KöMaL 2016 C1351

Trapezium $ABCD$ has an inscribed circle that touches the sides $AB$ , $BC$ , $CD$ and $DA$ at points $E$ , $F$ , $G$ and $H$ , respectively. The interior angle at vertex $B$ is $60^\circ$. Let $I$ denote the intersection of lines $AD$ and $FG$ , and let $K$ denote the midpoint of $FH$ . Prove that if $HE$ is parallel to $BC$ then $IK$ is also parallel to them.