**By: Erica Fletcher**

This is investigation 3 of part two of the final assignment for EMAT 6680. The goal was to select one additional item from the assignments or from explorations presented in class that you have not alreadyu written up. I chose to investigate the circumcenter from the earlier Assignment 4: centers of a triangle. One reason I chose this was because this is one of our Math I standards MM1G3e: understanding the different points of concurrency of a triangle. The more I investigate with the centers of a triangle the better I can explain the various properties to my students taking secondary school mathematics courses.

The CIRCUMCENTER (C) 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, C is on the perpendicular bisector of
each side of the triangle. Note: C may be outside of the triangle.

Goal: Construct the circumcenter C and explore its location for
various shapes of triangles. It is the center of the circumcircle (the circumscribed circle) of the
triangle.

Recall that the point where the three
perpendicular bisectors of a triangle meet is called the circumcenter.

One of several centers the triangle can have,
the circumcenter is the point where the perpendicular
bisectors
of a triangle intersect. The circumcenter is also the
center of the triangle's circumcircle - the circle that passes through
all three of the triangle's vertices. As you reshape the triangle above, notice
that the circumcenter may lie outside the triangle.

Look at the construction of the circumcenter below:

We see that when we have an obtuse triangle the circumcenter is outside of the triangle.

However, look at the construction below:

When we look at triangle ABC being an acute triangle, we notice the circumcenter always lie inside of the triangle.

Lastly, let’s see what happens when triangle ABC is a right triangle.

In the special case of a right triangle, the circumcenter (C in the figure at right) lies exactly at the
midpoint of the hypotenuse (longest side).

The circumcenter of a right triangle falls on the side opposite
the right angle.

Also
we notice that the circumcenter of a triangle ABC is
equidistant from each of the vertices of triangle ABC.

Click here if you would like to see
what happens to the circumcenter when we look at
various types of triangles.

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