Trilinears sec A sec 2A : sec B sec 2B : sec C sec 2CBarycentrics tan A sec 2A : tan B sec 2B : tan C sec 2C

This point solves a problem posed by Antreas Hatzipolakis in Hyacinthos #3130, June 25, 2001; see also Jean-Pierre Ehrmann, #3135, June 26, 2001. The problem and solution may be stated as follows. Let ABC be a triangle, La, Lb, Lc the perpendicular bisectors of sides BC, CA, AB, and AA', BB', CC' the altitudes of ABC, respectively. Let Ab be the point of intersection of AA' and Lb, and let Ac be the point of intersection of AA' and Lc. Let A" be the point of intersection of BAb and CAc. Define B" and C" cyclically. Then triangle A"B"C" is perspective to triangle ABC, with perspector X(847).

X(847) lies on these lines: 2,254 3,925 4,52 24,96 91,225 378,1105 403,1093

X(848)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (csc A)/(cot A - cot A'), where A' = 2πa/(a + b + c)Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = 1/(cot A - cot A'), A' = 2πa/(a + b + c)

X(848) point is introduced by Paul Yiu in Hyacinthos #2704, April 7, 2001 (see also #2708, April 10, 2001) as the solution X of the equation

angle BXC : angle CXA : angle AXB = a : b : c.

X(849)

Trilinears [a/(b + c)]2 : [b/(c + a)]2 : [c/(a + b)]2

Barycentrics a[a/(b + c)]2 : b[b/(c + a)]2 : c[c/(a + b)]2

Let D denote the circumcircle of triangle ABC. Let DA be the circle tangent to sideline BC and tangent to D at A. Define DB and DC cyclically. Let EA and FA be the points in

which circles DB and DC meet. Define EB, FB and EC, FC cyclically. Then lines EAFA, EBFB, ECFC are sidelines of a triangle A'B'C' homothetic to ABC, and X(849) is the center of homothety. See A. Hatzipolakis and P. Yiu, Hyacinthos #2056-2070, December, 2000.

X(849) lies on these lines: 32,163 36,58 110,595 249,1110 741,827 757,763

X(849) = isogonal conjugate of X(1089)

X(850) = BARYCENTRIC MULTIPLIER FOR KIEPERT HYPERBOLA

Trilinears (b2 - c2)/a3 : (c2 - a2)/b3 : (a2 - b2)/c3

Barycentrics (b2 - c2)/a2 : (c2 - a2)/b2 : (a2 - b2)/c2

The barycentric product of X(850) and the circumcircle is the Kiepert hyperbola.

X(850) lies on these lines: 2,647 99,476 110,685 297,525 316,512 325,523 340,520 669,804 670,892

X(850) = isotomic conjugate of X(110)X(850) = anticomplement of X(647)

X(851) = INTERCEPT OF EULER LINE AND POLE OF X(1)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = sin 2B sin(C - A) + sin 2C sin(B - A) Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A)f(A,B,C)

X(851) lies on these lines: 2,3 42,65 43,46 44,513 226,228

X(852) = INTERCEPT OF EULER LINE AND POLE OF X(3)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = sec C sin 2B sin(C - A) + sec B sin 2C sin(B - A) Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A)f(A,B,C)

X(852) lies on these lines: 2,3 216,373 520,647

X(853) = INTERCEPT OF EULER LINE AND POLE OF X(55)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B),where f(A,B,C) = sec2(C/2) sin 2B sin(C - A) + sec2(B/2) sin 2C sin(B - A) Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A)f(A,B,C)

X(853) lies on these lines: 2,3 657,663

X(854) = INTERCEPT OF EULER LINE AND POLE OF X(56)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B),where f(A,B,C) = csc2(C/2) sin 2B sin(C - A) + csc2(B/2) sin 2C sin(B - A) Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A)f(A,B,C)

X(854) lies on this line: 2,3

X(855) = INTERCEPT OF EULER LINE AND POLE OF X(57)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B),where f(A,B,C) = cot(C/2) sin 2B sin(C - A) + cot(B/2) sin 2C sin(B - A) Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A)f(A,B,C)

X(855) lies on these lines: 2,3 513,663

X(856) = INTERCEPT OF EULER LINE AND POLE OF X(63)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B),where f(A,B,C) = tan C sin 2B sin(C - A) + tan B sin 2C sin(B - A) Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A)f(A,B,C)

X(856) lies on these lines: 2,3 521,656

X(857) = INTERCEPT OF EULER LINE AND POLE OF X(75)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B),where f(A,B,C) = sin2C sin 2B sin(C - A) + sin2B sin 2C sin(B - A) Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A)f(A,B,C)

X(857) lies on these lines: 2,3 514,661

X(857) = inverse of X(379) in the orthocentroidal circle

X(858) = INTERCEPT OF EULER LINE AND POLE OF X(76)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B),where f(A,B,C) = sin3C sin 2B sin(C - A) + sin3B sin 2C sin(B - A) Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A)f(A,B,C)

X(858) lies on these lines: 2,3 50,230 67,524 125,511 126,625 316,691 325,523 842,925

X(858) = inverse of X(22) in the circumcircleX(858) = inverse of X(2) in the nine-point circleX(858) = complement of X(23)X(858) = anticomplement of X(468)

X(859) = INTERCEPT OF EULER LINE AND POLE OF X(81)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B),where f(A,B,C) = (sin A + sin B) sin 2B sin(C - A) + (sin A + sin C) sin 2C sin(B - A) Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A)f(A,B,C)

X(859) lies on these lines: 2,3 36,238 56,58 81,957 198,284 283,945 333,956

X(860) = INTERCEPT OF EULER LINE AND POLE OF X(82)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B),where f(A,B,C) = (tan A + tan B) sin 2B sin(C - A) + (tan A + tan C) sin 2C sin(B - A) Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A)f(A,B,C)

X(860) lies on these lines: 2,3 8,1068 10,201 34,997 240,522

X(861) = INTERCEPT OF EULER LINE AND POLE OF X(9)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B),where f(A,B,C) = tan(C/2) sin 2B sin(C - A) + tan(B/2) sin 2C sin(B - A) Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A)f(A,B,C)

X(861) lies on these lines: 2,3 650,663

X(862) = INTERCEPT OF EULER LINE AND POLE OF X(19)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B),where f(A,B,C) = cot C sin 2B sin(C - A) + cot B sin 2C sin(B - A) Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A)f(A,B,C)

X(862) lies on these lines: 2,3 661,663

X(863) = INTERCEPT OF EULER LINE AND POLE OF X(31)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B),where f(A,B,C) = csc2C sin 2B sin(C - A) + csc2B sin 2C sin(B - A) Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A)f(A,B,C)

X(863) lies on these lines: 2,3 667,788

X(864) = INTERCEPT OF EULER LINE AND POLE OF X(32)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B),where f(A,B,C) = csc3C sin 2B sin(C - A) + csc3B sin 2C sin(B - A) Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A)f(A,B,C)

X(864) lies on these lines: 2,3 669,688

X(865) = INTERCEPT OF EULER LINE AND POLE OF X(512)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B),where f(A,B,C) = sin 2B csc2C sin(C - A) csc(A - B) - sin 2C csc2B sin(B - A) csc(A - C)Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A)f(A,B,C)

X(865) lies on this line: 2,3 351,888

X(866) = INTERCEPT OF EULER LINE AND POLE OF X(513)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B),where f(A,B,C) = sin 2B sin(A - C)/(sin A - sin B) - sin 2C sin(A - B)/(sin A - sin C)Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A)f(A,B,C)

X(866) lies on these lines: 2,3 244,665

X(867) = INTERCEPT OF EULER LINE AND POLE OF X(514)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B),where f(A,B,C) = sin C sin 2B sin(A - C)/(sin A - sin B) - sin B sin 2C sin(A - B)/(sin A - sin C)Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A)f(A,B,C)

X(867) lies on these lines: 2,3 11,244

X(868) = INTERCEPT OF EULER LINE AND POLE OF X(523)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B),where f(A,B,C) = sin 2B sin(A - C) csc(A - B) - sin 2C sin(A - B) csc(A - C)Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A)f(A,B,C)

X(868) lies on these lines: 2,3 115,125 127,136

X(869)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = a2(b2 + c2 + bc) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(869) lies on these lines:1,2 6,292 31,32 38,980 55,893 100,731 101,743 192,1045 210,1107

X(869) = isogonal conjugate of X(870)X(869) = isotomic conjugate of X(871)

X(870)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = 1/[a2(b2 + c2 + bc)] Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(870) lies on these lines:1,76 2,292 6,75 34,331 56,85 58,274 86,871 106,789 767,825

X(870) = isogonal conjugate of X(869)X(870) = isotomic conjugate of X(984)

X(871)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = 1/[a4(b2 + c2 + bc)] Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(871) lies on these lines:2,561 75,700 76,335 86,870 310,982 675,789

X(871) = isotomic conjugate of X(869)

X(872)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = [a(b + c)]2 Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(872) lies on these lines:6,292 37,42 41,560 43,75 190,1045 386,984 688,798 740,1089

X(872) = isotomic conjugate of X(873)

X(873)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = [a(b + c)] - 2 Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(873) = isotomic conjugate of X(872)X(873) = isotomic conjugate of X(756)

