Prove that $HIGD$ is a parallelogram

Question

$EF$ is the mid segment of isosceles trapezoid $ABCD$. $AG$ is a height. The diagonals intersect $EF$ at points $H$ and $I$.

Prove $HIGD$ is a parallelogram.

I tried showing $HI=DG$ but to no avail.

Answer 1

Let $BK$ is an altitude of the trapezoid.

Thus, $DG=CK$ and $$HI=EI-EH=\frac{1}{2}DC-\frac{1}{2}AB=DG.$$