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Equal segment wanted

Source: 2022 Thailand MO Day 1 P1

June 15, 2022
geometry

Problem Statement

Let BCBC be a chord of a circle Γ\Gamma and AA be a point inside Γ\Gamma such that ∠BAC\angle BAC is acute. Outside △ABC\triangle ABC, construct two isosceles triangles △ACP\triangle ACP and △ABR\triangle ABR such that ∠ACP\angle ACP and ∠ABR\angle ABR are right angles. Let lines BABA and CACA meet Γ\Gamma again at points EE and FF, respectively. Let lines EPEP and FRFR meet Γ\Gamma again at points XX and YY, respectively. Prove that BX=CYBX=CY.
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