KöMaL 2008 A456

The point $D$ lies in the triangle ABC such that the circles inscribed in triangles ABD , BCD , and CAD pairwise touch each other. On lines $BC$, $CA$, $AB$, $AD$, $BD$, $CD$, denote the touching points by $A_1$, $B_1$, $C_1$, $A_2$, $B_2$, $C_2$, respectively. Let lines $B_1 C_2$ and $B_2 C_1$ meet at $E$, and let lines $A_1 C_2$ and $A_2 C_1$ meet at $F$. Show that the lines $AF$, $BE$, and $C_1 D$ are concurrent.