IMO 2007 Problem 4

In triangle $ABC$ the bisector of angle $BCA$ intersects the circumcircle again at $R$ , the perpendicular bisector of $BC$ at $P$ , and the perpendicular bisector of $AC$ at $Q$ . The midpoint of $BC$ is $K$ and the midpoint of $AC$ is $L$ . Prove that the triangles $RPK$ and $RQL$ have the same area.

Author: Marek Pechal, Czech Republic