Write-up #4

The Concurrency of the three Perpendicular Bisectors

of the Sides of a Triangle

by

Holly Anthony

Fall 2001

Problem: Prove that the three perpendicular bisectors of the sides of a triangle are concurrent.

Definitions:

A perpendicular bisector of a triangle is a line, ray, or segment that is perpendicular to a side of the triangle at the midpoint of the side.

When three or more lines intersect in the same point, they are called concurrent lines. The point of intersection of the lines is called the point of concurrency.

Let's examine the point of concurrency for the perpendicular bisectors of various triangles. This point of concurrency is called the circumcenter of the triangle.

Explorations:

Let's first look at an acute triangle.

In an acute triangle, the point of concurrency, or the circumcenter is inside the triangle. To explore this for yourself, choose the acute triangle file below.

Acute Triangle in GSP

Now, let's look at a right triangle.

In a right triangle, the point of concurrency, or the circumcenter is on the triangle. To manipulate this graph for yourself, choose the right triangle file below.

Right Triangle in GSP

Finally, let's look an an obtuse triangle.

In an obtuse triangle, the point of concurrency, or the circumcenter is outside the triangle. To explore this for yourself, choose the obtuse triangle file below.

Obtuse Triangle in GSP

Let's examine the measure of the lengths from the circumcenter to the vertices of the triangle. What do we notice?

Manipulate an Acute triangle showing measures from circumcenter to vertices

Manipulate a Right triangle showing measures from circumcenter to vertices

Manipulate an Obtuse triangle showing measures from circumcenter to vertices

It is interesting to note that in the above explorations, the perpendicular bisectors intersect at a point, the circumcenter, which appears to be equidistant from the vertices of the triangle. The above exploration with GSP measurements does not prove this.

However, this can be proved. To see a proof of this, choose the link below.

Proof of the Concurrency of Perpendicular Bisectors of a Triangle