IMO 1983 Problem 2

Let $A$ be one of the two distinct points of intersection of two unequal coplanar circles $C_1$ and $C_2$ with centers $O_1$ and $O_2$ respectively. One of the common tangents to the circles touches $C_1$ at $P_1$ and $C_2$ at $P_2$ , while the other touches $C_1$ at $Q_1$ and $C_2$ at $Q_2$ . Let $M_1$ be the midpoint of $P_1Q_1$ and $M_2$ the midpoint of $P_2Q_2$ . Prove that $\angle O_1AO_2=\angle M_1AM_2$ .