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Prove that some three points lie on a line

Source: Iran TST 2013:TST 2,Day 2,Problem 3

April 25, 2013
geometryincentercircumcirclequadraticsgeometric transformationhomothetytrigonometry

Problem Statement

Let ABCDABCD be a cyclic quadrilateral that inscribed in the circle ω\omega.Let I1,I2I_{1},I_{2} and r1,r2r_{1},r_{2} be incenters and radii of incircles of triangles ACDACD and ABCABC,respectively.assume that r1=r2r_{1}=r_{2}. let ω′\omega' be a circle that touches AB,ADAB,AD and touches ω\omega at TT. tangents from A,TA,T to ω\omega meet at the point KK.prove that I1,I2,KI_{1},I_{2},K lie on a line.
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