CONO-SUR 2009 Problem 3

Let $A$ , $B$ , and $C$ be three points such that $B$ is the midpoint of segment $AC$ and let $P$ be a point such that $<PBC=60$ . Equilateral triangle $PCQ$ is constructed such that $B$ and $Q$ are on different half=planes with respect to $PC$ , and the equilateral triangle $APR$ is constructed in such a way that $B$ and $R$ are in the same half-plane with respect to $AP$ . Let $X$ be the point of intersection of the lines $BQ$ and $PC$ , and let $Y$ be the point of intersection of the lines $BR$ and $AP$ . Prove that $XY$ and $AC$ are parallel.