FI-MO 2019 Problem 3

Let $ABCD$ be a cyclic quadrilateral whose side $AB$ is at the same time the diameter of the circle. The lines $AC$ and $BD$ intersect at point $E$ and the extensions of lines $AD$ and $BC$ intersect at point $F$ . Segment $EF$ intersects the circle at $G$ and the extension of segment $EF$ intersects $AB$ at $H$ . Show that if $G$ is the midpoint of $FH$ , then $E$ is the midpoint of $GH$ .