Notice that the orthocenter for triangle HBC was actually point A of the original triangle. In fact, the orthocenter for all of the triangles other than triangle ABC was simply the non-included point of triangle ABC. Next, we wish to contruct the circumcircles (a.k.a. circumscribed circles) for triangles ABC, HBC, HAB, and HAC.
Notice that the circumcircle for triangle ABC is in blue, triangle HBC in pink, triangle HAB in green, and triangle HAC is yellow.
Notice that triangle ABH is highlighted in blue and the circumcircle with center P that we constructed in shown in pink. Now we must use segment AB as a "mirror" and make some observations.
The reflection of triangle ABH about the segment AB is shown in green. One immediate observation is that center of the pink triangle and H' lie in the same half-plane.