BMO 2016 Short List Final problem

Problem

The point $M$ lies on the side $AB$ of the circumscribed quadrilateral $ABCD$. The points $I_1, I_2$, and $I_3$ are the incenters of $\triangle MBC$, $\triangle MCD$, and $\triangle MDA$. Show that the points $M, I_1, I_2$, and $I_3$ lie on a circle.