Question 13
Two circles $\omega_{1}$ and $\omega_{2}$, with center $O_{1}$ and $O_{2}$ respectively, meet at points $A$ and $B$. Let $X$ and $Y$ be points on $\omega_{1}$. Lines $X A$ and $Y A$ meet $\omega_{2}$ at $Z$ and $W$, respectively, such that $A$ lies between $X$ and $Z$ and between $Y$ and $W$. Let $M$ be the midpoint of $O_{1} O_{2}$, $S$ be the midpoint of $X A$ and $T$ be the midpoint of $W A$. Prove that $M S = M T$ if and only if $X, Y, Z$ and $W$ are concyclic.