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kite and angle bisector given, concurrency wanted

Source: INAMO Shortlist 2014 G4

July 13, 2019
geometryangle bisectorconcurrencyconcurrent

Problem Statement

Given an acute triangle ABCABC with AB<ACAB <AC. Points PP and QQ lie on the angle bisector of ∠BAC\angle BAC so that BPBP and CQCQ are perpendicular on that angle bisector. Suppose that point E,FE, F lie respectively at sides ABAB and ACAC respectively, in such a way that AEPFAEPF is a kite. Prove that the lines BC,PFBC, PF, and QEQE intersect at one point.
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