Problem

Source: Spain MO 2023 P6

Tags: geometry, Spain



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 $BG=CH$.