BALTIC-WAY 1997 Problem 12

Two circles $\mathcal{C}_{1}$ and $\mathcal{C}_{2}$ intersect in $P$ and $Q$. A line through $P$ intersects $\mathcal{C}_{1}$ and $\mathcal{C}_{2}$ again in $A$ and $B$, respectively, and $X$ is the midpoint of $A B$. The line through $Q$ and $X$ intersects $\mathcal{C}_{1}$ and $\mathcal{C}_{2}$ again in $Y$ and $Z$, respectively. Prove that $X$ is the midpoint of $Y Z$.