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Four circumcenters not on a circle.

Source: Brazilian Mathematical Olympiad 2021, Level 3, Problem 1

February 7, 2022
geometryBrazilian Math OlympiadBrazilconvex quadrilateral

Problem Statement

Let ABCDABCD be a convex quadrilateral in the plane and let OA,OB,OCO_{A}, O_{B}, O_{C} and ODO_{D} be the circumcenters of the triangles BCD,CDA,DABBCD, CDA, DAB and ABCABC, respectively. Suppose these four circumcenters are distinct points. Prove that these points are not on a same circle.
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