ELMO SL 2013 G9

Let $ABCD$ be a cyclic quadrilateral inscribed in circle $\omega$ whose diagonals meet at $F$ . Lines $AB$ and $CD$ meet at $E$ . Segment $EF$ intersects $\omega$ at $X$ . Lines $BX$ and $CD$ meet at $M$ , and lines $CX$ and $AB$ meet at $N$ . Prove that $MN$ and $BC$ concur with the tangent to $\omega$ at $X$ .

Proposed by Allen Liu