X(873) lies on these lines:2,799 81,239 86,310 261,552 689,741

X(874)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = (a2u2 - bcvw)/[a2u(bv - cw)], u : v : w = X(1) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(874) lies on these lines:1,75 99,670 100,789 190,646

X(874) = isogonal conjugate of X(875)X(874) = isotomic conjugate of X(876)

X(875)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = a2u(bv - cw)/(a2u2 - bcvw), u : v : w = X(1) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(875) lies on these lines:1,512 31,669 42,649 213,667 291,659 295,926

X(875) = isogonal conjugate of X(874)

X(876)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = a2u(bv - cw)/(a4u2 - bcvw), u : v : w = X(1) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(876) lies on these lines:1,512 10,514 37,513 75,523 291,891 292,659 295,928 335,900 741,759

X(876) = isogonal conjugate of X(874)

X(877)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = (a2u2 - bcvw)/[a2u(bv - cw)], u : v : w = X(4) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(877) lies on these lines: 4,69 99,112

X(877) = isogonal conjugate of X(878)X(877) = isotomic conjugate of X(879)

X(878)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = a2u(bv - cw)/(a2u2 - bcvw), u : v : w = X(4) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(878) lies on these lines: 3,525 25,669 32,512 98,804 184,647

X(878) = isogonal conjugate of X(879)

X(879)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = a2u(bv - cw)/(a4u2 - bcvw), u : v : w = X(4) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(879) lies on these lines: 3,525 4,512 6,523 54,826 66,924 67,526 69,520 74,98 287,895

X(879) = isotomic conjugate of X(877)

X(880)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = (a2u2 - bcvw)/[a2u(bv - cw)], u : v : w = X(6) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(880) lies on these lines: 6,76 99,670 886,892

X(880) = isogonal conjugate of X(881)X(880) = isotomic conjugate of X(882)

X(881)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = a2u(bv - cw)/(a2u2 - bcvw), u : v : w = X(6) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(881) lies on these lines: 39,512 351,694

X(881) = isogonal conjugate of X(880)

X(882)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = a2u(bv - cw)/(a4u2 - bcvw), u : v : w = X(6) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(882) lies on these lines: 6,688 39,512 76,826 141,523 691,805 694,888 733,755

X(882) = isotomic conjugate of X(880)

X(883)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = (a2u2 - bcvw)/[a2u(bv - cw)], u : v : w = X(7) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(883) lies on these lines: 7,8 190,644

X(883) = isogonal conjugate of X(884)X(883) = isotomic conjugate of X(885)

X(884)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = a2u(bv - cw)/(a2u2 - bcvw), u : v : w = X(7) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(884) lies on these lines: 21,885 31,649 41,663 55,650 56,667 105,659

X(884) = isogonal conjugate of X(883)

X(885)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = a2u(bv - cw)/(a4u2 - bcvw), u : v : w = X(7) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(885) lies on these lines: 1,514 7,513 9,522 21,884 104,105 673,900 919,929

X(885) = isotomic conjugate of X(883)

X(886)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = (a2u2 - bcvw)/[a2u(bv - cw)], u : v : w = X(512) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(886) lies on these lines: 99,669 512,670 880,892

X(886) = isogonal conjugate of X(887)X(886) = isotomic conjugate of X(888)

X(887)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = a2u(bv - cw)/(a2u2 - bcvw), u : v : w = X(512) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(887) lies on these lines: 99,670 187,237

X(887) = isogonal conjugate of X(886)

X(888)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = a2u(bv - cw)/(a4u2 - bcvw), u : v : w = X(512) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

As the isogonal conjugate of a point on the circumcircle, X(888) lies on the line at infinity.

X(888) lies on these lines: 30,511 351,865 694,882 805,8925

X(888) = isotomic conjugate of X(886)

X(889)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = (a2u2 - bcvw)/[a2u(bv - cw)], u : v : w = X(513) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(889) lies on these lines: 99,898 190,649 350,903 513,668

X(889) = isogonal conjugate of X(890)X(889) = isotomic conjugate of X(891)

X(890)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = a2u(bv - cw)/(a2u2 - bcvw), u : v : w = X(513) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(890) lies on these lines: 100,190 187,237

X(890) = isogonal conjugate of X(889)

X(891)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = a2u(bv - cw)/(a4u2 - bcvw), u : v : w = X(513) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

As the isogonal conjugate of a point on the circumcircle, X(891) lies on the line at infinity.

X(891) lies on these lines: 1,659 30,511 244,665 291,876

X(891) = isogonal conjugate of X(898)X(891) = isotomic conjugate of X(889)

X(892)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = (a2u2 - bcvw)/[a2u(bv - cw)]where u : v : w = X(523) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(892) lies on these lines: 99,523 111,381 290,895 316,524 670,850 805,888 880,886

X(892) = isogonal conjugate of X(351)X(892) = isotomic conjugate of X(690)

X(893)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = a/(a2 + bc) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(893) lies on these lines: 9,43 19,232 42,694 55,869 100,733 171,292 239,257

X(893) = isogonal conjugate of X(894)

X(894)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = (a2 + bc)/a Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(894) lies on these lines: 1,87 2,7 6,75 8,193 10,1046 37,86 42,1045 65,257 72,1010 81,314 92,608 141,320 213,274 256,291 273,458 287,651 312,940 319,524 536,1100

X(894) = isogonal conjugate of X(893)X(894) = isotomic conjugate of X(257)

X(895)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = u/(v2 + w2 - 2u2 - 2vw cos A + wu cos B + uv cos C),where u : v : w = X(3) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(895) lies on these lines: 4,542 6,110 54,575 65,651 66,193 67,524 69,125 74,511 287,879 290,892

X(895) = isogonal conjugate of X(468)

X(896)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = v2 + w2 - 2u2 - 2vw cos A + wu cos B + uv cos C,where u : v : w = X(3) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(896) lies on these lines: 1,21 9,750 44,513 57,748 162,240 171,756 238,244 518,902

X(896) = isogonal conjugate of X(897)

X(897)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = 1/(v2 + w2 - 2u2 - 2vw cos A + wu cos B + uv cos C),where u : v : w = X(3) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(897) lies on these lines: 1,662 10,190 19,162 37,100 65,651 75,799 158,823 225,653 691,759

X(897) = isogonal conjugate of X(896)

X(898)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = u/(v2 + w2 - 2u2 - 2vw cos A + wu cos B + uv cos C),where u : v : w = X(100) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(898) lies on these lines: 99,889 100,667 101,765 105,666 106,238 813,1023 840,1083

X(898) = isogonal conjugate of X(891)

X(899)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = v2 + w2 - 2u2 - 2vw cos A + wu cos B + uv cos C,where u : v : w = X(100) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(899) lies on these lines: 1,2 6,750 38,210 44,513 55,748 88,291 100,238 244,518

X(900)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = (v2 + w2 - 2u2 - 2vw cos A + wu cos B + uv cos C)/u,where u : v : w = X(101) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

As the isogonal conjugate of a point that lies on the circumcircle, X(900) lies on the line at infinity.

X(900) lies on these lines: 11,244 30,511 37,665 100,190 335,876 673,885

X(900) = isogonal conjugate of X(901)

X(901)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = u/(v2 + w2 - 2u2 - 2vw cos A + wu cos B + uv cos C),where u : v : w = X(101) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(901) lies on the circumcircle.

X(901) lies on these lines: 3,953 36,106 55,840 59,109 88,105 100,513 101,649 104,517 484,759 675,903

X(901) = isogonal conjugate of X(900)

X(902)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = a(v2 + w2 - 2u2 - 2vw cos A + wu cos B + uv cos C)/u,where u : v : w = X(101) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(902) lies on these lines: 1,89 6,31 35,595 36,106 44,678 100,238 109,840 165,614 187,237 518,896 739,813 750,1001

X(902) = isogonal conjugate of X(903)

X(903)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = u/[a(v2 + w2 - 2u2 - 2vw cos A + wu cos B + uv cos C)],where u : v : w = X(101) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(903) lies on these lines: 2,45 7,528 27,648 75,537 86,99 310,670 320,519 335,536 350,889 527,666 675,901 812,1022

X(903) = isogonal conjugate of X(902)X(903) = isotomic conjugate of X(519)

X(904)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = a2/(a2 + bc)Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(904) lies on these lines: 1,257 21,238 31,237 55,869 101,733 172,694

X(905)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = [wb2 - vc2 + a(wb - vc)]cos A, u : v : w = X(3) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(905) lies on these lines: 36,238 241,514 441,525 521,656 1053,1054

X(906)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = (cos A)/[wb2 - vc2 + a(wb - vc)], u : v : w = X(3) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(906) lies on these lines: 32,218 41,601 72,248 100,112 101,109 163,692 219,577

X(907)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = 1/[wb3 - vc3 + a2(wb - vc)], u : v : w = X(3) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(907) lies on this line: 98,620

X(908)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = [(wb + vc)/a - v - w]/a, u : v : w = X(3) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(908) lies on these lines: 1,998 2,7 4,78 5,72 8,946 10,994 11,518 12,960 80,519 92,264 100,516 119,517 153,515 214,535 377,936 392,495 514,661

X(908) = isogonal conjugate of X(909)

X(909)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = a/[(wb + vc)/a - v - w], u : v : w = X(3) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(909) lies on these lines: 9,48 19,604 55,184 163,284 333,662

X(909) = isogonal conjugate of X(908)

X(910)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = a[wb2 + vc2 - a(wb + vc)], u : v : w = X(3) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(910) lies on these lines: 3,169 6,57 9,165 19,25 32,1104 40,220 41,65 44,513 46,218 48,354 101,517 103,971 105,919 118,516 227,607 241,294

X(911)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = a/[wb2 + vc2 - a(wb + vc)], u : v : w = X(3) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(911) lies on these lines: 3,101 41,603 48,692 56,607 241,294

X(912)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = (wb + vc)/a - v - w, u : v : w = X(4) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

As the isogonal conjugate of a point on the circumcircle, X(912) lies on the line at infinity.

