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2nd intersection of circumcircles ACY ,BCX lies on bisector <BCA, AX=AB=BY

Source: 2021 Austrian Federal Competition For Advanced Students, Part 1 p2

May 31, 2021
geometryequal segmentsangle bisector

Problem Statement

Let ABCABC denote a triangle. The point XX lies on the extension of ACAC beyond AA, such that AX=ABAX = AB. Similarly, the point YY lies on the extension of BCBC beyond BB such that BY=ABBY = AB. Prove that the circumcircles of ACYACY and BCXBCX intersect a second time in a point different from CC that lies on the bisector of the angle ∠BCA\angle BCA.
(Theresia Eisenkölbl)
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