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circumcenter of EFG lies on (ABC)

Source: 2022 Greece TST p2

November 3, 2022
geometrycircumcircle

Problem Statement

Consider triangle ABCABC with AB<AC<BCAB<AC<BC, inscribed in triangle Γ1\Gamma_1 and the circles Γ2(B,AC)\Gamma_2 (B,AC) and Γ2(C,AB)\Gamma_2 (C,AB). A common point of circle Γ2\Gamma_2 and Γ3\Gamma_3 is point EE, a common point of circle Γ1\Gamma_1 and Γ3\Gamma_3 is point FF and a common point of circle Γ1\Gamma_1 and Γ2\Gamma_2 is point GG, where the points E,F,GE,F,G lie on the same semiplane defined by line BCBC, that point AA doesn't lie in. Prove that circumcenter of triangle EFGEFG lies on circle Γ1\Gamma_1.
Note: By notation Γ(K,R)\Gamma (K,R), we mean random circle Γ\Gamma has center KK and radius RR.
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