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Geometry with orthocenter II

Source: Philippine Mathematical Olympiad 2024 P7

February 24, 2024
geometryorthocentercircumcircleCircumcenter

Problem Statement

Let ABCABC be an acute triangle with orthocenter HH, circumcenter OO, and circumcircle Ω\Omega. Points EE and FF are the feet of the altitudes from BB to ACAC, and from CC to ABAB, respectively. Let line AHAH intersect Ω\Omega again at DD. The circumcircle of DEFDEF intersects Ω\Omega again at XX, and AXAX intersects BCBC at II. The circumcircle of IEFIEF intersects BCBC again at GG. If MM is the midpoint of BCBC, prove that lines MXMX and OGOG intersect on Ω\Omega.
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