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AP bisects BC, BD // CE, 3 circumcircles related

Source: 1st Final Mathematical Cup 2019 FMC , juniors p1

October 6, 2020
geometrycircumcircle

Problem Statement

Let ABCABC be a triangle and let D,ED, E are points on its circumscribed circle, such that DD lies on arc AB,EAB, E lies on arc ACAC (smaller arcs) and BD∥CEBD \parallel CE . Let the point F be the intersection of the lines DADA and CECE, and the intersection of the lines EAEA and BDBD is GG. Let PP be the second intersection of the circumscribed circles of △ABG\vartriangle ABG and △ACF\vartriangle ACF. Prove that the lineAP AP passes through the mid point of the side BCBC.
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