Problem

Source: Iran TST 2004

Tags: geometry proposed, geometry



Let $ M,M'$ be two conjugates point in triangle $ ABC$ (in the sense that $ \angle MAB=\angle M'AC,\dots$). Let $ P,Q,R,P',Q',R'$ be foots of perpendiculars from $ M$ and $ M'$ to $ BC,CA,AB$. Let $ E=QR\cap Q'R'$, $ F=RP\cap R'P'$ and $ G=PQ\cap P'Q'$. Prove that the lines $ AG, BF, CE$ are parallel.