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2 circumcircles and 1 line concurrent, symmetrix wrt sides, orthocenter

Source: 2019 Balkan MO Shortlist G8 - BMO

November 20, 2020
geometrySymmetricconcurrencyconcurrentorthocenter

Problem Statement

Given an acute triangle ABCABC, (c)(c) its circumcircle with center OO and HH the orthocenter of the triangle ABCABC. The line AOAO intersects (c)(c) at the point DD. Let D1,D2D_1, D_2 and H2,H3H_2, H_3 be the symmetrical points of the points DD and HH with respect to the lines AB,ACAB, AC respectively. Let (c1)(c_1) be the circumcircle of the triangle AD1D2AD_1D_2. Suppose that the line AHAH intersects again (c1)(c_1) at the point UU, the line H2H3H_2H_3 intersects the segment D1D2D_1D_2 at the point K1K_1 and the line DH3DH_3 intersects the segment UD2UD_2 at the point L1L_1. Prove that one of the intersection points of the circumcircles of the triangles D1K1H2D_1K_1H_2 and UDL1UDL_1 lies on the line K1L1K_1L_1.
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