AT-OEMO-FED1 2008 Problem 4

In a triangle $ABC$ let $E$ be the midpoint of the side $AC$ and $F$ the midpoint of the side $BC$ . Let $G$ be the foot of the perpendicular from $C$ to $AB$ . Show that $\vartriangle EFG$ is isosceles if and only if $\vartriangle ABC$ is isosceles.