IR-TST 2013 Problem 12

Let $ABCD$ be a cyclic quadrilateral that inscribed in the circle $\omega$ .Let $I_{1},I_{2}$ and $r_{1},r_{2}$ be incenters and radii of incircles of triangles $ACD$ and $ABC$ ,respectively.assume that $r_{1}=r_{2}$ . let $\omega'$ be a circle that touches $AB,AD$ and touches $\omega$ at $T$ . tangents from $A,T$ to $\omega$ meet at the point $K$ .prove that $I_{1},I_{2},K$ lie on a line.