UZ-MO 2012 Problem 5

Given points $A,B,C$ and $D$ lie a circle. $AC\cap BD=K$ . $I_1, I_2,I_3$ and $I_4$ incenters of $ABK,BCK,CDK,DKA$ . $M_1,M_2,M_3,M_4$ midpoints of arcs $AB,BC,CA,DA$ . Then prove that $M_1I_1,M_2I_2,M_3I_3,M_4I_4$ are concurrent.