X(912) lies on these lines:1,90 3,63 5,226 30,511 38,1064 65,68 222,1060 601,976 774,1066 960,993

X(912) = isogonal conjugate of X(915)

X(913)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = a/[(wb + vc)/a - v - w], u : v : w = X(4) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(913) lies on these lines: 19,101 25,692 27,662 571,608

X(913) = isogonal conjugate of X(914)

X(914)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = [(wb + vc)/a - v - w]/a, u : v : w = X(4) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(914) lies on these lines: 8,224 63,69 514,661

X(914) = isogonal conjugate of X(913)

X(915)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = 1/[(wb + vc)/a - v - w], u : v : w = X(4) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(915) lies on these lines: 3,48 19,101 21,925 24,108 28,110 34,46 99,286 242,929

X(915) = isogonal conjugate of X(912)

X(916)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = wb2 + vc2 - a(wb + vc), u : v : w = X(4) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

As the isogonal conjugate of a point on the circumcircle, X(916) lies on the line at infinity.

X(916) lies on these lines: 3,48 30,511 72,185 1037,1069

X(916) = isogonal conjugate of X(917)

X(917)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = 1/[wb2 + vc2 - a(wb + vc)], u : v : w = X(4) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(917) lies on the circumcircle.

X(917) lies on these lines: 4,101 27,110 92,100 109,278

X(917) = isogonal conjugate of X(916)

X(918)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = (w/b - v/c)/a2, u : v : w = X(4) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

As the isogonal conjugate of a point of the circumcircle, X(918) lies on the line at infinity.

X(918) lies on these lines: 30,511 63,654 190,644 1086,1111

X(918) = isogonal conjugate of X(919)X(918) = isotomic conjugate of X(666)

X(919)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = a2/(w/b - v/c), u : v : w = X(4) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(919) lies on the circumcircle.

X(919) lies on these lines: 6,840 99,666 100,650 101,663 103,672 104,294 105,910 106,1055 109,649 673,675 885,929

X(919) = isogonal conjugate of X(918)

X(920)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B),where f(A,B,C) = cos2B cos2C - cos2A Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = f(A,B,C) sin A

X(920) lies on these lines: 1,21 4,46 4,78 9,498 19,91 57,499 158,921 201,601 243,1075

X(921) = isogonal conjugate of X(920)

X(921)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B),where f(A,B,C) = 1/(cos2B cos2C - cos2A) Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = f(A,B,C) sin A

X(921) lies on these lines: 19,47 46,225 63,91 158,920

X(921) = isogonal conjugate of X(920)

X(922)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = a2(cos2B cos2C - 2 cos2A), Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(922) lies on these lines: 31,48 667,788

X(923)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = a2/(cos2B cos2C - 2 cos2A) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(923) lies on these lines: 1,662 31,163 42,101 213,692 691,741

X(924)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = bw - cv, u : v : w = X(5) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

As the isogonal conjugate of a point on the circumcircle, X(924) lies on the line at infinity.

X(924) lies on these lines: 30,511 50,647 66,879 669,684

X(924) = isogonal conjugate of X(925)

X(925)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = 1/(bw - cv), u : v : w = X(5) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(925) lies on the circumcircle.

X(925) lies on these lines: 2,136 3,847 4,131 20,68 21,915 22,98 91,759 94,96 648,933 842,858

X(925) = isogonal conjugate of X(924)X(925) = anticomplement of X(136)

X(926)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = bw - cv, u : v : w = X(7) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

As the isogonal conjugate of a point on the circumcircle, X(926) lies on the line at infinity.

X(926) lies on these lines: 30,511 55,654 101,692 295,875 657,663

X(926) = isogonal conjugate of X(927)

X(927)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = 1/(bw - cv), u : v : w = X(7) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(927) lies on the circumcircle.

X(927) lies on these lines: 7,840 100,693 101,514 103,516 109,658 813,1025

X(927) = isogonal conjugate of X(926)

X(928)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = bw - cv, u : v : w = X(11) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

As the isogonal conjugate of a point on the circumcircle, X(928) lies on the line at infinity.

X(928) lies on these lines: 30,511 101,109 102,103 116,124 117,118 151,152 295,876

X(928) = isogonal conjugate of X(929)

X(929)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = 1/(bw - cv), u : v : w = X(11) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(929) lies on the circumcircle.

X(929) lies on these lines: 101,522 102,516 103,515 109,514 242,915 885,919

X(929) = isogonal conjugate of X(928)

X(930)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = 1/(bw - cv), u : v : w = X(17) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(930) lies on the circumcircle.

X(930) lies on these lines: 2,137 3,252 4,128 74,550

X(930) = anticomplement of of X(137)

X(931)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = 1/(bw - cv), u : v : w = X(21) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(931) lies on the circumcircle.

X(931) lies on these lines: 100,645 101,643 108,648 109,662 111,941

X(932)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = 1/(bw - cv), u : v : w = X(43) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(932) lies on the circumcircle.

X(932) lies on these lines: 1,727 21,741 81,715 87,106 105,330 172,699 644,813 667,668

X(933)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = 1/(bw - cv), u : v : w = X(54) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(933) lies on the circumcircle.

X(933) lies on these lines: 4,137 54,74 98,275 250,759 270,759 648,925

X(934)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = 1/(bw - cv), u : v : w = X(56) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(934) lies on the circumcircle.

X(934) lies on these lines: 1,103 3,972 7,104 56,105 77,102 100,658 101,651 106,269 644,1025 675,1088 727,1106 741,1042 759,1014

X(935)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = 1/(bw - cv), u : v : w = X(67) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(935) lies on the circumcircle.

X(935) lies on these lines: 4,842 67,74 98,186 110,525 111,468 112,523 250,827 378,477

X(936)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = a3 - a2(b + c) - a(b - c)2 + (b + c)3 Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(936) lies on these lines: 1,2 3,9 40,960 56,210 57,72 63,404 165,411 223,1038 226,443 269,307 377,908 581,966 984,988

X(936) = isogonal conjugate of X(937)

X(937)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = 1/[a3 - a2(b + c) - a(b - c)2 + (b + c)3] Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(937) lies on these lines: 1,329 6,40 31,1103 34,196 56,223

X(937) = isogonal conjugate of X(936)

X(938)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = bc[a4 - 2a3(b + c) - 4a2bc + (b - c)(b2 - c2)(2a - b - c)] Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(938) lies on these lines: 1,2 4,7 20,57 29,81 40,390 56,411 63,452 65,497 354,388 355,1056 517,1058 774,986 944,999

X(938) = isogonal conjugate of X(939)X(939) = anticomplement of X(936)

X(939)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = a/[a4 - 2a3(b + c) - 4a2bc + (b - c)(b2 - c2)(2a - b - c)] Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(939) lies on these lines: 3,269 34,55 56,212

X(939) = isogonal conjugate of X(938)

X(940)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = a2 + a(b + c) + 2bcBarycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(940) lies on these lines: 1,3 2,6 31,1001 37,63 42,750 58,405 72,975 222,226 312,894 386,474 387,443 518,612

X(940) = isogonal conjugate of X(941)

X(941)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = 1/[a2 + a(b + c) + 2bc]Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(941) lies on these lines: 1,573 2,314 6,21 8,37 9,42 81,967 84,581 111,931

X(941) = isogonal conjugate of X(940)

X(942)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = 2abc + (b + c)(a - b + c)(a + b - c)Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(942) lies on these lines: 1,3 2,72 4,7 5,226 6,169 8,443 10,141 11,113 28,60 30,553 34,222 37,579 42,1066 58,1104 63,405 78,474 212,582 238,1046 277,1002 279,955 284,1100 355,388 496,946 750,976 758,960 962,1058 1042,1064

X(942) = isogonal conjugate of X(943)X(942) = inverse of X(36) in the incircleX(942) = complement of X(72)

X(943)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = 1/[2abc + (b + c)(a - b + c)(a + b - c)]Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(943) lies on these lines: 1,201 3,7 4,12 8,405 21,72 28,228 35,79 80,950 100,442 500,651 968,1039 1001,1058

X(943) = isogonal conjugate of X(942)

