IR-TST 2004 Problem 4

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.