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MN and NP are perpendicular if and only if PN is bisector

Source: Romanian NMO 2006, Grade 7, Problem 1

April 18, 2006
geometryangle bisector

Problem Statement

Let ABCABC be a triangle and the points MM and NN on the sides ABAB respectively BCBC, such that 2⋅CNBC=AMAB2 \cdot \frac{CN}{BC} = \frac{AM}{AB}. Let PP be a point on the line ACAC. Prove that the lines MNMN and NPNP are perpendicular if and only if PNPN is the interior angle bisector of ∠MPC\angle MPC.
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