1. Construct any triangle ABC.

2. Construct the Orthocenter of triangle ABC, call it H.

3. Construct the Orthocenter of triangles HAB, HAC and HBC.

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

Given any triangle ABC with orthocenter H, the circumcircle of triangle ABC is congruent to the circumcircles of triangle HAB, HAC and HBC.

I WILL BEGIN by proving that circle Q (the circumcircle of triangle HBC) is congruent to circle T ( the circumcircle of triangle ABC). I want to show that the segments BQ and BT (the radii of the two circles) are congruent.

Because Q is the circumcenter of triangle HBC, it lies on, j, the perpindicular bisector of segment AC (by definition).

Similarly, T is the circumcenter of triangle ABC and so it must also lie on, j, the perpindicular bisector of segment AC.

P is the point of intersection of line j and segment BC. Construct segment TQ which lies on line j, the perpindicular bisector of segment AC. Let's form four triangles: BPQ, BPT, BQC and BTC.

By similar argument, circle S is congruent to circle T, circle R is congruent to circle T and therefore, all four circles are congruent to each other (by transitivity). QED.