SHARYGIN 2017 Problem 15

Let $ABC$ be an acute-angled triangle with incircle $\omega$ and incenter $I$ . Let $\omega$ touch $AB, BC$ and $CA$ at points $D, E, F$ respectively. The circles $\omega_1$ and $\omega_2$ centered at $J_1$ and $J_2$ respectively are inscribed into A $DIF$ and $BDIE$ . Let $J_1J_2$ intersect $AB$ at point $M$ . Prove that $CD$ is perpendicular to $IM$ .