Question 2
Two circles $C,C'$ have centers $O,O'$, respectively, with intersecting reference lines $OX$ and $O'X'$. Points $M,M'$ move on $C,C'$ so that the counterclockwise directed angles $\angle XOM$ and $\angle X'O'M'$ are equal. Determine the locus of the midpoint of $MM'$. Suppose $OM$ and $O'M'$ intersect at $Q$. Prove that the circle through $Q,M,M'$ always passes through a fixed point.