Assignment 8: Altitudes and Orthocenters
For this assignment I decided to look into the orthocenter of a triangle, and the orthocenters of the 'sub-triangles' formed by the altitudes. The orthocenter itself is constructed by finding the three altitudes of each side. Below are some of my investigations.
Orthocenter of triangle ABC:
As I varied the shape of the triangle from an acute triangle to an obtuse triangle I found that the orthocenter was found outside of the triangle. This is because in order to find the altitudes of an obtuse triangle, you must extend the sides causing the intersection to be located outside of the triangle. When the triangle was right, the orthocenter was on the vertex of the right angle.
Orthocenter of HBC, HBA, and HAC:
When the triangle ABC was acute, all of the sub-triangles orthocenters were found directly on the opposite vertex of ABC. This is a very interesting finding, and I was not expecting this! As expected though, when the triangle ABC became obtuse, the sub-triangles orthocenters were located outside of the triangle ABC.
These findings are thought provoking, and would be a good way to get students to explore the centers of triangles. Rather than stopping at orthocenters, I would suggest having students do the same kind of exploration with the other centers of a triangle, and maybe even Euler's line. Having students manipulate the images will help them to understand the material in a more concrete manner.