MK-MO 2013 Problem 3

Acute angle triangle is given such that $BC$ is the longest side. Let $E$ and $G$ be the intersection points from the altitude from $A$ to $BC$ with the circumscribed circle of triangle $ABC$ and $BC$ respectively. Let the center $O$ of this circle is positioned on the perpendicular line from $A$ to $BE$ . Let $EM$ be perpendicular to $AC$ and $EF$ be perpendicular to $AB$ . Prove that the area of $FBEG$ is greater than the area of $MFE$ .