IT-CESENATICO 2013 Problem 5

$ABC$ is an isosceles triangle with $AB=AC$ and the angle in $A$ is less than $60^{\circ}$ . Let $D$ be a point on $AC$ such that $\angle{DBC}=\angle{BAC}$ . $E$ is the intersection between the perpendicular bisector of $BD$ and the line parallel to $BC$ passing through $A$ . $F$ is a point on the line $AC$ such that $FA=2AC$ ( $A$ is between $F$ and $C$ ). Show that $EB$ and $AC$ are parallel and that the perpendicular from $F$ to $AB$ , the perpendicular from $E$ to $AC$ and $BD$ are concurrent.