PMO 2016 Problem 5

Pentagon $ABCDE$ is inscribed in a circle. Its diagonals $AC$ and $BD$ intersect at $F$. The bisectors of $\angle BAC$ and $\angle CDB$ intersect at $G$. Let $AG$ intersect $BD$ at $H$, let $DG$ intersect $AC$ at $I$, and let $EG$ intersect $AD$ at $J$. If $FHGI$ is cyclic and $$JA \cdot FC \cdot GH = JD \cdot FB \cdot GI,$$ prove that $G$, $F$ and $E$ are collinear.