PONCELET PARABOLA PIROUETTES

DAN REZNIK

Abstract. We describe some three-dozen curious phenomena manifested byparabolas inscribed or circumscribed about certain Poncelet triangle families.Despite their pirouetting motion, parabolas’ focus, vertex, directrix, etc., willoften sweep or envelop rather elementary loci such as lines, circles, or points.Most phenomena are unproven though supported by solid numerical evidence(proofs are welcome). Some yet unrealized experiments are posed as “challenges”(results are welcome!).

Keywords locus, Poncelet, ellipse, inscribed, circumscribed, parabola, per-spector, focus, vertex.

MSC 51M04 and 51N20 and 51N35 and 68T20

1. Introduction

We visit three-dozen surprising Euclidean phenomena manifested by parabolasdynamically inscribed or circumscribed about certain Poncelet families of triangles.As shown in Figure 1, these are triangles simultaneously inscribed to an outer conicand circumscribed about an inner one. Loci and invariants of such families areexplored in [5, 10, 11].

Referring to Figure 2, every triangle is associated with a 1d family of circumpa-rabolas wich pass through the three vertices. These can be swept as (i) the imageunder isogonal conjugation of lines tangent to the circumcircle, or (ii) as the image

Date: October, 2021.

X3 X2

Figure 1. Left: Poncelet triangles inscribed in a circle (fixed circumcircle) and circumscribing aconic. The family has a common circumcenter X3. Right: a Poncelet triangles interscribed be-tween two concentric, homothetic ellipses, where the outer (resp. inner) is the Steiner circumellipse(resp. inellipse). The barycenter X2 is stationary at the center.

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FV

Q

X3

P1

P2

P3F

V

Q

X2

P1

P2

P3

Figure 2. Left: A triangle’s circumparabola (red) passes thru the 3 vertices and is the isogonalimage of a line tangent to the circumcircle at a point Q. Also shown is the vertex V and thedirectrix (dashed red). Right: Alternatively, a circumparabola is also the isotomic image of a linetangent to the Steiner ellipse at a point Q.

X3

P1

P2

P3

F

V

X2

Π

P1

P2

P3

F

V

Figure 3. Left: An inparabola (red) is tangent to a reference triangle’s sides (blue). Its focusF lies on the circumcircle. Also shown is the vertex V and the directrix (dashed red). Curiously,the latter is parallel to the Simson line S which passes through V [1]. Right: The vertices ofthe antipolar triangle T ′ with respect to an inparabola (red) are the tangency points of sidelinesof a reference triangle T ’ (blue) with an inparabola P. As before, the focus F of P lies on thecircumcircle; T and T ′ are perspective at a point Π on the Steiner ellipse.

under isotomic conjugation of lines tangent to the Steiner ellipse1. Recall that theisogonal (resp. isotomic) conjugate of a point P on the plane of a triangle T = ABCis the intersection of reflections of PA, PB, and PC about the angle bisectors (resp.medians) [16].

Similarly, every triangle is associated with a 1d family of inscribed parabolas orinparabolas, tangent to each of the sidelines, see Figure 3(left). The focus F (resp.Brianchon point Π) always lies on the circumcircle (resp. Steiner ellipse) [16]. Soto generate all inparabolas one can either (i) sweep F over the circumcircle, or (ii)sweep Π over the Steiner circumellipse.

1This is the unique circumellipse centered on the barycenter X2 [16, Circumconic].

PONCELET PARABOLA PIROUETTES 3

Experimental Thrust and a Preview of Results. Using tools of graphicalsimulation first (and numerical verification second), we look for salient phenomenamanifested by in- or circumparabolas to certain “hand-picked” Poncelet families,namely, where the outer conic is either a circle or the Steiner ellipse itself, seeFigures 1 and 4.

Specifically, for circle- (resp. Steiner-) inscribed Poncelet, we fix the focus (resp.Brianchon point) on the outer conic. As we traverse Poncelet triangles in a givenfamily, we observe parabolas’ classical components – vertices, Brianchon points,directrix, centroids of polar triangle – will more than often sweep (or envelop) conics,circles, lines, and/or points. This is similar in spirit to [15].

Most results are presented below as (unproven) observations. When certainpatterns emerge over several families, we generalize them: Conjecture 1, Theorem 1,Conjecture 2, Conjecture 3.

Article structure. In Sections 2 and 3 we describe circumparabola phenomenaover both circle- and Steiner-inscribed Poncelet families. In Sections 4 and 5 we turnour attention to inparabola phenomena, over similarly-inscribed triangle families.A summary appears in Section 6 as well as a link to narrated videos of someexperiments.

