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.

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.

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.

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