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semicircle tangent to isosceles triangle

Source: JBMO Shortlist 2003

October 12, 2017
geometryJBMO

Problem Statement

Let ABCABC be an isosceles triangle with AB=ACAB = AC. A semi-circle of diameter [EF][EF] with E,Fāˆˆ[BC]E, F \in [BC], is tangent to the sides AB,ACAB,AC in M,NM, N respectively and AEAE intersects the semicircle at PP. Prove that PFPF passes through the midpoint of [MN][MN].
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