2010 OIM Problems/Problem 5

Problem

Let $ABCD$ be a cyclic quadrilateral whose diagonals $AC$ and $BD$ are perpendicular. Let $O$ be the circumcenter of $ABCD$ , $K$ be the intersection of the diagonals, $L \ne O$ be the intersection of the circumferences circumscribed to $OAC$ and $OBD$ , and $G$ the intersection of the diagonals of the quadrilateral whose vertices are the midpoints of $ABCD$ . Prove that $O, K, L$ and $G$ are aligned.