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concurrent circumcircles related to 3 cyclic quadrilaterals, circumcenter

Source: Sharygin 2011 Final 10.7

March 31, 2019
geometrycyclic quadrilateralCircumcenterconcurrentcirclesaltitudescircumcircle

Problem Statement

Point OO is the circumcenter of acute-angled triangle ABCABC, points A1,B1,C1A_1,B_1, C_1 are the bases of its altitudes. Points A,B,CA', B', C' lying on lines OA1,OB1,OC1OA_1, OB_1, OC_1 respectively are such that quadrilaterals AOBC,BOCA,COABAOBC', BOCA', COAB' are cyclic. Prove that the circumcircles of triangles AA1A,BB1B,CC1CAA_1A', BB_1B', CC_1C' have a common point.