Kristen Robinson

EMAT 4680

24 August 2000

Homework Assignment 1: Problem 2

By moving all three vertices of triangle ABC, several conclusions could be drawn about the four centers of the triangle. I discovered that the orthocenter and the circumcenter are the only centers which can move outside of the triangle. The centroid and the incenter remain in the triangle. In addition, the orthocenter and the circumcenter move opposite in direction from each other.

The orthocenter is constructed by creating a perpendicular line through a vertex and the side opposite it. The intersection of these three perpendicular lines is the orthocenter of the triangle. It is possible for this intersection to lie outside of the triangle, as the lines are continuous and not segmented by the vertices or sides of the triangle. The circumcenter is the center of the circumscribed circle containing the vertices of the triangle. This circle is created by finding the intersection of the three lines which are perpendicular to a side and its midpoint. Therefore, these lines are continuous and their intersection can lie outside of the triangle.

The centroid is constructed by drawing a segment from a vertex and the midpoint of the opposite side. The intersection of these three segments is the centroid of the triangle. Because the segments are constricted by the vertices and sides of the triangle, the centroid must lie inside the triangle. The incenter is the center of the inscribed circle touching all three sides of the triangle. Since the circle is inscribed in the triangle, the center of the circle must also be inside the triangle at all times. Therefore, the incenter must lie inside the triangle.