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incenter lies on a line, 3 excircles related

Source: 2019 XXII All-Ukrainian Tournament of Young Mathematicians named after M. Y. Yadrenko, Qualifying p11

May 5, 2021
geometryincenterexcirclesexcircleUkrainian TYM

Problem Statement

Let ωa,ωb,ωc\omega_a, \omega_b, \omega_c be the exscribed circles tangent to the sides a,b,ca, b, c of a triangle ABCABC, respectively, Ia,Ib,Ic I_a, I_b, I_c be the centers of these circles, respectively, Ta,Tb,TcT_a, T_b, T_c be the points of contact of these circles to the line BCBC, respectively. The lines TbIcT_bI_c and TcIbT_cI_b intersect at the point QQ. Prove that the center of the circle inscribed in triangle ABCABC lies on the line TaQT_aQ.
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