IOM 2017 Problem 6

et $ABCDEF$ be a convex hexagon which has an inscribed circle and a circumcribed. Denote by $\omega_{A}, \omega_{B},\omega_{C},\omega_{D},\omega_{E}$ and $\omega_{F}$ the inscribed circles of the triangles $FAB, ABC, BCD, CDE, DEF$ and $EFA$ , respecitively. Let $l_{AB}$ , be the external of $\omega_{A}$ and $\omega_{B}$ ; lines $l_{BC}$ , $l_{CD}$ , $l_{DE}$ , $l_{EF}$ , $l_{FA}$ are analoguosly defined. Let $A_1$ be the intersection point of the lines $l_{FA}$ and $l_{AB}$ , $B_1, C_1, D_1, E_1, F_1$ are analogously defined. Prove that $A_1D_1, B_1E_1, C_1F_1$ are concurrent.