CGMO 2015 Problem 1

Let $\triangle ABC$ be an acute-angled triangle with $AB>AC$ , $O$ be its circumcenter and $D$ the midpoint of side $BC$ . The circle with diameter $AD$ meets sides $AB,AC$ again at points $E,F$ respectively. The line passing through $D$ parallel to $AO$ meets $EF$ at $M$ . Show that $EM=MF$ .