## EMAT 6680 Assignment #4

## Triangle Centers

*by: Kent Wiginton*

*3. 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.

*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.

By using Geometer Sketch Pad we can explore
this center of a triangle. The definition of the Circumcenter
(C) we can construct the perpendicular bisectors of two different
sides and then construct a point at their interscetion. Next we
construct the circle by center and radius construction. And volla!
we have the circumcenter and the circumcircle of our triangle
(ABD).

One interesting fact about the circumcenter
can be observed by picking on vertex and moving it to create a
new triangle. We will notice that C will move perpendicular to
the opposite side of the triangle, no matter which vertex one
picks.

**Try it in GSP**

If we look at the movement of C, then we have
to consider different types of triangles. First I checked to see
when C is inside the triangle and when C is outside the triangle.
Anytime that triangle ABD has all acute angles then the circumcenter
C is always inside the triangle. When we have a right triangle
then C is on the midpoint opposite the 90 degree angle. And when
ABD is an obtuse triangle, or if triangle ABD has at least one
obtuse angle then C will never be inside the triangle

.

When we move the triangle around and if we
trace the Circumcenter we will notice that C will move along the
perpendicular bisector of the opposite side.

**Give it a try and see what
you can find out.**

**Return**