Question 23
(USS 1) IMO2. Let $K$ be one of the two intersection points of the circles $W_1$ and $W_2$. Let $O_1$ and $O_2$ be the centers of $W_1$ and $W_2$. The two common tangents to the circles meet $W_1$ and $W_2$ respectively in $P_1$ and $P_2$, the first tangent, and $Q_1$ and $Q_2$, the second tangent. Let $M_1$ and $M_2$ be the midpoints of $P_1Q_1$ and $P_2Q_2$, respectively. Prove that
$$\angle O_1KO_2=\angle M_1KM_2.$$
$$\angle O_1KO_2=\angle M_1KM_2.$$