X(944)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = bc[3a4 - 2a3(b + c) + (b - c)2(2ab + 2ac - 2bc - b2 - c2 - 2a2)] Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(944) lies on these lines: 1,4 2,355 3,8 10,631 20,145 30,962 40,376 48,281 80,499 84,1000 150,348 390,971 392,452 938,999 958,1006

X(944) = isogonal conjugate of X(945)X(944) = anticomplement of X(355)

X(945)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = a/[3a4 - 2a3(b + c) + (b - c)2(2ab + 2ac - 2bc - b2 - c2 - 2a2)] Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(945) lies on these lines: 78,517 283,859

X(945) = isogonal conjugate of X(944)

X(946)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = bc[a3(b + c) + (b - c)2(a2 - ab - ac - b2 - c2 - 2bc)] Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(946) lies on these lines: 1,4 2,40 3,142 5,10 7,84 8,908 11,65 29,102 30,551 46,499 56,1012 79,104 165,631 238,580 355,381 392,442 496,942 546,952 951,1067

X(946) = isogonal conjugate of X(947)X(946) = complement of X(40)

X(947)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = a/[a3(b + c) + (b - c)2(a2 - ab - ac - b2 - c2 - 2bc)] Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(947) lies on these lines: 29,515 40,77 48,282 73,102 219,572 581,1036 950,1067 951,1066

X(947) = isogonal conjugate of X(946)

X(948)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = bcvw[a3 - a2(b + c) + a(b + c)2 - (b - c)(b2 - c2)], where u : v : w = X(9) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(948) lies on these lines: 1,4 2,85 6,7 37,347 57,169 142,269 220,329 307,966 342,393

X(948) = isogonal conjugate of X(949)

X(949)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = au/[a3 - a2(b + c) + a(b + c)2 - (b - c)(b2 - c2)], where u : v : w = X(9) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(949) lies on these lines: 1,607 2,294 3,41 6,77 48,1037 78,220

X(949) = isogonal conjugate of X(948)

X(950)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = bc(b + c - a)[2a3 + (b + c)(a2 + (b - c)2)] Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(950) lies on these lines: 1,4 8,9 10,55 11,214 20,57 29,284 30,553 35,1006 65,516 72,519 80,943 142,377 145,329 281,380 389,517 440,1104 947,1067

X(950) = isogonal conjugate of X(951)

X(951)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = a/[(b + c - a)[2a3 + (b + c)(a2 + (b - c)2)]] Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(951) lies on these lines: 29,226 56,219 57,78 73,284 77,738 946,1067 947,1066

X(951) = isogonal conjugate of X(950)

X(952)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = bc[2a4 + 2a3(b + c) - a2(b2 - 4bc + c2) + (2a - b - c)(b - c)(b2 - c2)] Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

As the isogonal conjugate of a point on the circumcircle, X(952) lies on the line at infinity.

X(952) lies on these lines: 1,5 3,8 4,145 10,140 30,511 40,550 150,664 182,996 390,1000 546,946 547,551 572,594

X(952) = isogonal conjugate of X(953)

X(953)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = a/[2a4 + 2a3(b + c) - a2(b2 - 4bc + c2) + (2a - b - c)(b - c)(b2 - c2)] Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(953) lies on the circumcircle.

X(953) lies on these lines: 3,901 36,109 100,517 104,513 110,859

X(953) = isogonal conjugate of X(952)

X(954)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = a5 + (b + c)[2a2(b2 + c2 - a2 + bc) - (b - c)2(2bc + ab + ac)] Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(954) lies on these lines: 1,6 3,7 4,390 10,480 21,144 55,226 142,474 971,1012 999,1006

X(954) = isogonal conjugate of X(955)

X(955)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = 1/[a5 + (b + c)[2a2(b2 + c2 - a2 + bc) - (b - c)2(2bc + ab + ac)]] Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(955) lies on these lines: 57,991 278,354 279,942

X(955) = isogonal conjugate of X(954)

X(956)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = a3 - a(b - c)2 - 2bc( b + c) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(956) lies on these lines: 1,6 2,495 3,8 10,56 21,145 55,519 63,517 183,668 210,997 333,859 388,442 452,1058

X(956) = isogonal conjugate of X(957)

X(957)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = 1/[a3 - a(b - c)2 - 2bc( b + c)] Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(957) lies on these lines: 2,392 57,995 81,859

X(957) = isogonal conjugate of X(956)

X(958)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = (b + c - a)(a2 + ab + ac + 2bc) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(958) lies on these lines: 1,6 2,12 3,10 8,21 28,281 36,474 40,1012 48,965 64,65 78,210 104,631 198,966 243,318 452,497 944,1006

X(958) = isogonal conjugate of X(959)X(958) = complement of X(388)

X(959)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = 1/[(b + c - a)(a2 + ab + ac + 2bc)] Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(959) lies on these lines: 1,573 2,65 6,961 7,274 8,181 28,608 56,81 57,1042 193,330

X(959) = isogonal conjugate of X(958)

X(960)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = (b + c - a)(b2 + c2 + ab + ac) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(960) lies on these lines: 1,6 2,65 3,997 5,10 8,210 12,908 19,965 21,60 36,191 40,936 46,474 55,78 56,63 113,123 221,1038 241,1042 329,388 758,942 912,993 978,986

X(960) = isogonal conjugate of X(961)X(960) = complement of X(65)

X(961)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = 1/[(b + c - a)(b2 + c2 + ab + ac)] Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(961) lies on these lines: 1,572 2,12 6,959 57,1106 65,81 105,1104 108,429 274,1014

X(961) = isogonal conjugate of X(960)

X(962)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = bc[a4 + 2a3(b + c) - 4a2bc - (b + c)(b - c)2(2a + b + c)] Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(962) lies on these lines: 1,7 2,40 4,8 30,944 55,411 65,497 145,515 149,151 278,412 382,952 392,443 484,499 942,1058

X(962) = isogonal conjugate of X(963)X(962) = anticomplement of X(40)

X(963)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = a/[a4 + 2a3(b + c) - 4a2bc - (b + c)(b - c)2(2a + b + c)] Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(963) lies on these lines: 3,200 33,56 48,220 55,603

X(963) = isogonal conjugate of X(962)

X(964)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = bc[a4 + (b + c)( a3 + ab2 + ac2 + abc + (b + c)(a2 + bc))] Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(964) lies on these lines: 1,321 2,3 6,8 10,31

X(965)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = a4 - a3(b + c) - a2(b2 + c2) + a(b + c)3 + 2bc(b + c)2 Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(965) lies on these lines: 2,6 3,9 10,219 19,960 37,78 48,958 284,405 474,579

X(966)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = bc[a2 - 2a(b + c) - (b + c)2] Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(966) lies on these lines: 2,6 4,9 8,37 45,346 198,958 307,948 443,579 572,631 581,936

X(966) = isogonal conjugate of X(967)

X(967)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = a/[a2 - 2a(b + c) - (b + c)2] Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(967) lies on these lines: 3,42 25,58 27,393 37,63 81,941

X(967) = isogonal conjugate of X(966)

X(968)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = a2 - 2a(b + c) - (b + c)2 Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(968) lies on these lines: 1,21 9,42 19,25 35,975 45,210 165,750 200,756 614,1001 943,1039

X(968) = isogonal conjugate of X(969)

X(969)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = 1/[a2 - 2a(b + c) - (b + c)2] Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(969) lies on these lines: 7,225 10,69 19,81 37,63 65,77 158,286

X(969) = isogonal conjugate of X(968)

X(970)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = a[a3(b + c)2 + a(ab + ac - 2bc)(b2 + c2) - bc(b3 + c3) - a(b4 + c4) - (b5 + c5)] Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(970) lies on these lines: 1,181 3,6 5,10 21,51 40,43 185,411

X(971)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = a4(b + c) - 2a3(b2 + c2 - bc) + 2a(b - c)2(b2 + c2 + bc) - (b - c)2(b + c)3 Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

As the isogonal conjugate of a point on the circumcircle, X(971) lies on the line at infinity.

X(971) lies on these lines: 3,9 4,7 5,142 6,990 20,72 30,511 33,222 37,991 103,910 165,210 390,944 954,1012

X(971) = isogonal conjugate of X(972)

X(972)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = 1/[a4(b + c) - 2a3(b2 + c2 - bc) + 2a(b - c)2(b2 + c2 + bc) - (b - c)2(b + c)3] Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(972) lies on the circumcircle.

X(972) lies on these lines: 3,934 40,101 55,108 100,329 109,165

X(972) = isogonal conjugate of X(971)

X(973) = 1st EHRMANN POINT

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = u[a10 - 3a8s + a6(2b4 + 3b2c2 + 2c4) + a4s(2b4 - b2c2 + 2c4) + a2(b2 - c2)2(3b4 + 5b2c2 + 3c4) + s(b2 - c2)2(b4 - b2c2 + c4)], where u : v : w = X(51), s = b2 + c2 Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

For constructions of X(973) and X(974), see Hyacinthos message 3695, Sept. 1, 2001, and related messages.

X(973) lies on these lines: 5,15 6,24 68,568

X(974) = 2nd EHRMANN POINT

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = [sa10 - a8(3b4 - 2b2c2 + 3c4) + a6s(2b4 - 3b2c2 + 2c4) + a4(b2 - c2)2(2b4 - 7b2c2 + 2c4) - 3a2s(b2 - c2)2(b4 - 3b2c2 + c4) + b2 - c2)4(b4 + b2c2 + c4)] cos A, where s = b2 + c2 Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

For constructions of X(973) and X(974), see Hyacinthos message 3695, Sept. 1, 2001, and related messages.

