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2 circles and a line concurrent, incenters (2013 Kyiv City MO Round2 9.5 10.3)

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August 5, 2020
geometryconcurrencyconcurrentincenter

Problem Statement

Given a triangle ABC ABC , AD AD is its angle bisector. Let E,F E, F be the centers of the circles inscribed in the triangles ADC ADC and ADB ADB , respectively. Denote by ω \omega , the circle circumscribed around the triangle DEF DEF , and by Q Q , the intersection point of BE BE and CF CF , and H,J,K,M H, J, K, M , respectively the second intersection point of the lines CE,CF,BE,BF CE, CF, BE, BF with circle ω \omega . Let ω1,ω2\omega_1, \omega_2 the circles be circumscribed around the triangles HQJ HQJ and KQM KQM Prove that the intersection point of the circles ω1,ω2\omega_1, \omega_2 different from Q Q lies on the line AD AD .
(Kivva Bogdan)
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