by Jan White
1. Construct any triangle ABC.
2. Construct the Orthocenter H of triangle ABC.
3. Construct the Orthocenter of triangle HBC.
4. Construct the Orthocenter of triangle HAB.
5. Construct the Orthocenter of triangle HAC.
6. Construct the Circumcircles of triangles ABC, HBC, HAB, and HAC.
7. Conjectures? Proofs?
What would happen if any vertex of the triangle ABC was move to where the orthocenter H is located? Where would H then be located?
When Vertex B is moved to the orthocenter of ABC then AHB and CHB collapse and the circumcircle of AHC now coincides with the circumcircle of ABC.
8. Construct the nine point circles for triangles ABC, HBC, HAC, and HAB. Conjecture? Proof?
The nine point circle for all four triangles are the same since each of them has two common vertexes and the orthocenter of ABC is the vertex of the other three triangles.