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Extension of a classic config

Source: Serbia TST 2022 P2

April 20, 2023
geometry

Problem Statement

Given is a triangle ABCABC with circumcircle γ\gamma. Points E,FE, F lie on AB,ACAB, AC such that BE=CFBE=CF. Let (AEF)(AEF) meet γ\gamma at DD. The perpendicular from DD to EFEF meets γ\gamma at GG and ADAD meets EFEF at PP. If PGPG meets γ\gamma at JJ, prove that JEJF=AEAF\frac {JE} {JF}=\frac{AE} {AF}.
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