Concurrency Points

Circumcenter, Incenter

by

Troy Morris

I was interested in doing this essay because I teach about concurrency points each semester in my Geometry Course at the high school. I thought it would be helpful to my students to be able to look at this site to view the constructions of these points and the proofs of a couple of them. I will be doing the proofs in two-column form because this is the method taught in most high school texts. This will be helpful in my students' understanding of each step used.

CIRCUMCENTER-The circumcenter is found by constructing the perpendicular bisectors of each side of a given triangle, then locating the intersection point.

Construction-Given any triangle ABC, use GSP to construct the perpendicular bisectors of each side of the triangle. Locate the intersection point and label it D. This point is the circumcenter.

Points G,E, and F are all midpoints and D is the circumcenter. Notice the circumcenter is on the interior of the triangle, which is acute. To see what happens to point D when the triangle is not acute, click here.

Circumcircles - circles formed by using the circumcenter as the center of the circle and the distance from the circumcenter to any vertex the radius.

Theorem : The perpendicular bisectors of the sides of a triangle meet at a point, the circumcenter, which is equidistant from the vertices of the triangle. In the figure below, DA = DB = DC.

INCENTER- The incenter is found by constructing the angle bisectors of a triangle, then locating the point at which they intersect.

Construction-Given any triangle ABC, use GSP to bisect the three angles of the triangle. Label their intersection point I. This point is the incenter.

If you used GSP to construct the incenter, grab a vertex of the triangle and change it to a right triangle and then to an obtuse. What happens to the incenter? Does it stay inside the triangle or does it move in and out like the circumcenter? To take the easy way out, click here.

Theorem-The bisectors of the angles of a triangle meet at a point,I, that is equidistant from the sides of the triangle. In the figure below, IX = IY = IZ.