Orthocenter and equal circles

Orthocenter (or the intersection point of altitudes) has an interesting construction. Take three equal circles, and make them pass through one point H. Their other point of intersection creates a triangle ABC. Turns out, H is the orthocenter of ABC.

In this process, we all create an equilateral (but not necessarily equiangular) hexagon. Here is a three-part discussion of this problem:

Part 1

Part 2

Part 3

Part 4