# Circumcenter: Definition, Formula & Construction

Instructor: Beverly Maitland-Frett

Beverly has taught mathematics at the high school level and has a doctorate in teaching and learning.

This lesson will examine the circumcenter of a triangle. We will discuss the definition, location and demonstrate how to construct and identify the circumcenter.

## Definition of the Circumcenter

Let's begin with the following scenario: three communities are located at points L, M and N as seen in Figure. 1. If the mayor of the town wants to build a hospital so that each community will travel the same distance to the hospital, where would the exact location of the hospital be?

In order to find the exact location of the hospital, we would have to find the circumcenter. The circumcenter is the point where all the perpendicular bisectors of a triangle meet.

Perpendicular bisectors are lines segments, or lines, that bisect another line to form right angles. In other words perpendicular bisectors cut other lines in half and form right angles as they do. We could also say that the perpendicular bisector intersect other lines at their midpoints and form right angles. They all mean the same thing, the thing to remember is that perpendicular means right angle, and bisect means to cut into two equal parts.

## Constructing a Perpendicular Bisector

We can construct a bisector by folding, using a compass, or by using geometry software. As shown in Figure 2, I constructed the perpendicular bisector of line segment AB using geometer's sketchpad, but you can use a compass and pencil following these steps:

1) Draw a line segment AB.

2) Open your compass to a little more than half the line segment.

3) Place your compass at point A and swing an arc above and below the line segment AB.

4) Place your compass at point B and swing an arc above and below the line segment.

5) Where the two arcs meet forms the line for your perpendicular bisector.

## Constructing the Circumcenter

Let us revert to our original scenario. Let's connect points L, M and N to form a triangle and we are going to draw the perpendicular bisector for each side just like we did for line segment AB. If you observe Figure 3, you will see that I constructed all my perpendicular bisectors. Notice that all three perpendicular bisectors meet at point O. Therefore, this point is our circumcenter, and would be the location for the hospital. Therefore, once you construct the perpendicular bisector of the three sides of the triangle, the circumcenter will automatically become obvious.

So, how do we know that this point O is the same distance from points L, M and N? Well, take a look at Figure 4, it's the same triangle, but there are two important differences:

1) The circumcenter is also the center of a circle. This circle goes around the triangle and notice that it touches the three vertices of the triangle, points L, M and N. This is part of the reason the point is called the circumcenter, 'circum' means around, like circumference.

2) Line segments OL, OM and ON represented by the dotted orange lines are all radii of the same circle. Remember the radius of a circle extends from the center of the circle to any point on the circumference. Therefore people will travel the same distance.

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