Assignment 4: Circumcenter of a triangle

By Dorothy Evans

The CIRCUMCENTER of a triangle is the point in the plane equidistant from the three vertices of the triangle. Since a point equidistant from two points lies on the perpendicular bisector of the segment determined by the two points, the circumcenter (labeled below) is the point of concurrency of the three perpendicular bisectors of each side of the triangle. The basic construction of the circumcenter is to identify the midpoints of the original triangle.  In the picture above the original triangle is triangle DACD.

The circumcenter was constructed by identifying the midpoints of the segments AC, CD, and DA.  Then a perpendicular line was drawn through the midpoints perpendicular to the side segment.

This particular example is of an acute scalene triangle.  Let’s now explore what happens to the circumcenter in other types of triangles.

First let’s look at a right triangle. Notice that the circumcenter is now on the segment AD of the triangle ACD

I wonder what would happen if the right triangle was an isosceles right triangle? Notice this time the line from the midpoint of AD perpendicular to AD now passes through the vertex C of triangle ACD.

Now that we have explored acute and right triangles let’s now look at obtuse triangles. Wow … What happened?

The circumcenter is now outside of the triangle ACD.

Hmmm, what would happen if the obtuse triangle was isosceles? Now how cool is that?

The circumcenter of an obtuse isosceles triangle is outside the triangle and the perpendicular bisector passes through the obtuse angle of the triangle.

Let’s also note where the circumcenter gets its name.  The circumcenter is also the center of circle that the triangle is circumscribed inside of.  You may be asking yourself, “What does circumscribed mean?”  Let’s look at the picture of our triangle and it’s circumcircle. Notice that the triangle is inside the circle and the circumcenter is in the center of the circle.  From our first statement we proclaimed that the circumcenter is equidistant from each vertex therefore we must conclude that we can create a circle where the segment from the circumcenter to each vertex is a radii of the circle.

Now it’s your turn click here to open a GSP sketch to investigate the circumcenter of a circle

The bigger question is “Why is the circumcenter equidistant from each vertex and though we have shown it, how would we prove it?”