Question 20
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 $AB$. 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 $YZ$.