KöMaL 2014 A630

The konvex quadrilateral $ABCD$ has an inscribed circle with center $I$ . The rays $AB$ and $DC$ meet at point $F$ , the rays $AD$ and $BC$ meet at point $G$ . Let $\mathcal{E}$ be the ellipse with foci $F$ and $G$ that passes through points $B$ and $D$ , and let $\mathcal{H}$ be the hyperbola branch with foci $F$ and $G$ that passes through points $A$ and $C$ . Denote by $P$ and $Q$ the intersections of $\mathcal{E}$ and $\mathcal{H}$ . Show that the points $P$ , $Q$ and $I$ are collinear.