KöMaL 2011 B4388

$D$ is an interior point of side $AB$ of an acute-angled triangle ABC . The line drawn through $D$ parallel to side $AC$ intersects side $BC$ at $E$, and the line drawn through $D$parallel to side $BC$ intersects side $AC$ at $F$. The other intersection of the circles ADF and BDE is $G$. Prove that ABEF is a cyclic quadrilateral if and only if $G$ lies on the line segment $CD$. (Suggested by Sz. Miklós, Herceghalom)