2. Circumparabolas as isogonal images

In this section we consider circumparabolas which are isogonal images of a fixedline tangent to the circumcircle. We call these “isogonal CPs” for short.

Specifically, below we mention properties of such parabolas over certain Poncelettriangle families inscribed in a circle C and circumscribing an inner ellipse E ′. LetR denote the radius of the outer circle. Shown in Figure 4 are four such familiesstudied, and defined as follows:

• Inellipse: E ′ is a concentric ellipse with semi-axes a, b. (C, E ′) admit Poncelettriangles if a+ b = R [5].

• Bicentric (also known as Chapple’s porism): E ′ is a circle of radius r. Let d =|OI| = |X1X3| denote the distance between fixed incenter and circumcenter.The condition for Poncelet triangle admissibility is d =

√R(R− 2r).

• MacBeath porism: E ′ is the MacBeath inellipse [16], whose foci are X3

and X4 and center X5, with center on X5, the center of 9-point circle.Curiously, it is equivalent to the family of excentral triangles to Bicentricones [10, 11, 6].

• Brocard porism: E ′ is the Brocard inellipse [16], whose foci are the twostationary Brocard points of the family [2, 14]. These triangles conserveBrocard angle and are also known as the N = 3 harmonic family [3].

2.1. Focus Locus. For a fixed triangle, the locus of the focus over all possiblecircumparabolas is a complicated quintic [8]. However here triangles are Poncelet-varying. Referring to Figure 5:

Observation 1. Over the bicentric family, the locus of the focus of isogonal CPsis a straight line.

Observation 2. Over the bicentric family, the locus of the barycenter X2 of thepolar triangle is a straight line parallel to the locus of the focus.

4 DAN REZNIK

X3

inellipse

X3 X1

bicentric

X3 X4X5

MacBeath

X3

Ω1

Ω2

X39

Brocard

Figure 4. The four circle-inscribed Poncelet triangle families considered herein.

Challenge 1. Describe the envelope of the linear focus locus over the bicentricfamily over all tangents to the circumcircle (pre-images of a given isogonal CPfamily).

2.2. Directrix Envelope. Referring to Figure 5:

Observation 3. Over bicentric family, the envelope of the directrix of isogonal CPsis a parabola with focus on the center X1 of the inscribed circle.

Referring to Figure 6, over the inellipse family, neither the locus of the focus northat of the vertex are low degree curves, however:

Observation 4. Over the inellipse family, the envelope of the directrix of isogonalCPs is a parabola.

In fact:

PONCELET PARABOLA PIROUETTES 5

X 2

F V

Fenv

Q

X3 X 1

P1

P2

P3

Figure 5. Over the bicentric family, the locus of the focus of isogonal CPs (red) is a straightline (green), as is that of the barycenter X2 of the polar triangle (orange). The two linear lociare parallel. The envelope of the directrix (dashed red) is a parabola (cyan) whose focus Fenv

coincides with X1.

Observation 5. Over both the MacBeath and Brocard families, the envelope of thedirectrix of isogonal CPs are parabolas.

In turn, this gives credence to:

Conjecture 1. Over any Poncelet triangle family inscribed in a circle, the envelopeof directrices of isogonal CPs is a parabola.

Challenge 2. For each circle-inscribed family (other than the bicentric one), de-scribe the locus of the focus of the parabolic directrix envelope over all tangents tothe circumcircle which are isogonal pre-images of CPs.

2.3. Perspectors. The perspector Π of a conic is the point at which a referencetriangle and the polar triangle with respect to said conic are perspective. Interest-ingly, The perspectors of all circumparabolas to a fixed triangle sweep the Steinerinellipse [12].

Referring to Figure 7:

Observation 6. Over both the bicentric and MacBeath families, the locus of theperspector of isogonal CPs is an ellipse.

Referring to Figure 8:

Observation 7. Over the Brocard family, the locus of the perspector of isogonalCPs is a circle.

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F

V

Fenv

Q

X3

P1

P2

P3

Figure 6. Over the inellipse family, the locus of the focus and vertex of isogonal CPs (red) arecurves of degree higher than 2 (red and magenta, respectively). The envelope of the directrix(dashed red) is a parabola (cyan), its focus at a point Fenv .

