Problem

Source: 2018 Iran MO 3rd Geometry 2

Tags: geometry, incenter



Two intersecting circles $\omega_1$ and $\omega_2$ are given.Lines $AB,CD$ are common tangents of $\omega_1,\omega_2$($A,C \in \omega_1 ,B,D \in \omega_2$) Let $M$ be the midpoint of $AB$.Tangents through $M$ to $\omega_1$ and $\omega_2$(other than $AB$) intersect $CD$ at $X,Y$.Let $I$ be the incenter of $MXY$.Prove that $IC=ID$.