OIM 1989 Problem 4

The incircle of the triangle $ABC$ is tangent to sides $AC$ and $BC$ at $M$ and $N$ , respectively. The bisectors of the angles at $A$ and $B$ intersect $MN$ at points $P$ and $Q$ , respectively. Let $O$ be the incentre of $\triangle ABC$ . Prove that $MP\cdot OA=BC\cdot OQ$ .