Mathematica competitions in Croatia

Let $△ABC$ be a triangle with an obtuse angle at vertex $B$, let $D$ and $E$ be midpoints of the segments $\overline{AB}$ and $\overline{AC}$ respectively, let $F$ be a point on the segment $\overline{BC}$ such that $\angle BFE=90^{\circ }$, and let $G$ be a point on the segment $\overline{DE}$ such that $\angle BGE=90^{\circ }$. Prove that the points $A,F$ and $G$ are collinear if and only if $2|BF|=|CF|$. (Matija Bašić)