Question 42
Triangle $ABC$ is inscribed in circle $\omega$. Point $D$ is midpoint of side $AC$, and point $M$ lies on segment $BD$ with $DM = 2BM$. Ray $AM$ meets side $BC$ at $E$, and ray $CM$ meets side $BA$ at $F$. Ray $FE$ intersects $\omega$ at $N$. Suppose that $AM \perp CM$. Prove that $ADEF$ is cyclic if and only if line $AN$ bisects segment $BC$.