KöMaL 2013 A579

The circle $k_1$ is internally tangent to the circle $k$ which is externally tangent to $k_2$. The common external tangents of $k_1$ and $k_2$ are $u$ and $v$. The line $u$ is tangent to $k_1$ and $k_2$ at $P$ and $Q$, respectively, and meets $k$ at $A$ and $B$ in such a way that $B$ lies between $P$ and $Q$. Analogously, the line $v$ is tangent to $k_1$ and $k_2$ at $R$ and $S$, respectively, and meets $k$at $C$ and $D$ in such a way that $D$ lies between $R$ and $S$and $k_1$ is tangent to that arc $BD$ of $k$ which does not contain $A$ and $C$. Show that $\frac {AB\cdot AD} {AP\cdot AQ} = \frac {CB\cdot CD} {CR\cdot CS} .$