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Circle passing C with center B

Source: 2017 Korea Winter Program Practice Test 2 #6

August 14, 2019
geometry

Problem Statement

ABCABC is an obtuse triangle satisfying ∠A>90∘\angle A>90^\circ, and its circumcenter OO and circumcircle ω1\omega_1. Let ω2\omega_2 be a circle passing CC with center BB. ω2\omega_2 meets BCBC at DD. ω1\omega_1 meets ADAD and ω2\omega_2 at EE and F(≠C)F(\neq C), respectively. AFAF meets ω2\omega_2 at G(≠F)G(\neq F). Prove that the intersection of CECE and BGBG lies on the circumcircle of AOBAOB.
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