Question 12
be $A$ one of the two distinct intersection points of two different circles $C_{1}$, $C_{2}$ of centers $O_{1}$, $O_{2}$, respectively. One of the common tangents to the two circles touches $C_{1}$ on $P_{1}$ y a $C_{2}$ on $P_{2}$, while the other touches $C_{1}$ on $Q_{1}$ y a $C_{2}$ on $Q_{2}$. be $M_{1}$ the midpoint of $P_{1} Q_{1}$ y $M_{2}$ the midpoint of $P_{2} Q_{2}$. show that $\widehat{O_{1} A O_{2}} = \widehat{M_{1} A M_{2}}$.