KöMaL 2010 A505

In a cyclic quadrilateral ABCD , the points $O_1$ and $O_2$ are the incenters of triangles ABC and ABD , respectively. The line $O_1 O_2$ meets $BC$ and $AD$ at $E$ and $F$, respectively. (a) Show that there exists a circle $k$ which touches the lines $BC$ and $AD$ at $E$ and $F$, respectively. (b) Prove that $k$ also touches the circumcirlce of ABCD .