X(974) lies on these lines: 5,113 6,74

X(975)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = a3 + a2(b + c) + a(b2 + c2 + 4bc) + (b + c)3 Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(975) lies on these lines: 1,2 3,37 9,58 28,33 35,968 46,750 57,201 72,940 226,1038 312,1010

X(976)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = a3 + (b + c)(b2 + c2) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(976) lies on these lines: 1,2 3,38 21,983 31,72 37,41 66,73 100,986 210,1104 244,474 404,982 405,756 601,912 750,942 1060,1066

X(976) = isogonal conjugate of X(977)

X(977)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = 1/[a3 + (b + c)(b2 + c2)] Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(977) lies on these lines: 22,56 58,982 106,833

X(977) = isogonal conjugate of X(976)

X(978)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = a2(b + c) + a(b2 - bc + c2) - bc(b + c) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(978) lies on these lines: 1,2 3,238 9,39 21,748 31,404 40,1050 46,1054 56,979 57,1046 58,87 72,982 171,474 266,361 631,1064 651,1106 960,986

X(978) = isogonal conjugate of X(979)

X(979)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = 1/[a2(b + c) + a(b2 - bc + c2) - bc(b + c)] Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(979) lies on these lines: 10,87 43,58 56,978

X(979) = isogonal conjugate of X(978)

X(980)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = a2(b2 + bc + c2) + (b2 + c2)(bc + ca + ab) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(980) lies on these lines: 1,3 2,39 32,81 38,869 63,213

X(980) = isogonal conjugate of X(981)

X(981)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = 1/[a2(b2 + bc + c2) + (b2 + c2)(bc + ca + ab)] Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(981) lies on these lines: 6,314 8,213 21,32 256,573

X(981) = isogonal conjugate of X(980)

X(982)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = b2 - bc + c2 Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(982) lies on these lines: 1,3 2,38 7,256 43,518 58,977 63,238 72,978 81,985 222,613 226,262 240,278 257,330 310,871 312,726 404,967 758,995 846,1001

X(982) = isogonal conjugate of X(983)

X(983)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = 1/[b2 - bc + c2] Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(983) lies on these lines: 1,182 7,171 8,238 21,976 55,256

X(983) = isogonal conjugate of X(982)

X(984)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = b2 + bc + c2 Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(984) lies on these lines: 1,6 2,38 8,192 10,75 21,976 43,210 55,846 63,171 100,753 101,761 201,388 240,281 386,872 519,751 936,988

X(984) = isogonal conjugate of X(985)X(984) = isotomic conjugate of X(870)

X(985)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = 1/[b2 + bc + c2] Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(985) lies on these lines: 1,32 2,31 6,291 58,274 81,982 105,825 279,1106 727,789

X(985) = isogonal conjugate of X(984)

X(986)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = a(b2 + bc + c2) + b3 + c3 Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(986) lies on these lines: 1,3 4,240 6,1046 8,38 10,75 43,72 100,976 194,257 291,337 386,758 405,846 474,1054 774,938 960,978

X(986) = isogonal conjugate of X(987)

X(987)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = 1/[a(b2 + bc + c2) + b3 + c3] Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(987) lies on these lines: 3,256 4,171 7,1106 8,31 9,32 58,314

X(987) = isogonal conjugate of X(986)

X(988)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = a3 - a2(b + c) - (3a + b + c)(b2 + c2) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(988) lies on these lines: 1,3 9,39 21,614 38,78 77,1106 84,256 404,612 936,984

X(988) = isogonal conjugate of X(989)

X(989)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = 1/[a3 - a2(b + c) - (3a + b + c)(b2 + c2)] Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(989) lies on these lines: 21,612 40,256 84,171

X(989) = isogonal conjugate of X(988)

X(990)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = a5 - a4(b + c) - 2a3bc - a(b - c)2(b2 + c2) + (b - c)2(b + c)3 Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(990) lies on these lines: 1,7 3,37 6,971 33,57 58,84 165,612 226,1040

X(991)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = a[a3(b + c) - a2(b2 - bc + c2) - a(b + c)(b2 + c2) + (b - c)(b3 - c3)] Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(991) lies on these lines: 1,7 3,37 6,971 33,57 58,84 165,612 226,1040

X(992)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = a3(b + c) + a2(b2 + c2) - abc(b + c) - bc(b + c)2 Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(992) lies on these lines: 2,6 9,39 44,583 238,1009

X(993)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = a3 - a(b2 + c2) - bc(b + c) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(993) lies on these lines: 1,21 2,36 3,10 8,35 9,48 32,1107 55,519 56,226 75,99 87,106 238,995 495,529 516,1012 527,551 912,960

X(993) = isogonal conjugate of X(994)

X(994)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = 1/[a3 - a(b2 + c2) - bc(b + c)] Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(994) lies on these lines: 10,908 31,759 37,517 65,386 75,758

X(994) = isogonal conjugate of X(993)

X(995)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = a(ab + ac - bc + b2 + c2) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(995) lies on these lines: 1,2 3,595 6,101 31,36 56,58 57,957 238,993 581,1104 609,1055 758,982 991,1064

X(995) = isogonal conjugate of X(996)

X(996)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = 1/[a(ab + ac - bc + b2 + c2)] Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(996) lies on these lines: 2,106 6,519 8,58 10,56 182,952

X(996) = isogonal conjugate of X(995)

X(997)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = a3 - a2(b + c) - a(b - c)2 + (b + c)(b2 + c2) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(997) lies on these lines: 1,2 3,960 9,48 21,90 34,860 36,63 46,404 55,392 56,72 57,758 65,474 141,1060 210,956 518,999

X(997) = isogonal conjugate of X(998)

X(998)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = 1/[a3 - a2(b + c) - a(b - c)2 + (b + c)(b2 + c2)] Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(998) lies on these lines: 1,908 6,517 46,58 106,614

X(998) = isogonal conjugate of X(997)

X(999)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = a(a2 + 4bc - b2 - c2) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(999) lies on these lines: 1,3 2,495 4,496 5,388 6,101 7,104 8,474 11,381 12,499 20,1058 30,497 63,392 77,1057 78,1059 81,859 145,404 329,405 376,390 518,997 527,551 601,1106 938,944 954,1006

X(999) = isogonal conjugate of X(1000)

X(1000)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = 1/[a(a2 + 4bc - b2 - c2)] Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(1000) lies on these lines: 1,631 7,517 8,392 9,519 21,145 55,104 79,388 80,497 84,944 390,952

X(1000) = isogonal conjugate of X(999)

X(1001)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = a2 - a(b + c) - 2bc Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(1001) lies on these lines: 1,6 2,11 3,142 7,21 8,344 31,940 35,474 42,748 63,354 182,692 388,452 527,551 529,1056 614,968 750,902 846,982 943,1058

X(1001) = isogonal conjugate of X(1002)

X(1002)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = 1/[a2 - a(b + c) - 2bc] Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(1002) lies on these lines: 1,672 2,210 6,105 8,274 28,607 42,57 55,81 65,279 145,330 277,942

X(1002) = isogonal conjugate of X(1001)

X(1003)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = bc(3a4 - a2b2 - a2c2 + 2b2c2) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(1003) lies on these lines: 2,3 6,99 32,538 183,187

X(1004)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = a5 - 2a4(b + c) + 2a2(b3 + c3) - a(b2 + c2)2 + 2bc(b - c)(b2 - c2) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(1004) lies on these lines: 2,3 7,100 46,200 63,210 65,224

X(1005)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = a5 - 2a4(b + c) - a3bc + a2(2b3 + 2c3 + b2c + bc2) - a(b4 + c4 - b3c - bc3 - 4b2c2) + bc(b - c)(b2 - c2) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(1005) lies on these lines: 2,3 9,100 55,329 108,342

X(1006)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = a6 - a5(b + c) - a4(2b2 + bc + 2c2) + 2a3(b3 + c3) + a2(b4 + c4 + 2bc(b2 + c2) + 2b2c2) - a(b5) + c5 - bc(b3 + c3)) - bc(b2 - c2) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(1006) lies on these lines: 1,201 2,3 9,48 35,950 36,226 54,72 238,1064 944,958 954,999

X(1007)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = bc[a4 - 4a2(b2 + c2) + 3b4 - 2b2c2) + 3c4] Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(1007) lies on these lines: 2,6 4,99 305,311 315,631 316,376 317,459

X(1008)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = bc[a5(b + c) + a4(b + c)2 + a3(b + c)(b2 + bc + 2c2) + a2(b2 + c2) + bc)2 + abc(b + c)3 + b2c2(a + b)2] - a(b5) + c5 - bc(b3 + c3)) - bc(b2 - c2) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(1008) lies on these lines: 1,76 2,3

X(1009)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = a4(b + c) + 2a3bc - a2(b3 + c3) + bc(b + c)(b2 + c2) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(1009) lies on these lines: 1,39 2,3 72,894 283,1065 518,583

