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Angle condition, circles and concurrent lines

Source: 5th Memorial Mathematical Competition "Aleksandar Blazhevski - Cane" - Senior - Problem 4

January 29, 2024
geometry

Problem Statement

Let DD be a point inside △ABC\triangle ABC such that ∠CDA+∠CBA=180∘.\angle CDA + \angle CBA = 180^{\circ}. The line CDCD meets the circle ⊙ABC\odot ABC at the point EE for the second time. Let GG be the common point of the circle centered at CC with radius CDCD and the arc AC⌢\overset{\LARGE \frown}{AC} of ⊙ABC\odot ABC which does not contain the point BB. The circle centered at AA with radius ADAD meets ⊙BCD\odot BCD for the second time at FF. Prove that the lines GE,FD,CBGE, FD, CB are concurrent or parallel.
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