Question 57
In the scalene triangle $ABC$ with incentre $I$, the line $AI$ meets the circumcircle again at the point $D$, and $J$ is the point such that $D$ is the midpoint of $IJ$. Consider points $E$ and $F$ on the 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 $BG = CH$.