KöMaL 2013 B4583

The points $D$ and $E$ lie on the line segment $AB$. In the same half plane, a regular triangle is drawn over each of the line segments $AD$, $DB$, $AE$ and $EB$. The third vertices are $F$, $G$, $H$ and $I$, respectively. Prove that if the lines $FI$ and $GH$ are not parallel, then their intersection lies on the line $AB$.