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Concurrent: line thru orthocenter & circumcenter, BC & other

Source: XV Rioplatense Mathematical Olympiad (2006), Level 3

August 9, 2011
geometrycircumcircleparallelogramgeometric transformationreflectionperpendicular bisectorBritishMathematicalOlympiad

Problem Statement

The acute triangle ABCABC with AB≠ACAB\neq AC has circumcircle Γ\Gamma, circumcenter OO, and orthocenter HH. The midpoint of BCBC is MM, and the extension of the median AMAM intersects Γ\Gamma at NN. The circle of diameter AMAM intersects Γ\Gamma again at AA and PP. Show that the lines APAP, BCBC, and OHOH are concurrent if and only if AH=HNAH = HN.
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