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Triangles ABC and A_1B_1C_1 with same incircle and circumcircle

Source: Saint Petersburg olympiad 2024, 10.6

September 22, 2024
geometrycircumcircle

Problem Statement

Inscribed hexagon AB1CA1BC1AB_1CA_1BC_1 is given. Circle ω\omega is inscribed in both triangles ABCABC and A1B1C1A_1B_1C_1 and touches segments ABAB and A1B1A_1B_1 at points DD and D1D_1 respectively. Prove that if ∠ACD=∠BCD1\angle ACD = \angle BCD_1, then ∠A1C1D1=∠B1C1D\angle A_1C_1D_1 = \angle B_1C_1D.
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