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concyclic from Philippines, equal angles and circumcircle related

Source: 21st Philippine Mathematical Olympiad 2019 p4

January 1, 2020
geometryConcyclicequal anglescircumcircle

Problem Statement

In acute triangle ABCABC with ∠BAC>∠BCA\angle BAC > \angle BCA, let PP be the point on side BCBC such that ∠PAB=∠BCA\angle PAB = \angle BCA. The circumcircle of triangle APBAP B meets side ACAC again at QQ. Point DD lies on segment APAP such that ∠QDC=∠CAP\angle QDC = \angle CAP. Point EE lies on line BDBD such that CE=CDCE = CD. The circumcircle of triangle CQECQE meets segment CDCD again at FF, and line QFQF meets side BCBC at GG. Show that B,D,FB, D, F, and GG are concyclic
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