RS-SMO 2023 Problem 6

Given is a triangle $ABC$ with incenter $I$ and circumcircle $\omega$ . The incircle is tangent to $BC$ at $D$ . The perpendicular at $I$ to $AI$ meets $AB, AC$ at $E, F$ and the circle $(AEF)$ meets $\omega$ and $AI$ at $G, H$ . The tangent at $G$ to $\omega$ meets $BC$ at $J$ and $AJ$ meets $\omega$ at $K$ . Prove that $(DJK)$ and $(GIH)$ are tangent to each other.