Π

FV

Q

X3 X1

P1

P2

P3

Π FV

Q

X3 X1

P1

P2

P3

Figure 7. Over both the bicentric (left) and MacBeath (right) families, the locus of the perspectorΠ of isogonal CPs (red) is an ellipse (gold).

Let ΠQ denote the locus of the perspector of CPs isogonal to a line tangent tothe circumcircle at Q.

Challenge 3. Over all Q, describe the locus of the center of ΠQ generated overbicentric, MacBeath, and Brocard families.

PONCELET PARABOLA PIROUETTES 7

Π F

V

Q

X3 X39

P1

P2

P3

Figure 8. Over the Brocard family, the the locus of the perspector Π of isogonal CPs (red) is acircle (gold).

3. Circumparabolas as isotomic images

In this section we consider circumparabolas which are isotomic images of afixed line L tangent to the Steiner (circum)ellipse. We call these “isotomic CPs”for short. Below we enumerate some salient properties of such parabolas over afamily of Poncelet triangles interscribed between two homothetic ellipses E and E ′,see Figure 1(right). Recall these are precisely the Steiner circum- and inellipse,respectively, whose center is X2. Since this family is the affine image of equilateralsinterscribed between two concentric circles, it is area-constant. Indeed, it conservesa myriad of other quantities such as sum of squared sidelengths, Brocard angle, etc.[5].

Referring to Figure 9, the following has been kindly proved by B. Gibert [9]:

Proposition 1. Over the homothetic family, all isotomic CPs are tangent to thereflection of L with respect to the common center X2. Said CPs envelop an ellipseaxis-aligned with the pair and tangent to E at Q and to E ′ at Q′ where Q is whereL touches E and Q′ is the intersection of QX2 with E ′ farthest from Q.

Referring to Figure 10, one notices that over said family, the locus of either thefocus or vertex of isotomic CPs are sinuous curves. However:

Observation 8. Over the homothetic family, the envelope of the directrix of isotomicCPs is a parabola.

Furthermore:

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Q

X2

O

Figure 9. Over the homothetic family, all isotomic CPs are tangent to the reflection of the tangentline L with respect to the common center X2. This family of circumparabolas envelops an ellipse(green) axis-aligned with the homothetic pair and with center at the midpoint of Q and the distalintersection of line QX2 with the caustic.

Observation 9. Over the homothetic family, the locus of the barycenter X2 of thepolar triangles with respect to isotomic CPs is a line parallel to L.

Observation 10. Over the homothetic family, the perspector Π of isotomic CPs isstationary on the Steiner inellipse and collinear with X2 and the touchpoint Q of Lon the outer Steiner.

Challenge 4. Over all tangents to the Steiner which are pre-images of isotomicCPs, describe the locus of the focus of the parabolic directrix envelope swept over thehomothetic family.

3.1. Locus of Generatrix Intersection. Referring to Figure 11, consider boththe isogonal and isotomic pre-images of some circumparabola to a triangle T . Asmentioned above, these are lines tangent to the circumcircle and Steiner ellipse,respectively. Let Z denote their intersection, and Q and R denote the tangencypoints, respectively.

Observation 11. Q, R, and the Steiner Point X99 are collinear.

Recall the Kiepert parabola is an inscribed conic with focus on X110 whosedirectrix is the Euler line of a triangle [16, Kiepert parabola]. Still referring toFigure 11, the following has been kindly proved by B. Gibert [9]:

Proposition 2. Over the 1d family of circumparabolas to a fixed triangle, the locusof Z is the isogonal image of the Kiepert parabola.

PONCELET PARABOLA PIROUETTES 9

Π

X 2

F

V

Fenv

Q

X2

Figure 10. Over the homothetic family, the locus of the focus and vertex of isotomic CPs (red)are sinuous curves (green and magenta, respectively). Interestingly, the locus of the barycenter X2

of the polar triangle (orange) is a straight line parallel to L. The envelope of the directrix (dashedred) is a parabola (cyan); Fenv indicates its focus for the case shown. Finally, the perspector Π ofsaid CPs is stationary on the inellipse and collinear with the tangency point Q of L and X2.

4. Inparabolas over Circle-Inscribed Poncelet

In this section we describe loci and envelope phenomena manifested by inparabolasP over circle-inscribed Poncelet families (Figure 4), where there focus F is a fixedpoint on the circumcircle.

Let V (resp. C) denote the vertex of P (resp. the reflection of F on V , i.e., theprojection of F or V on the directrix), see Figure 12. Recall since the Simson lineS is parallel to the directrix and tangent to P at V , V is the projection of F onsaid line. So any properties of V mentioned below are properties of projections ofF on S.

