**Lets begin this exercise by constructing
any triangle abc, constructing the orthocenter H of abc and then
the orthocenters of hbc,hab, and hac.**

**Given this picture, we can see by careful
inspection rather than construction that the orthocenter of triangle
HAB must be C. Likewise the orthocenter of HBC must be A, and
the orthocenter of HAC must be B.Now let us proceed to construct
the circumcircles of these triangles.**

**It would appear that all the circumscribed
circles are congruent. indeed this seems to be provable by the
fact that they all describe congruent arc lengths at their intersection
points, for example the light and dark green circles, both of
them describing equal ac lenghts which pass through points a and
b.**