Final Round of National Olympiad

A convex quadrilateral $ABCD$ where $\angle DAB+\angle ABC<180^{\circ }$ is given on a plane. Let $E$ be a point different from the vertices of the quadrilateral on the line interval $AB$ such that the circumcircles of triangles $AED$ and $BEC$ intersect inside the quadrilateral $ABCD$ at point $F$. Point $G$ is defined so that $\angle DCG=\angle DAB$, $\angle CDG=\angle ABC$ and triangle $CDG$ is located outside quadrilateral $ABCD$. Prove that the points $E$, $F$, $G$ are collinear.