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concyclic wanted, altitudes, orthocenter, circumcircle and tangents related

Source: Ukraine TST 2018 p9

April 29, 2020
geometrycircumcircleConcyclicorthocenteraltitudes

Problem Statement

Let AA1,BB1,CC1AA_1, BB_1, CC_1 be the heights of triangle ABCABC and HH be its orthocenter. Liune ℓ\ell parallel to ACAC, intersects straight lines AA1AA_1 and CC1CC_1 at points A2A_2 and C2C_2, respectively. Suppose that point B1B_1 lies outside the circumscribed circle of triangle A2HC2A_2 HC_2. Let B1PB_1P and B1TB_1T be tangent to of this circle. Prove that points A1,C1,PA_1, C_1, P, and TT are cyclic.
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