Altitudes and Orthocenters
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.
a. Just from viewing the object above, it appears that all of the circumcircles have the same area (or radius). A proof of this could include congruent arcs through 2 points, eventually going around triangle ABC and including all parts of the 4 circumcircles.
b. The orthocenters of triangles AHB, BHC, CHA lie on the vertices of triangle ABC (a proof by contradiction may be best in this case).