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three lines throught vertices and circumcenters are concurrent

Source: 2021 Oral Moscow Geometry Olympiad grades 8-9 p4

May 17, 2021
geometrycyclic quadrilateralcircumcircle

Problem Statement

On the diagonal ACAC of cyclic quadrilateral ABCDABCD a point EE is chosen such that ∠ABE=∠CBD\angle ABE = \angle CBD. Points O,O1,O2O,O_1,O_2 are the circumcircles of triangles ABC,ABEABC, ABE and CBECBE respectively. Prove that lines DO,AO1DO,AO_{1} and CO2CO_{2} are concurrent.
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