Gallatly studied the envelope of Simson lines over the bicentric family, proving itis a fixed point [4].

Unlike Section 2 which was organized around parabola objects (vertex, directrix,etc.), here we group phenomena family-by-family.

4.1. The inellipse family. The inellipse family appears in Figure 4(top left).Referring to Figure 12, over this family, one observes:

Observation 12. The locus of V is a circle passing through F and tangent to thePoncelet caustic at a point U .

Let ρ denote the radius of the locus of V .

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X2

X3

R

Q

Z

X99

X110

Figure 11. A particular circumparabola (red) is shown of a triangle T (blue). Its isogonal (resp.isotomic) pre-image is a line tangent at Q to the circumcircle (resp. at R to the Steiner ellipse).These meet at a point Z. Over the family of circumparabolas of T , the locus of Z is a curve (green)which is the isogonal image of the Kiepert parabola (pink), with focus on X110 and directrix theEuler line X2X3 (not shown) [16]. Also shown is the curious fact that Q, R and the Steiner pointX99 are collinear.

Corollary 1. The locus of C is a circle of radius 2ρ centered on U .

Still referring to Figure 12, let W denote the reflection of F on U .

Observation 13. The envelope of the directrices (resp. Simson lines) is W (resp.U).

Over all foci. We can regard Observation 12 as associating to each F a circularlocus, and more specifically a center O to that locus, as well as a fixed point Wabout which directrices turn. Referring to Figure 13:

Corollary 2. Over all F on the circumcircle, the locus of the touchpoint U of thecircular locus of inparabola vertices is the caustic itself.

PONCELET PARABOLA PIROUETTES 11

X3

C

O

U

W

P1

P2

P3

F

V

Figure 12. A Poncelet triangle (blue) is shown in the inellipse family, as well as the inparabola P(red) with respect to a fixed point F on the circumcircle; V and C denote its vertex its projectionon the directrix (dashed orange), respectively. Also shown is the projection C of the focus on thedirectrix (dashed orange). The Simson line S (dark green) is parallel to the directrix and tangentto P at V . Over the Poncelet family, (i) the locus V is a circle (magenta) passing through F andtangent to the caustic at a point U ; O indicates its center. (ii) The locus of C is a twice-sized circle(orange) also containing F and centered at U . All directrices (resp. Simson lines) pass through W(resp. U), the reflection of F on U .

Observation 14. Over all F on the circumcircle, the locus of O is an ellipseconcentric and axis-aligned with the caustic of the inellipse family.

Observation 15. Over all F on the circumcircle, the locus of W is a circleconcentric with the two Poncelet conics.

4.2. Bicentric family. Referring to Figure 14(left), all observations pertaining tothe circumcircle family remain true, namely:

Observation 16 (Bicentric combo). Over the Bicentric family, the locus of bothV and C are circles, and that of all Simson lines (resp. directrices) pass through afixed point U (resp. W ), where U is antipodal to F on the locus of V , and W is thereflection of F on U .

Notice that unlike in the inellipse case, here the locus of C is notangent to thecaustic. Referring to Figure 14(right), over all F :

Observation 17. Over all foci F of inparabolas, the locus of the center O of the(circular) loci of the vertex is an ellipse whose minor axis runs along X1X3 andwhose center is that segment’s midpoint X1385.

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X3

C

O

U

WP1

P2

P3 F

V

Figure 13. Over all F on the circumcircle, the centers O of circular vertex loci (magenta circle)sweep an ellipse (dashed magenta) concentric and axis aligned with the caustic. The envelope Wof the directrices sweep a concentric circle (dashed orange).

Observation 18. The locus of U is an ellipse internally tangent to the caustic,with minor axis along X1X3, and centered on X1.

Observation 19. The locus of W is a circle with center on the X1X3 axis.

4.3. MacBeath family. Referring to Figure 15, the claims in Observation 16 arealso valid for the MacBeath family. Recall a well-known fact: the orthocenter X4

of a triangle lies on the directrix of any inscribed parabola [1]. Notice that in thisfamily, X4 is stationary at one of the caustic’s foci. Therefore:

Corollary 3. Over the MacBeath family, the envelope of the directrix of inparabolaswith focus any point F on the circumcircle, is the X4 focus of the caustic.

Observation 20. Over all F , the locus of both O and U are circles. The former iscentered on the midpoint X140 of the X3X5 segment. The latter is concentric withthe caustic on X5.

