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Source: Bulgaria EGMO TST 2019 Day 2 Problem 2

February 3, 2023
geometrycyclic quadrilateralcircumcircle

Problem Statement

Let ABCDABCD be a cyclic quadrilateral with circumcircle ω\omega centered at OO, whose diagonals intersect at HH. Let O1O_1 and O2O_2 be the circumcenters of triangles AHDAHD and BHCBHC. A line through HH intersects ω\omega at M1M_1 and M2M_2 and intersects the circumcircles of triangles O1HOO_1HO and O2HOO_2HO at N1N_1 and N2N_2, respectively, so that N1N_1 and N2N_2 lie inside ω\omega. Prove that M1N1=M2N2M_1N_1 = M_2N_2.
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