X(1010)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = bc[a2 + (b + c)2]/(b + c) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(1010) lies on these lines: 1,75 2,3 8,81 10,58 72,894 283,1065 312,975 759,833

X(1011)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = a[a3(b + c) + a2bc - a(b + c)(b2 + c2) - bc(b + c)2] Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(1011) lies on these lines: 2,3 6,31 9,228 35,43 51,573 184,572

X(1012)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = a7 - a5r + 2a4bc(b + c) + a3s - a(b + c)2(b2 - c2)2 - 2bc(b + c)(b2 - c2)2, where r = 3b2 - 2bc + 3c2 and s = 3b4 + 2b2c2 + 3c4 Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(1012) lies on these lines: 1,84 2,3 7,104 40,958 55,515 56,946 63,517 268,281 516,993 954,971

X(1013)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = a8 - a7u1 - a6u2 + a5u1u2 - a4(b4 - 4b2c2 + c4)+ a3u1u3 + a2u2u3 - au1u2u3 - 2b2c2u3, u1 = b + c, u2 = b2 + c2, u3 = (b2 - c2)2

Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(1013) lies on these lines: 2,3 6,162 7,108 33,63 55,92 100,281

X(1014)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = 1/[(b + c)(b + c - a)] Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(1014) lies on these lines: 7,21 28,279 57,77 58,269 60,757 69,404 261,552 272,1088 274,961 332,1037 759,934

X(1015)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B),where f(A,B,C) = sin A sin2(A/2) [1 - cos(B - C)] Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = f(A,B,C) sin A

X(1015) lies on these lines: 1,39 2,668 6,101 11,115 32,56 36,187 37,537 55,574 76,330 214,1100 216,1060 244,665 350,538 812,1086

X(1015) = isogonal conjugate of X(1016)X(1015) = complement of X(668)

X(1016)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B),where f(A,B,C) = 1/[sin A sin2(A/2) [1 - cos(B - C)]] Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = f(A,B,C) sin A

X(1016) lies on these lines: 8,1083 99,813 100,667 190,514 238,519 512,660 644,666

X(1016) = isogonal conjugate of X(1015)X(1016) = complement of X(1086)

X(1017)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = a(b + c - 2a)2 Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(1017) lies on these lines: 6,101 44,214

X(1018)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = (b + c)/(b - c) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(1018) lies on these lines: 1,39 9,80 40,728 63,544 99,813 100,101 163,643 190,646 346,573 519,672 664,1025

X(1018) = isogonal conjugate of X(1019)

X(1019)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = (b - c)/(b + c) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(1019) lies on these lines: 1,512 36,238 58,1027 81,1022 99,813 239,514 759,840

X(1019) = isogonal conjugate of X(1018)

X(1020)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = (cos B + cos C)/(cos B - cos C) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(1020) lies on these lines: 1,185 57,1986 101,651 108,109 190,658 269,292 347,573 648,1021

X(1020) = isogonal conjugate of X(1021)

X(1021)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = (cos B - cos C)/(cos B + cos C) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(1021) lies on these lines: 1,647 239,514 243,522 333,1024 521,650 648,1020

X(1021) = isogonal conjugate of X(1020)

X(1022)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = (b - c)/(2a - b - c) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(1022) lies on these lines: 1,513 2,514 81,1019 89,649 105,106 291,876 812,903

X(1022) = isogonal conjugate of X(1023)

X(1023)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = (2a - b - c)/(b - c) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(1023) lies on these lines: 1,6 100,101 813,898

X(1023) = isogonal conjugate of X(1022)

X(1024)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = (b - c)(b + c - a)/[b2 + c2 - a(b + c)] Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(1024) lies on these lines: 6,513 9,522 55,650 57,649 333,1021 673,812

X(1024) = isogonal conjugate of X(1025)

X(1025)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = [b2 + c2 - a(b + c)]/[(b - c)(b + c - a)] Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(1025) lies on these lines: 2,7 56,1083 100,109 190,658 644,934 664,1018 813,927

X(1025) = isogonal conjugate of X(1024)

X(1026)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = [b2 + c2 - a(b + c)]/(b - c) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(1026) lies on these lines: 1,2 55,1083 100,101 664,668 666,1027

X(1026) = isogonal conjugate of X(1027)

X(1027)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = (b - c)/[b2 + c2 - a(b + c)] Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(1027) lies on these lines: 1,514 6,513 56,667 58,1019 105,106 292,659 666,1026

X(1027) = isogonal conjugate of X(1026)

X(1028)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = 2a2bc + [(b2 - c2)2 - a4]/(b + c - a) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(1028) is the center of the conic that passes through the six points of tangency of the excircles with the sidelines of triangle ABC. (Paul Yiu, "The Clawson Point and Excircles," Dec., 1999.)

X(1028) lies on this line: 6,19

X(1029)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = 1/[a + 2(a + b + c) cos A] Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(1029) lies on these lines: 10,191 115,593 319,321

X(1029) = isogonal conjugate of X(1030)X(1029) = cyclocevian conjugate of X(1)

X(1030)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = a + 2(a + b + c) cos A Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(1030) lies on these lines: 3,6 35,37 36,1100 45,198 55,199 100,594

X(1030) = isogonal conjugate of X(1029)

X(1031)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = 1/[a(w2u2 + u2v2 - v2w2) + 2uvw(au + bv + cw) cos A], where u : v : w = X(6) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(1031) lies on this line: 141,384

X(1031) = cyclocevian conjugate of X(6)

X(1032)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = 1/[a(w2u2 + u2v2 - v2w2) + 2uvw(au + bv + cw) cos A], where u : v : w = X(20) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(1032) lies on this line: 20,394

X(1032) = isogonal conjugate of X(1033)X(1032) = cyclocevian conjugate of X(20)

X(1033)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = a(w2u2 + u2v2 - v2w2) + 2uvw(au + bv + cw) cos A, where u : v : w = X(20) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(1033) lies on these lines: 6,64 19,56 25,393 55,204

X(1033) = isogonal conjugate of X(1032)

X(1034)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = 1/[a(w2u2 + u2v2 - v2w2) + 2uvw(au + bv + cw) cos A], where u : v : w = X(329) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(1034) lies on these lines: 2,271 20,78

X(1034) = isogonal conjugate of X(1035)X(1034) = cyclocevian conjugate of X(329)

X(1035)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = a(w2u2 + u2v2 - v2w2) + 2uvw(au + bv + cw) cos A, where u : v : w = X(329) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(1035) lies on these lines: 3,223 6,603 25,34 55,64 222,581

X(1035) = isogonal conjugate of X(1034)

X(1036)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B),where f(A,B,C) = 1/(1 + cos B cos C) Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)

X(1036) lies on these lines: 1,25 3,31 4,1065 21,332 29,497 41,219 55,78 56,77 73,1037 282,380 581,947 1058,1067 1059,1066

X(1036) = isogonal conjugate of X(388)

X(1037)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B),where f(A,B,C) = 1/(1 - cos B cos C) Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)

X(1037) lies on these lines: 1,1041 3,1066 4,1067 29,388 48,949 55,77 56,78 73,1036 219,604 332,1014 916,1069 1056,1065 1057,1064

X(1037) = isogonal conjugate of X(497)

X(1038)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B),where f(A,B,C) = cos A + cos A cos B cos C Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)

X(1038) lies on these lines: 1,3 2,34 4,1076 9,478 20,33 21,1041 38,1106 63,210 69,73 72,222 172,577 221,960 223,936 225,377 226,975 278,443 388,612 1068,1074

X(1038) = isogonal conjugate of X(1039)

X(1039)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B),where f(A,B,C) = 1/(cos A + cos A cos B cos C) Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)

X(1039) lies on these lines: 1,25 4,1096 7,34 8,33 9,607 21,1040 29,314 65,1041 943,968

X(1039) = isogonal conjugate of X(1038)

X(1040)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B),where f(A,B,C) = cos A - cos A cos B cos C Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)

X(1040) lies on these lines: 1,3 2,33 4,1074 20,34 21,1039 63,212 78,345 226,990 243,1096 497,614 1068,1076

X(1040) = isogonal conjugate of X(1041)

X(1041)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B),where f(A,B,C) = 1/(cos A - cos A cos B cos C) Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)

X(1041) lies on these lines: 1,1037 7,33 8,34 9,608 19,294 21,1038 65,1039

X(1041) = isogonal conjugate of X(1040)

X(1042)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B),where f(A,B,C) = (1 - cos A)(cos B + cos C) Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)

X(1042) lies on these lines: 1,7 31,56 34,207 42,65 57,959 241,960 517,1066 604,608 741,934 942,1064

X(1042) = isogonal conjugate of X(1043)

X(1043)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B),where f(A,B,C) = 1/[(1 - cos A)(cos B + cos C)] Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)

X(1043) lies on these lines: 1,75 8,21 20,64 27,306 29,33 58,519 72,190 81,145 99,103 200,341 220,346 239,1104 280,285 283,643 286,322

X(1043) = isogonal conjugate of X(1042)

X(1044)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B),where f(A,B,C) = cos B + cos C - cos A + cos B cos C - cos A cos B - cos A cos C Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)