Referring to Figure 16:

Observation 21. Over the MacBeath family, the locus of the circumcenter X3 ofpolar triangles with respect to inparabolas with fixed focus F on the circumcircle isa line. Over all F said X3 loci envelop a conic whose major axis coincides withthe MacBeath’s major axis. Said conic has one focus on on the center X5 of theMacBeath inconic.

PONCELET PARABOLA PIROUETTES 13

X1X3

C

O

U

WP1

P2

P3

F

V

X1X3

C

X1385

O

U

WP1

P2

P3

F

V

Figure 14. Left: A Poncelet triangle (blue) is shown in the bicentric family. Consider inparabolasP (red) with focus at a fixed point F on the circumcircle. As in the inellipse family, the locusof both V and C are circles containing F (magenta and orange); over the Poncelet family, alldirectrices pass through a fixed point W which is diametrically opposite to F on the C locus.Right: over all F on the circumcircle, the locus of O, the center of vertex loci, is an ellipse(dashed pink) with minor axis along the X3X1 line and centered at their midpoint X1385. Thelocus of U is a second axis-aligned ellipse (dashed light blue) centered on X1. Finally, the locus ofW is a circle centered on the X1X3 line.

Observation 22. Over the MacBeath family, the locus of the Brianchon point ofinparabolas with fixed focus F on the circumcircle is an ellipse. In general, the locusof the center of said ellipses is not a conic.

4.4. Brocard family. Referring to Figure 17, the claims in Observation 16 arealso valid for the Brocard family. Furthermore:

Observation 23. The locus of the directrix fixed point W is a circle with centercollinear with the centers of the two Poncelet conics, i.e., on the X3X39 line.

Observation 24. Over all F , the locus of the center O of circular vertex loci is anellipse whose minor axis coincides with that of the Brocard inellipse, centered at themidpoint of X3X39.

Observation 25. Over all F , the locus of the fixed point U of Simson lines is anellipse axis-aligned and concentric with the Brocard inellipse, to which it is tangentinternally at both co-vertices.

Referring to Figure 18:

Observation 26. Over the Brocard family, the locus of the Brianchon point Π ofinparabolas with fixed focus F on the circumcircle is an circle. Over all F , the locusof the center of this circle is a conic whose major axis is along the X3X39 line.

4.5. General circle-inscribed Poncelet. Consider a circle-inscribed Poncelettriangle family where the inner conic is some generic nested ellipse. Let F be a fixedpoint on the circumcircle. As mentioned above, the Simson line with respect to Fis tangent to the inparabola with focus on F at its vertex V . So V can be regardedas the perpendicular projection of F onto the Simson line [7].

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X5X3

C

X140

O

U

W=X4

P1

P2

P3

FV

Figure 15. Over the MacBeath family, the locus of V and U are still circles (solid pink andorange, respectively). Interestingly, for any position of F or any Poncelet triangle, the directrixof the inscribed parabola (red) will pass through the caustic’s right focus, labeled W = X4. Theenvelope of Simson lines (dashed dark green) is U , the antipode of F on the locus of V . Over allF , the locus of O is a circle (dashed green) centered at the midpoint X140 of the X3X2 segment,and that of U (the fixed point of the Simson) is a circle concentric and internally tangent to thecaustic.

Referring to Figure 19:

Theorem 1. Over an arbitrary Poncelet triangle family inscribed in a circle, thelocus of the perpendicular projection of F onto the Simson line is a circle.

The following proof was kindly provided by Alexey Zaslavsky [17].

Proof. Identify the circumcircle with the unit circle in the complex plane. Let f1,f2 be the complex numbers corresponding to the foci of the inconic, and set F = 1.Let a, b, c denote the sidelengths. Then we have a+ b+ c = f1 + f2 + f1f2abc and

PONCELET PARABOLA PIROUETTES 15

Π

X3

X5X3 W=X4

P1

P2

P3

F

V

Figure 16. Over the MacBeath family, the locus of the Brianchon point Π of inparabolas P withfixed focus at F is a conic (gold). Over all F the locus of their centers is an oval (dashed gold).Over the MacBeath family, the locus of the circumcenter X3 of polar triangles with respect to Pis a line (solid teal). Over all F said loci envelop a conic whose major axis coincides with thecaustic’s, and with one of the foci on X5

ab+bc+ca = f1f2 +(f1 + f2)abc. The projection of F onto AB is (1+a+b−ab)/2;that onto BC and CA are obtained cyclically. From this obtain that the projectionV of F onto the Simson line is V = (1 + k− kabc)/2, where k = f1 + f2 − f1f2, i.e.,this point moves along a circle. �

Proposition 3. The locus of the isogonal conjugate V ′ of V is a line tangent tothe circumcircle at the antipode of F .

