Assignment 8: Altitudes and Orthocenters

by

Melissa Silverman
Begin this investigation by constructing any Triangle ABC. Construct the orthocenter of Triangle ABC. If you construct the orthocenters of triangles HAC, HAB, and HBC, their orthocenters fall on the vertices of the triangle. In the picture below, H is the orthocenter of Triangle ABC. Note how the orthocenters of triangles HAC, HBC, and HAB are the same as the vertices.

What can we say about the circumcenters of triangles ABC, HAB, HAC, and HBC? The circumcenter of Triangle ABC is inside the triangle. The circumcenters of the other three triangles all fall outside of Triangle ABC. Connecting the four circumcenters and the three vertices create a shape that looks like a cube.

How to the circumcenters relate to the circumcircle of Triangle ABC? Do they fall on the circumcircle? Constructing the circumcircle of Triangle ABC reveals that two of the circumcenters are outside the circumcircle, but one is inside the circumcircle. The circumcenters do not fall on the circumcircle and do not seem to be related to the circumcircle.

What about the circumcircles of triangles HAC, HBC, and HAC? Constructing the circumcircles of these three triangles shows that circumcircles go through the orthocenter and two of the vertices of Triangle ABC. Reflecting the circumcircle of Triangle ABC across the sides of Triangle ABC will result in the circumcircles of triangle HAC, HBC, and HAB.