OMM 2011 Problem 6

Let $\mathcal{C}_1$ and $\mathcal{C}_2$ be two circumferences intersecting at points $A$ and $B$ . Let $C$ be a point on line $AB$ such that $B$ lies between $A$ and $C$ . Let $P$ and $Q$ be points on $\mathcal{C}_1$ and $\mathcal{C}_2$ respectively such that $CP$ and $CQ$ are tangent to $\mathcal{C}_1$ and $\mathcal{C}_2$ respectively, $P$ is not inside $\mathcal{C}_2$ and $Q$ is not inside $\mathcal{C}_1$ . Line $PQ$ cuts $\mathcal{C}_1$ at $R$ and $\mathcal{C}_2$ at $S$ , both points different from $P$ , $Q$ and $B$ . Suppose $CR$ cuts $\mathcal{C}_1$ again at $X$ and $CS$ cuts $\mathcal{C}_2$ again at $Y$ . Let $Z$ be a point on line $XY$ . Prove $SZ$ is parallel to $QX$ if and only if $PZ$ is parallel to $RX$ .