X(1044) lies on these lines: 1,7 43,46

X(1045)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = -u2 + v2 + w2 + vw + wu + uv, u : v : w = X(2) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(1045) lies on these lines: 1,75 9,43 40,511 42,894 190,872 192,869

X(1046)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = -u2 + v2 + w2 + vw + wu + uv, u : v : w = X(3) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(1046) lies on these lines: 1,21 6,986 10,894 40,511 43,46 57,978 72,171 238,942 484,1048

X(1047)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = -u2 + v2 + w2 + vw + wu + uv, u : v : w = X(4) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(1047) lies on these lines: 1,29 43,46

X(1048)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = -u2 + v2 + w2 + vw + wu + uv, u : v : w = X(5) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(1048) lies on these lines: 1,564 484,1046

X(1049)

Trilinears A : B : C Barycentrics A sin A : B sin B : C sin C

X(1049) = isogonal conjugate of X(1085)

X(1050)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = -u2 + v2 + w2 + vw + wu + uv, u : v : w = X(8) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(1050) lies on these lines: 1,341 40,978

X(1051)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = -u2 + v2 + w2 + vw + wu + uv, u : v : w = X(37) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(1051) lies on these lines: 1,748 6,846 81,1054 165,572

X(1052)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = -u2 + v2 + w2 + vw + wu + uv, u : v : w = X(100) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(1052) lies on these lines: 1,765 238,517 513,1054

X(1053)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = -u2 + v2 + w2 + vw + wu + uv, u : v : w = X(101) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(1053) lies on these lines: 1,1110 238,517 905,1054

X(1054)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = -u2 + v2 + w2 + vw + wu + uv, u : v : w = X(513) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(1054) lies on these lines: 1,88 2,846 43,57 46,978 81,1051 105,165 474,986 513,1052 905,1053

X(1055)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = a(b2 + c2 - 2a2 + ab + ac - 2bc) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(1055) lies on these lines: 6,41 36,101 106,919 187,237 609,995

X(1056)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B),where f(A,B,C) = 2 + cos B cos C Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)

X(1056) lies on these lines: 1,4 2,495 7,517 8,443 29,1059 30,390 55,376 56,631 145,377 329,392 355,938 529,1001 1037,1065

X(1056) = isogonal conjugate of X(1057)

X(1057)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B),where f(A,B,C) = 1/(2 + cos B cos C) Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)

X(1057) lies on these lines: 29,1058 73,1059 77,999 78,392 497,1065 1037,1064

X(1057) = isogonal conjugate of X(1056)

X(1058)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B),where f(A,B,C) = 2 - cos B cos C Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)

X(1058) lies on these lines: 1,4 2,496 3,390 8,392 20,999 20,1057 55,631 56,376 149,377 452,956 517,938 942,962 943,1001 1036,1067

X(1058) = isogonal conjugate of X(1059)

X(1059)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B),where f(A,B,C) = 1/(2 - cos B cos C) Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)

X(1059) lies on these lines: 29,1056 73,1057 78,999 388,1067 1036,1066

X(1059) = isogonal conjugate of X(1058)

X(1060)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B),where f(A,B,C) = 2 + sec B sec C Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)

X(1060) lies on these lines: 1,3 5,34 21,1063 30,33 68,73 72,394 141,997 201,255 216,1015 222,912 377,1068 495,612 601,774 976,1066

X(1060) = isogonal conjugate of X(1061)

X(1061)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B),where f(A,B,C) = 1/(2 + sec B sec C) Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)

X(1061) lies on these lines: 1,24 8,406 21,1062 33,80 34,79 65,1063

X(1061) = isogonal conjugate of X(1060)

X(1062)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B),where f(A,B,C) = 2 - sec B sec C Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)

X(1062) lies on these lines: 1,3 5,33 21,1061 30,34 394,1069 496,614 602,774

X(1062) = isogonal conjugate of X(1063)

X(1063)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B),where f(A,B,C) = 1/(2 - sec B sec C) Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)

X(1063) lies on these lines: 1,378 8,475 21,1060 33,79 34,80 65,1061

X(1063) = isogonal conjugate of X(1062)

X(1064)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B),where f(A,B,C) = 1 + cos A (cos B + cos C) Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)

X(1064) lies on these lines: 1,4 3,31 38,912 40,386 42,517 102,112 104,256 238,1006 631,978 942,1042 991,995 1037,1057

X(1064) = isogonal conjugate of X(1065)

X(1065)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B),where f(A,B,C) = 1/[1 + cos A (cos B + cos C)] Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)

X(1065) lies on these lines: 3,388 4,1036 102,226 283,1010 284,515 497,1057 1037,1056

X(1065) = isogonal conjugate of X(1064)

X(1066)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B),where f(A,B,C) = 1 - cos A (cos B + cos C) Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)

X(1066) lies on these lines: 1,4 3,1037 42,942 222,601 517,1042 774,912 947,951 976,1060 1036,1059

X(1066) = isogonal conjugate of X(1067)

X(1067)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B),where f(A,B,C) = 1/[1 - cos A (cos B + cos C)] Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)

X(1067) lies on these lines: 3,496 4,1037 388,1059 946,951 947,950 1036,1058

X(1067) = isogonal conjugate of X(1066)

X(1068)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B),where f(A,B,C) = 1 - sec A (cos B + cos C) Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)

X(1068) lies on these lines: 1,4 8,860 24,108 92,406 155,651 281,451 318,475 377,1060 429,495 1038,1074 1040,1076

X(1068) = isogonal conjugate of X(1069)

X(1069)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B),where f(A,B,C) = 1/[1 - sec A (cos B + cos C)] Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)

X(1069) lies on these lines: 1,90 11,68 394,1062 496,613 916,1037

X(1069) = isogonal conjugate of X(1068)

X(1070)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B),where f(A,B,C) = 1 + cos B cos C (cos B + cos C) Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)

X(1070) lies on these lines: 1,4 55,1076 56,1074

X(1071)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = a(b2 + c2 - a2)[a3(b + c) - a2(b - c)2 + (b2 - c2)2] Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(1071) appears in Hyacinthos message #3849, Paul Yiu, Sept. 19, 2001.

X(1072)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B),where f(A,B,C) = 1 - cos2B cos C - cos B cos2C Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)

X(1072) lies on these lines: 1,4 55,1074 56,1076

X(1073)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B),where f(A,B,C) = (cot A)/(cos A - cos B cos C) Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)

X(1073) lies on these lines: 2,253 3,64 9,223 222,268

X(1074)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B),where f(A,B,C) = cos A + cos2B cos C + cos B cos2C Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)

X(1074) lies on these lines: 1,224 3,225 4,1040 55,1072 56,1070 1038,1068

X(1075)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B),where f(A,B,C) = cos B cos C (cos2C cos2A + cos2A cos2B - cos2B cos2C) Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)

X(1075) lies on these lines: 4,15 155,450 216,631 243,920 648,1092

X(1076)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B),where f(A,B,C) = cos A - cos2B cos C - cos B cos2C Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)

X(1076) lies on these lines: 3,225 4,1038 55,1070 56,1072 1040,1068

X(1077)

Trilinears 1/A : 1/B : 1/C Barycentrics (sin A)/A : (sin B)/B : (sin C)/C

X(1078) = REFLECTION OF X(64) IN X(3)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B),where f(A,B,C) = cos A - 1/(cos A - cos B cos C) Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)

X(1078) lies on these lines: 1,84 3,64 4,6 20,394 25,185 30,155 40,219 195,382

X(1079) = REFLECTION OF X(84) IN X(3)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B),where f(A,B,C) = cos A - 1/(cos B + cos C - cos A - 1) Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)

X(1079) lies on these lines: 1,4 3,9 20,78 40,64 63,411 165,191 224,908 386,990 975,991 1045,1047

X(1080)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), wheref(A,B,C) = csc(B - C) [sin 2B cos(C - ω) sin(C - π/3) - sin 2C cos(B - ω) sin(B - π/3)]

Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

Coordinates for X(1080) are obtained from those of X(383) by changing π/3 to - π/3; contributed by Edward Brisse.

X(1080) lies on these lines: 2,3 13,98 14,262 183,622 298,511 325,621

X(1080) = inverse of X(383) in the orthocentroidal circle

X(1081)

Trilinears sec(A/2) csc(A/2 - π/3) : sec(B/2) csc(B/2 - π/3) : sec(C/2) csc(C/2 - π/3) Barycentrics sin A csc(A/2 - π/3) : sin B csc(B/2 - π/3) : sin C csc(C/2 - π/3)

Coordinates for X(1081) are obtained from those of X(554) by changing π/3 to - π/3; contributed by Edward Brisse.

X(1081) lies on these lines: 1,30 7,559 13,226 75,298

X(1082)

Trilinears (sec A/2) sin(A/2 - π/3) : (sec B/2) sin(B/2 - π/3) : (sec C/2) sin(C/2 - π/3) Barycentrics (sin A/2) sin(A/2 - π/3) : (sin B/2) sin(B/2 - π/3) : (sin C/2) sin(C/2 - π/3)

Coordinates for X(1082) are obtained from those of X(559) by changing π/3 to - π/3; contributed by Edward Brisse.

