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isosceles and perpendicular, right isosceles (2016 Romanian NMO grade VII P4)

Source:

June 1, 2020
geometryisoscelesperpendicularanglesright triangle

Problem Statement

Consider the isosceles right triangle ABCABC, with ∠A=90o\angle A = 90^o and the point M∈(BC)M \in (BC) such that ∠AMB=75o\angle AMB = 75^o. On the inner bisector of the angle MACMAC take a point FF such that BF=ABBF = AB. Prove that:
a) the lines AMAM and BFBF are perpendicular; b) the triangle CFMCFM is isosceles.
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