OME 2023 Problem 6

In an acute scalene triangle $ABC$ with incenter $I$ , the line $AI$ intersects the circumcircle again at $D$ , and let $J$ be a point such that $D$ is the midpoint of $IJ$ . Consider points $E$ and $F$ on line $BC$ such that $IE$ and $JF$ are perpendicular to $AI$ . Consider points $G$ on $AE$ and $H$ on $AF$ such that $IG$ and $JH$ are perpendicular to $AE$ and $AF$ , respectively. Prove that $BG=CH$ .