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if 2 of 6 quadrilaterals are concyclic, then all 6 are concyclic

Source: JBMO Shortlist 2017 G5

July 25, 2018
geometryquadrilateralConcyclic

Problem Statement

A point PP lies in the interior of the triangle ABCABC. The lines AP,BPAP, BP, and CPCP intersect BC,CABC, CA, and ABAB at points D,ED, E, and FF, respectively. Prove that if two of the quadrilaterals ABDE,BCEF,CAFD,AEPF,BFPDABDE, BCEF, CAFD, AEPF, BFPD, and CDPECDPE are concyclic, then all six are concyclic.
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