Problem:

1. Construct any triangle ABC.

2. Construct the Orthocenter H of triangle ABC.

The Orthocenter of a Triangle is constructed
by finding the intersection of any two altitudes of a triangle.

3. Construct the Orthocenter of triangle HAB.

Analysis: Note that triangle HBA is an obtuse triangle. It is evident that obtuse triangles have exterior Orthocenters. In contrast, acute triangles have interior Orthocenters. In this case the Orthocenter of HAB is coincident with vertex C.

4. Construct the Orthocenter of triangle HBC.

5. Construct the Orthocenter of triangle HAC.

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

Conjectures:

All four circumcircles are congruent animation

The orthocenters of the three triangles formed
by creating the orthocenter of the original triangle lie on the
vertices of the original triangle.

The lines from the center of each circumcircle
going to the orthocenter H of the original triangle and the two
closest vertices of the original triangle form an interesting
figure.