CGMO 2010

[Marius Stănean]

In $\triangle ABC$ we have $AB=AC , D$ is the midpoint of $BC.\; E$ is a point outside $\triangle ABC$ such that $CE \perp AB$ and $BE=BD.\; M$ is the midpoint of $BE$. Point $F$ lies on the minor arc of $AD$ (on the circumcircle of $\triangle ABD$) and $MF$ is perpendicular to $BE$. Prove that $ED\perp DF$.