BG-MO 2014 Problem 3

Let $ABCD$ be a quadrilateral inscribed in a circle $k$ . $AC$ and $BD$ meet at $E$ . The rays $\overrightarrow{CB}, \overrightarrow{DA}$ meet at $F$ . Prove that the line through the incenters of $\triangle ABE\,, \triangle ABF$ and the line through the incenters of $\triangle CDE\,, \triangle CDF$ meet at a point lying on the circle $k$ .

Proposed by N. Beluhov