Question 129
Let $ABC$ be a triangle, $\omega$ be its circumcircle and $I$ be its incentre. Let the line $BI$ meet $AC$ at $E$ and $\omega$ at $M$ for the second time. Let the line $CI$ meet $AB$ at $F$ and $\omega$ at $N$ for the second time. Let the circumcircles of $BFI$ and $CEI$ meet at $K$ for the second time. Prove that the lines $BN$, $CM$, $AK$ are concurrent.