Proof. Let V ′ denote the isogonal conjugate of V . This satisfies V +V ′ + ¯V V ′abc =a+ b+ c, and we can see that V ′ + V̄ ′ = −2. �

Referring to Figure 20:

Conjecture 2. Over any Poncelet triangle family inscribed in a circle, the envelopeof Simson lines (dashed purple) is a point W antipodal to F on the circular locus ofV .

Conjecture 3. Over all F , the locus of W is an ellipse concentric with the inconic.

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Ω1

Ω2

X39X3

C

Y

O

U

W

P1

P2

P3

F

V

Figure 17. Over the Brocard family, the locus of V and U are again circles (solid pink and orange,respectively). Over all F , (i) the locus of the fixed point W of the directrices is a circle with centeron the minor axis of the Brocard inellipse E′; (ii) the locus of O is an ellipse (dashed green) whoseminor axis coincides with that of E′ and its center is at the midpoint Y of X3 and X39; (iii)the locus of U is an ellipse axis-aligned and concentric with E′, tangent to the latter at the bothco-vertices.

5. Inparabolas over Steiner-Inscribed Poncelet

A well-known fact is that while the focus to inparabolas lie on the circumcircle,the Brianchon point must lie on the Steiner ellipse [16, Brianchon point]. Let Π be afixed point on the outer ellipse of the homothetic Poncelet triangle family. Referringto Figure 21:

Observation 27. Over the homothetic family, the locus of inparabolas whoseBrianchon point is a fixed point Π on the outer ellipse is a circle.

Interestingly:

Observation 28. Over the homothetic family, the locus of the barycenter of polartriangles with respect to inparabolas with fixed Brianchon point Π on the outer ellipseis a circle is a line.

We suggest:

Challenge 5. Describe the envelope of the directrix (and/or Simson line) over thehomothetic family with a fixed Π on the Steiner ellipse.

PONCELET PARABOLA PIROUETTES 17

Π

X3

Ω1

Ω2

X39X3

P1

P2

P3 F

V

Figure 18. Over the Brocard family, the locus of X3 of the polar triangle (teal) with respect toinparabolas with fixed focus F is not a conic (teal). The locus of the Brianchon point Π is a circle(gold). Interestingly, over all F the locus of the center of Π loci is a conic whose major axis isalong the X3X39 line.

Challenge 6. Describe the locus of the center of the focus locus over all Π on theSteiner ellipse.

6. Conclusion

Narrated videos of some phenomena appear in a Youtube playlist [13]. We invitereaders to both contribute proofs and/or work out the challenges proposed above.

Acknowledgements

We would like to thank A. Akopyan, L. Gheorghe, B. Gibert, P. Moses, and A.Zaslavsky for their infinite patience and invaluable assistance.

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[3] Casey, J. (1888). A sequel to the first six books of the Elements of Euclid. Dublin: Hodges,Figgis & Co. Fifth edition. 3

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Figure 19. Over a circle-inscribed Poncelet triangle family (blue), where the inconic is in generalposition, the locus of the vertex V of inparabolas with focus on a fixed point F on the circumcircleis still a circle (magenta). Over all F , the centers of said circles (green) sweep an ellipse (dashedmagenta), not concentric nor axis-aligned with either Poncelet conics.

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com. 2, 3, 8, 10, 16[17] Zaslavsky, A. (2021). Private communication. 14

PONCELET PARABOLA PIROUETTES 19

V

F

W

O

Figure 20. Over circle-inscribed Poncelet with a generic inconic, the envelope of Simson lines(tangents to inparabolas at V ) is a point W which is the antipode of F on the circular locus of V .Over all F , W sweeps an ellipse (orange) concentric though not-axis aligned with the inconic.

20 DAN REZNIK

X2

Π

P1

P2

P3

F

V

Figure 21. Let Π be a fixed point on the outer (Steiner) ellipse of a Poncelet triangle family (blue)interscribed between two homothetic families. Let P be an inparabola (red) whose Brianchon is Π.Over the Poncelet family, the foci of P sweep a circle (green), while the vertex sweeps a non-conic(orange). Interestingly, the locus of the barycenter X2 of the polar triangle (dashed teal) is astraight line (solid teal).