Triangle Centers - Overview

Thousands of years ago, when the Greek philosophers were laying the first foundations of geometry, someone was experimenting with triangles. They bisected two of the angles and noticed that the angle bisectors crossed. They drew the third bisector and surprised to find that it too went through the same point. They must have thought this was just a coincidence. But when they drew any triangle they discovered that the angle bisectors always intersect at a single point!   This must be the 'center' of the triangle. Or so they thought.

Triangle Centers - Overview

After some experimenting they found other surprising things. For example the altitudes of a triangle also pass through a single point (the orthocenter). But not the same point as before. Another center! Then they found that the medians pass through yet another single point. Unlike, say a circle, the triangle obviously has more than one 'center'.The points where these various lines cross are called the triangle's points of concurrency.

Sections 5.2-5.4

Points of Concurrency

There are many points of concurrency in a triangle, however, we will only Learn about four of them:the incenter, the circumcenter, the orthocenter,And the centroid.

Concurrent When three or

more lines intersect in one point they are concurrentH

A

B

C

D

G

I

Point of Concurrency The point at

which three or more lines intersect is the point of concurrency

H

A

B

C

D

G

I

Point of concurrency

The perpendicular bisectors of the sides of a triangle are concurrent at a pointequidistant from the vertices.

P

O

NM

J

L

PL = 5.52 cm

KP = 5.52 cm

JP = 5.52 cm

Circumcenter

The point of concurrency of the perpendicular bisectors is called the circumcenter of the triangle.

P

O

NM

J

KL

Circumcenter

P

O

NM

J

KL

The circle is circumscribed about the triangle.

The circumcenter is the center of the circumscribed circle.

P

O

NM

J

KL

The circle is circumscribed about the triangle.

To circumscribe a circle about a triangle, place your compass point on the circumcenter and the radius should extend to any of the three vertices. Now, construct your circle.

The towns of Adamsville, Brooksville, andCartersville want to build a library that isequidistant from the three towns.Trace thediagram and show where they should buildthe library?

Draw segmentsconnecting the towns.Build the library at theintersection point of the perpendicularbisectors of the segments. The perpendicular bisectors of the sides of a triangle are concurrent at a point equidistant from the vertices.

The bisectors of the angles of a triangle are concurrent at a point equidistantfrom the sides.

D

A

B C

Incenter

The point of concurrency of the angle bisectors of a triangle is called the incenter of the triangle.

E

D

A

B C

incenter

The circle is inscribed in the triangle

E

D

A

B C

The center of the inscribed circle is the incenter of the circle. The center is equallyDistant from the three sides. (Remember: it is the perpendicular distance from the Center that determines the point at which the circle touches the side of the triangle, so construct a perpendicular from the incenter to a side of the triangle and use this as the radius of your inscribed circle.

Example of the incenterQuestion about Pools: The Jacksons want to install the largest possible circular pool in theirtriangular backyard.Where would the largest possible pool be located?Answer:Locate the center of the pool at the point of concurrency of the angle bisectors.This point is equidistant from the sides of the yard. If you choose any other point as the center of the pool, it will be closer to at least one of the sides of the yard,and the pool will be smaller.

A median of a triangle is a segmentwhose endpoints are a vertex and themidpoint of the opposite side.

Midpoint

Median

Centroid

In a triangle, the point of concurrency of the medians is the centroid.

The point is also called the center of gravity of a triangle because it is the point where a triangular shape will balance.

There is a special property of a centroid

Look closely:

What is the relationship between the centroid to the midpoint and the centroid to the vertex?

Altitudes of triangles An altitude of a triangle is the perpendicular

segment from a vertex to the line containing the opposite side. Unlike angle bisectors and medians, an altitude of a triangle can be a side of a triangle or it may lie outside the triangle.

The lines containing the altitudes of a triangle are concurrent at the orthocenter of the triangle.

orthocenter

FYI: Useless Information

The intersection of the three altitudes of a triangle is called the orthocenter. The name was invented by Besant and Ferrers in 1865 while walking on a road leading out of Cambridge, England in the direction of London (Satterly 1962).

Try this:Find the center of the circle that you can circumscribe about a right triangle with these vertices:

You need an equation for the perpendicular bisectors to circumscribe a circle about a triangle

You only need to find the equation for two sides. I would use sides AB and BC since they are horizontal and vertical. Much easier. First, find the midpoints of those sides. Now find the equation of the perpendicular bisectors. Then find the intersection point of the two equations.Or…since this is a right triangle, we should know that the

circumcenter is at the midpoint of the hypotenuse. Much easier to find!

8

6

4

2

-2

-4

-6

-8

-10 -5 5 10

BA

C

5) Altitude 6)Median 7) Perpendicular Bisector 8) Angle Bisector

Euler Line

ORTHOCENTER

CIRCUMCENTER

CENTROIDINCENTER

A

B

C

The incenter is NOT ALWAYS on the Euler line. The other three points ARE always on the Euler line.

Euler Line

The incenter will be on the Euler line when the triangle is isosceles. The other three points ARE always on the Euler line. In an equilateral triangle, the 4 points will always coincide.

On the Euler Line above, point O is the circumcenter, point G is the centroid, point F is the center of the nine point circle and point H is the orthocenter.

GO is ½ of HG

OG is 1/3 of HO

OF is ½ of HO

FG is 1/6 of HO

Know all vocabulary, know facts about an angle bisector and a perpendicular bisector. Know how to construct each of these points of concurrency. Know how to inscribe and circumscribe a circle in relation to a triangle.