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AP, BC and OH are concurrent iff AH=HN

Source: French TST 2012

August 2, 2012
geometrycircumcircletrigonometryparallelogramgeometric transformationreflectionratio

Problem Statement

Let ABCABC be an acute-angled triangle with AB≠ACAB\not= AC. Let Γ\Gamma be the circumcircle, HH the orthocentre and OO the centre of Γ\Gamma. MM is the midpoint of BCBC. The line AMAM meets Γ\Gamma again at NN and the circle with diameter AMAM crosses Γ\Gamma again at PP. Prove that the lines AP,BC,OHAP,BC,OH are concurrent if and only if AH=HNAH=HN.
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