RMM 2012 Problem 2

Given a triangle $A B C$, let $D, E$, and $F$ respectively denote the midpoints of the sides $B C, C A$, and $A B$. The circle $B C F$ and the line $B E$ meet again at $P$, and the circle $A B E$ and the line $A D$ meet again at $Q$. Finally, the lines $D P$ and $F Q$ meet at $R$. Prove that the centroid $G$ of the triangle $A B C$ lies on the circle $P Q R$.