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Circumcircles of ABC, ADE, AO_1O_2 have a common point

Source: Czech-Polish-Slovak Match 2018, Problem 2

July 2, 2018
geometrycircumcircle

Problem Statement

Let ABCABC be an acute scalene triangle. Let DD and EE be points on the sides ABAB and ACAC, respectively, such that BD=CEBD=CE. Denote by O1O_1 and O2O_2 the circumcentres of the triangles ABEABE and ACDACD, respectively. Prove that the circumcircles of the triangles ABC,ADEABC, ADE, and AO1O2AO_1O_2 have a common point different from AA.
Proposed by Patrik Bak, Slovakia
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