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Geometry with circle centered at arc midpoint

Source: Greece National Olympiad 2024, Problem 2

February 24, 2024
geometry

Problem Statement

Let ABCABC be a triangle with AB<AC<BCAB<AC<BC with circumcircle Γ1\Gamma_1. The circle Γ2\Gamma_2 has center DD lying on Γ1\Gamma_1 and touches BCBC at EE and the extension of ABAB at FF. Let Γ1\Gamma_1 and Γ2\Gamma_2 meet at K,GK, G and the line KGKG meets EFEF and CDCD at M,NM, N. Show that BCNMBCNM is cyclic.
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