Proof that the orthocenter of a triangle is the incenter of the triangles' orthic triangle.

Quadrilateral APHR is cyclic since opposite angles APH and ARH are supplementary (in fact they are 90deg each).
Hence angle PAH = angle PRH.
Similarly angle QRH = angle QCH.

Quadrilateral APQC is cyclic since AC subtends angle APC = angle AQC (both = 90 deg).

It follows that angle PAQ = angle PCQ and by the earlier discussion that angle PRH = angle QRH. ie HR bisects angle PRQ.

Similarly HP bisects angle RPQ and HQ bisects angle PQR and H is the incenter of triangle PQR!