X(1082) lies on these lines: 1,3 7,554 13,226 298,319

X(1083)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = a4 - a3(b + c) - a2)bc + 2abc(b + c) - bc(b2 + c2) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(1083) lies on a circle related to the 1st and 2nd Brocard points; Hyacinthos message #4053, Paul Yiu, Oct. 4, 2001.

X(1083) lies on these lines: 1,6 3,667 8,1016 55,1026 56,1025 105,644 840,898

X(1083) = inverse of X(667) in the orthocentroidal circle

X(1084)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = a3(b2 - c2)2 Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

The line f(a,b,c)x + f(b,c,a)y + f(c,a,b)z = 0 is tangent to the circumcircle at X(99).

X(1084) lies on these lines: 2,670 6,694 39,597 115,804 351,865

X(1085)

Trilinears A2 : B2 : C2 Barycentrics A2 sin A : B2 sin B : C2 sin C

X(1086)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = bc(b - c)2 Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = (b - c)2

The line f(a,b,c)x + f(b,c,a)y + f(c,a,b)z = 0 is tangent to the circumcircle at X(101).

X(1086) lies on these lines: 1,528 2,45 6,7 8,599 10,537 11,244 37,142 44,527 53,273 57,1020 75,141 115,116 220,277 239,320 812,1015 918,1111

X(1086) = isotomic conjugate of X(1016)X(1086) = complement of X(190)

X(1087) = TRILINEAR SQUARE OF X(5)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B),where f(A,B,C) = cos2(B - C) Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)

X(1087) lies on these lines: 1,564 31,91 92,255

X(1088) = TRILINEAR SQUARE OF X(7)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B),where f(A,B,C) = sec4(A/2) Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)

X(1088) lies on these lines: 2,85 7,354 57,658 86,269 234,555 272,1014 305,341 675,934

X(1088) = isotomic conjugate of X(200)

X(1089) = TRILINEAR SQUARE OF X(10)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = b2c2(b + c)2 Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(1089) lies on these lines: 1,312 8,80 10,321 76,334 190,191 200,318 244,596 345,498 594,762 740,872

X(1089) = isogonal conjugate of X(849)X(1089) = isotomic conjugate of X(757)

X(1090) = TRILINEAR SQUARE OF X(11)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B),where f(A,B,C) = [1 - cos(B - C)]2 Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)

X(1090) lies on these lines: 5,1091 11,523

X(1091) = TRILINEAR SQUARE OF X(121)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B),where f(A,B,C) = [1 + cos(B - C)]2 Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)

X(1091) lies on these lines: 5,1090 12,1109

X(1092) = TRILINEAR CUBE OF X(3)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B),where f(A,B,C) = cos3A Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)

X(1092) lies on these lines: 2,578 3,49 4,801 20,110 24,511 54,69 68,125 140,343 156,550 450,1093 648,1075

X(1092) = isogonal conjugate of X(1093)

X(1093) = TRILINEAR CUBE OF X(4)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B),where f(A,B,C) = sec3A Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)

X(1093) lies on these lines: 3,1105 4,51 5,264 24,107 155,648 158,225 393,800 403,847 436,578 450,1092

X(1093) = isogonal conjugate of X(1092)

X(1094) = TRILINEAR SQUARE OF X(15)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B),where f(A,B,C) = sin2(A + π/3) Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)

X(1094) lies on these lines: 15,36 48,163

X(1095) = TRILINEAR SQUARE OF X(16)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B),where f(A,B,C) = sin2(A - π/3) Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)

X(1095) lies on these lines: 16,36 48,163

X(1096) = TRILINEAR SQUARE OF X(19)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B),where f(A,B,C) = tan2A Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)

X(1096) lies on these lines: 1,29 4,1039 19,31 33,42 34,207 63,240 107,741 213,607 243,1040 278,614 281,612

X(1096) = isogonal conjugate of X(326)

X(1097) = TRILINEAR SQUARE OF X(20)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B),where f(A,B,C) = (cos A - cos B cos C)2A Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)

X(1097) lies on these lines: 1,75 31,775

X(1098) = TRILINEAR SQUARE OF X(21)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B),where f(A,B,C) = 1/(cos B - cos C)2 Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)

X(1098) lies on these lines: 3,662 8,643 21,60 29,270 58,86 65,409 81,1104

X(1099) = TRILINEAR SQUARE OF X(30)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B),where f(A,B,C) = (cos A - 2 cos B cos C)2A Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)

X(1099) lies on these lines: 1,564 75,811 162,255

X(1100)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = 2avw + cv2 + bw2 + u(bv + cw), u : v : w = X(37) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(1100) is the midpoint of the bicentric pair y : z : x and z : x : y, where x : y : z = X(37)

X(1100) lies on these lines: 1,6 2,319 36,1030 48,354 65,604 71,583 81,593 86,239 214,1015 284,942 517,572 519,594 536,894 820,836

X(1100) = complement of X(319)

X(1101) = TRILINEAR SQUARE OF X(110)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B),where f(A,B,C) = csc2(B - C) Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)

X(1101) lies on these lines: 59,60 163,798 656,662

X(1101) = isogonal conjugate of X(1109)

X(1102) = TRILINEAR CUBE OF X(63)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B),where f(A,B,C) = cot3A Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)

X(1102) lies on these lines: 63,304 255,326

X(1103) = TRILINEAR SQUARE OF X(40)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B),where f(A,B,C) = (cos B + cos C - cos A - 1)2A Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)

X(1103) lies on these lines: 1,2 31,937 40,221 46,269 165,255

X(1104)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = 2avw + cv2 + bw2 + u(bv + cw), u : v : w = X(72) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(1104) is the midpoint of the bicentric pair y : z : x and z : x : y, where x : y : z = X(72)

X(1104) lies on these lines: 1,6 11,429 25,34 31,65 32,910 58,942 81,1098 105,961 210,976 229,593 239,1043 440,950 517,580 581,995

X(1105)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B),where f(A,B,C) = (sec A)/(cos2B + cos2C) Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)

X(1105) lies on these lines: 3,1093 4,801 20,393 185,648 225,412 243,411 378,847

X(1105) = isogonal conjugate of X(185)

X(1106) = TRILINEAR SQUARE OF X(56)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B),where f(A,B,C) = (1 + cos A)2A Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)

X(1106) lies on these lines: 3,1037 7,987 31,56 32,604 34,244 36,255 38,1038 57,961 58,269 77,988 279,985 388,750 601,999 651,978 727,934

X(1106) = isogonal conjugate of X(341)

X(1107)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 2avw + cv2 + bw2 + u(bv + cw), where u : v : w = X(213) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(1107) is the midpoint of the bicentric pair y : z : x and z : x : y, where x : y : z = X(213)

X(1107) lies on these lines: 1,6 2,330 10,39 32,993 75,194 210,869 239,257

X(1108)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 2avw + cv2 + bw2 + u(bv + cw), where u : v : w = X(219) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(1108) is the midpoint of the bicentric pair y : z : x and z : x : y, where x : y : z = X(219)

X(1108) lies on these lines: 1,6 2,322 19,56 104,112 241,347 278,393 517,579

X(1108) = complement of X(322)

X(1109) = TRILINEAR SQUARE OF X(523)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B),where f(A,B,C) = csc2(B - C) Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)

X(1109) lies on these lines: 1,564 11,523 12,1091 31,92 75,799 91,255

X(1109) = isogonal conjugate of X(1101)

X(1110) = TRILINEAR SQUARE OF X(101)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = [a/(b - c)]2 Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(1110) lies on these lines: 1,1053 36,59 101,663 249,849 667,692

X(1110) = isogonal conjugate of X(1111)

X(1111) = TRILINEAR SQUARE OF X(514)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = [(b - c)/a]2 Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(1111) lies on these lines: 1,85 7,80 75,537 76,334 269,273 348,499 918,1086

X(1111) = isogonal conjugate of X(1110)X(1111) = isotomic conjugate of X(765)

X(1112)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = a[a4(b2 + c2) - 2a2(b4 + c4) + b6 + c6 ]/(b2 + c2 - a2) Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(1112) is the center of the conic that passes through the vertices of the cevian triangles of X(4) and X(648), and also through the centers X(i) for i = 4, 113, 155, 193. (Paul Yiu, Oct. 16, 2001, as contributing editor for "Conics associated with a cevian nest," Forum Geometricorum 1 (2001) 141-150; see Example 2.)

X(1112) lies on these lines: 4,94 25,110 51,125 52,113 389,974 428,542 468,511

X(1113)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = (a'/a")1/3, a' = [ca(a2 - c2 + bd][ba(a2 - b2) + cd], a" = [(b2 - c2)(b2 + c2 - a2]2, d = (a6 + b6 + c6 + 3a2b2c2 - S)1/2, S = b2c2(b2 + c2) + c2a2(c2 + a2) + a2b2(a2 + b2)

Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(1113) is a point of intersection of the Euler line and the circumcircle. Its antipode is X(1114).

X(1113) lies on this line: 2,3

X(1114)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),where f(a,b,c) = (a'/a")1/3, where a' = [ca(a2 - c2 - bd][ba(a2 - b2] - cd], a" and d as for X(1113)

Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(1114) is a point of intersection of the Euler line and the circumcircle. Its antipode is X(1113).

X(1114) lies on this line: 2,3

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