Let $I$ be the incentre of triangle $ABC$. Let $D,E,F$ be the intersections between the incenter and sides $BC,CA,AB$ respectively. Let $M$ be the midpoint of $EF$ and let $Q$ be the second intersection between $AD$ and the incircle. Show that $MIDQ$ is a cyclic quadrilateral.
I have tried some angle-chasing, but no result. Any help?