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perpendicular wanted, right isosceles, square (2014 Romanian NMO grade VII P4)

Source:

May 28, 2020
geometrysquareperpendicularisoscelesright triangle

Problem Statement

Outside the square ABCDABCD is constructed the right isosceles triangle ABDABD with hypotenuse [AB][AB]. Let NN be the midpoint of the side [AD][AD] and M=CE∩AB{M} = CE \cap AB, P=CN∩AB{P} = CN \cap AB , F=PE∩MN{F} = PE \cap MN. On the line FPFP the point QQ is considered such that the [CE[CE is the bisector of the angle QCBQCB. Prove that MQ⊥CFMQ \perp CF.
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