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Conditional geometry

Source: JBMO 2023 Problem 4

June 26, 2023
geometrycircumcircleCircumcenter

Problem Statement

Let ABCABC be an acute triangle with circumcenter OO. Let DD be the foot of the altitude from AA to BCBC and let MM be the midpoint of ODOD. The points ObO_b and OcO_c are the circumcenters of triangles AOCAOC and AOBAOB, respectively. If AO=ADAO=AD, prove that points AA, ObO_b, MM and OcO_c are concyclic.
Marin Hristov and Bozhidar Dimitrov, Bulgaria
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