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collinear wanted, <A=60^o, semicircle of diameter BC. another circle related

Source: Indonesia INAMO Shortlist 2010 G5

August 27, 2021
geometrycollinear

Problem Statement

Given an arbitrary triangle ABCABC, with ∠A=60o\angle A = 60^o and AC<ABAC < AB. A circle with diameter BCBC, intersects ABAB and ACAC at FF and EE, respectively. Lines BEBE and CFCF intersect at DD. Let Γ\Gamma be the circumcircle of BCDBCD, where the center of Γ\Gamma is OO. Circle Γ\Gamma intersects the line ABAB and the extension of ACAC at MM and NN, respectively. MNMN intersects BCBC at PP. Prove that points AA, PP, OO lie on the same line.
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