Question 128
Let $k_{1}$ and $k_{2}$ be two circles that touch each other externally at point $P$. A third circle $k$ touches $k_{1}$ in $B$ and $k_{2}$ in $C$, so that $k_{1}$ and $k_{2}$ lie inside $k$. Let $A$ be one of the intersections of $k$ with the common tangent of $k_{1}$ and $k_{2}$ through $P$. The straight lines $A B$ and $A C$ intersect $k_{1}$ and $k_{2}$ again into $R$ and $S$. Show that $R S$ is a common tangent of $k_{1}$ and $k_{2}$.