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tangent circles wanted, incircle and 3 circumcircles related

Source: 2014 XVII All-Ukrainian Tournament of Young Mathematicians named after M. Y. Yadrenko, Qualifying p23

May 7, 2021
geometrytangent circlesUkrainian TYM

Problem Statement

The inscribed circle ω\omega of triangle ABCABC with center II touches the sides AB,BC,CAAB, BC, CA at points C1,A1,B1C_1, A_1, B_1. The circle circumsrcibed around △AB1C1\vartriangle AB_1C_1 intersects the circumscribed circle of ABCABC for second time at the point KK. Let MM be the midpoint BCBC, LL be the midpoint of B1C1B_1C_1. The circle circumsrcibed around △KA1M\vartriangle KA_1M cuts intersects ω\omega for second time at the point TT. Prove that the circumscribed circles of triangles KLTKLT and LIMLIM are tangent.
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