Erin Mueller

Altitudes and Orthocenters

1. Construct any triangle ABC.

2. Construct the Orthocenter H of triangle ABC.

3. Construct the Orthocenter of triangle HBC.

Even though we cannot see it, the orthocenter is the vertex A.

4. Construct the Orthocenter of triangle HAB.

The Orthocenter of triangle HAB is the vertex C. What do you think
the Orthocenter of triangle HAC will be?

5. Construct the Orthocenter of triangle HAC.

If you guessed the vertex B, you would be correct.

6. Construct the Circumcircles of triangles ABC, HBC, HAB, and
HAC.

7. From the above drawing, we can conjecture that all of the
circles are the same size and thus they have equal radii.