AustriaMO2011

Two circles $k_1$ and $k_2$ with radii $r_1$ and $r_2$ are externally tangent in $Q$. The other end-points of the diameter through $Q$ are named $P$ on $k_1$ and $R$ on $k_2$. We choose two points $A$ and $B$, one on each of the arcs $PQ$ on $k_1$. (PBQA is convex.) Furthermore, $C$ is the second common point of the line $AQ$ and $k_2$, and $D$ is the second common point of $BQ$ with $k_2$. The lines $PB$ and $RC$ intersect in $U$ and $PA$ and $RD$ intersect in $V$. Show that a point $Z$ exists, that is common to all possible lines $UV$.