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