Školjka

%V0 Let $ABCD$ be a convex quadrilateral with $BA$ different from $BC$. Denote the incircles of triangles $ABC$ and $ADC$ by $k_{1}$ and $k_{2}$ respectively. Suppose that there exists a circle $k$ tangent to ray $BA$ beyond $A$ and to the ray $BC$ beyond $C$, which is also tangent to the lines $AD$ and $CD$. Prove that the common external tangents to $k_{1}$ and $k_{2}$ intersects on $k$. Author: Vladimir Shmarov, Russia