Parallels to sidelines

Let ABC be a triangle.

Denote:

Ab = AO /\ BH

Ac = AO /\ CH

A' = (Parallel to AC through Ab) /\ (Parallel to AB trough Ac)

Similarly:

Bc = BO /\ CH

Ba = BO /\ AH

B' = (Parallel to BA through Bc) /\ (Parallel to BC through Ba)

and

Ca = CO /\ AH

Cb = CO /\ BH

C' = (Parallel to CB through Ca) /\ (Parallel to CA through Cb)

The triangles ABC, A'B'C' are perspective.

Perspector?

Also, if we replace H with O and O with H, the triangles are perspective:

Generalization:

P,P* = two isogonal conjugate points (instead of O,H)

APH, 20 December 2011