DK-GEORGMOHR 2012 Problem 5

In the hexagon $ABCDEF$ , all angles are equally large. The side lengths satisfy $AB = CD = EF = 3$ and $BC = DE = F A = 2$ . The diagonals $AD$ and $CF$ intersect each other in the point $G$ . The point $H$ lies on the side $CD$ so that $DH = 1$ . Prove that triangle $EGH$ is equilateral.