Question 54
Given a non-isosceles triangle $ABC$, let $D$, $E$ and $F$ denote the midpoints of the sides $BC$, $CA$ and $AB$ respectively. The circumcircle of triangle $BCF$ and the line $BE$ meet again at $P$, and the circumcircle of $ABE$ and the line $AD$ meet again in $Q$. Finally, the lines $DP$ and $FQ$ meet at $R$. Prove that the centroid $G$ of the triangle $ABC$ lies on the circle $PQR$.
