Triangle Centers

Frank KoegelSummer Institute 2007

What are the properties of a median in a triangle?

A median in a triangle is the segment that joins a vertex with the midpoint of the opposite side.

There are three medians in a triangle.

How many medians are in a triangle?

Let’s use GeoGebra to create medians in triangles!

Medians are concurrent

The medians in a triangle are concurrent (i.e., they meet in one interior point of the triangle.) centroid

Mc

Mb

Ma

A

C

B

The point of concurrency is the centroid of a triangle.

Are the six triangles formed by the medians

similar? The medians

split the triangle in six smaller

triangles. Mc

centroid

Ma

Mb

A

C

B

Mc

centroid

A

C

B

Do the blue and the yellow triangles have the same area?

A=xA=x

The distance from the centroid to AB is the –common--height for both triangles.

(remember that Mc is a midpoint)

Mc

A

B

C

Notice that triangles CAMc and CMcB have the same area as well (again the base is the same and the height is the same).

A=yA=y

A=zA=z

A=xA=x

Mc

A

B

C

So the orange triangle and the pink triangle have the same area (since we already proved that the yellow triangle and the blue triangle have the same area).

A=xA=x

A=z/2

A=z/2

A=z/2

A=z/2

And this in turn shows that all six little triangles have the same area.

A=zA=z

We can divide the orange and pink into two congruent halves for the same reason we were able to divide triangle AGB

G

Let’s use GeoGebra to create angle bisectors with triangles!

Angle Bisectors

Angle bisectors are concurrent

• The angle bisectors in a triangle are concurrent (i.e., they meet in one interior point of the triangle.)

• The point of concurrency is the incenter of a triangle.

The Angle Bisectors in a Triangle

The incenter is the center of a triangle's incircle. It can be found as the intersection of the angle bisectors.

(A angle bisector is a line that bisects an angle into two congruent triangles.)

GeoGebra

Let’s look at a file on

using a circle on a segment.

The Perpendicular Bisectors in a Triangle

The circumcenter is the center of a triangle's circumcircle. It can be found as the intersection of the perpendicular bisectors.

(A perpendicular bisector is a line that forms a right angle with one of the triangle's sides and intersects that side at its midpoint.)

Perpendicular bisectors are concurrent

• The perpendicular bisectors in a triangle are concurrent (i.e., they meet in one interior point of the triangle.)

• The point of concurrency is the circumcenter of a triangle.

The circumcenter is equidistant from any pair of the triangle's points, and all points on the perpendicular bisectors are equidistant from those points of the triangle.

In the special case of a right triangle, the circumcenter (C in the figure at right) lies exactly at the midpoint of the hypotenuse (longest side).

Where would the circumcenter beon a right triangle?

A triangle is acute (all angles smaller than a right angle) if and only if the circumcenter lies inside the triangle; it is obtuse (has an angle bigger than a right one) if and only if the circumcenter lies outside, and it is a right triangle if and only if the circumcenter lies on one of its sides (namely on the hypotenuse). This is one form of Thales' theorem.

Let’s use GeoGebra to create altitudes with triangles!

The Altitudes in a Triangle

The orthocenter is the intersection of the altitudes in a triangle.

(An altitude in a triangle is the perpendicular distance from a vertex to the base opposite.)

Altitudes are concurrent

• The altitudes in a triangle are concurrent.

• The point of concurrency is the orthocenter of a triangle.

Let’s use GeoGebra to create the Fermat Point!

Fermat Point

The sum PA+PB+PC is thesmallest distance possiblefrom the three original vertices. (Angles must be less than 120 degrees)

The interior angles APB, BPC and APC are congruent

• Let’s combine some of our constructions

• Our goal is to find all nine points on the 9 point circle

9 Point Circle

We need 3 sets of points:We need 3 midpoints of the triangle.

We need the midpoints of the triangles vertices and the orthocenter.

We need 3 altitudes of the triangle.

When you put any two of these points on a circle, you should find that all 9 are on that circle!

Symmedian PointLet’s try to create this!

What are the requirements for a triangle center?Homogeneity

Cyclicity (trilinear coordinates)

Bisymmetry

Homogeneity

This refers to similar triangles having similarly placed centers.

Cyclicity Trilinear coordinates

The reference point (P) has proportional directed distance from the center to the sideline.

There are 27 widely accepted triangle centers, with 15 classical ones, and 12 more recent additions.

But according to Wolfram’s website, the most recent update shows 2676 centers.

Sunshine State Standards

MA.C.2.4: The student visualizes and illustrates ways in which shapes can be combined, subdivided, and changed.

MA.C.2.4.1: The student understands geometric concepts such as perpendicularity, parallelism, tangency, congruency, similarity, reflections, symmetry, and transformations including flips, slides, turns, enlargements, rotations, and fractals.