VMO 2016 Problem 3

Let $ABC$ be an acute triange with $B,C$ fixed. Let $D$ be the midpoint of $BC$ and $E,F$ be the foot of the perpendiculars from $D$ to $AB,AC$ , respectively. a) Let $O$ be the circumcenter of triangle $ABC$ and $M=EF\cap AO, N=EF\cap BC$ . Prove that the circumcircle of triangle $AMN$ passes through a fixed point; b) Assume that tangents of the circumcircle of triangle $AEF$ at $E,F$ are intersecting at $T$ . Prove that $T$ is on a fixed line.