Circumcenter

By: Carly Cantrell

The circumcenter of a
triangle is the point in the plane equidistant from the three vertices of the
triangle.

How do we find the circumcenter?
IÕm glad you asked.

A point that is equidistant from two other points must
lie on the perpendicular bisector of the segment determined by the two points.

So, the circumcenter is the
intersection of the three perpendicular bisectors of each side of the triangle.

We can use GSP to construct the circumcenter.

First, construct any triangle and the midpoints.

Next, construct perpendicular lines to the segments
and through the corresponding midpoints.

Now, where the three perpendicular lines intersect is
the circumcenter!

CircumcenterÕs location for a right triangle:

CircumcenterÕs location for an equilateral triangle:

Now, in certain situations the circumcenter
is located outside of the triangle. For example, an obtuse
triangle.

ThatÕs all neat and whatnot but what is extra special
about this triangle center? Well, once you find the circumcenter
then you are able to construct the circumcircle. The circumcenter is the center of the triangleÕs circumcircle.

LetÕs explore more about the circumcenter
by proving that the perpendicular bisectors to the sides of the triangle are
concurrent.

Proof:

Throughout the proof, refer to the picture below:

Where D, E, and F are midpoints, and the dotted lines
are the perpendicular bisectors.

I have constructed the segments AG and GC.

AF = FC, by the midpoint definition.

Angles AFG and CFG are 90 degrees because of
perpendicular construction.

Thus, triangles AFG and CFG are congruent by the SAS
postulate.

Hence, AG = CG.

Next, I constructed the segment BG.

Using the same postulate as above, I can confidently
claim that BG = CG.

Using substitution, AG = CG and BG = CG, then AG = BG.

Thus, we proved that the perpendicular bisectors have
a point of concurrency, which is the circumcenter.