SHARYGIN 2023 Problem 2

The diagonals of a rectangle $ABCD$ meet at point $E$ . A circle centered at $E$ lies inside the rectangle. Let $CF$ , $DG$ , $AH$ be the tangents to this circle from $C$ , $D$ , $A$ ; let $CF$ meet $DG$ at point $I$ , $EI$ meet $AD$ at point $J$ , and $AH$ meet $CF$ at point $L$ . Prove that $LJ$ is perpendicular to $AD$ .