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Perpendicular to quadrilateral diagonal

Source: 2016 IMO Shortlist G6

July 19, 2017
geometryIMO Shortlist

Problem Statement

Let ABCDABCD be a convex quadrilateral with ∠ABC=∠ADC<90∘\angle ABC = \angle ADC < 90^{\circ}. The internal angle bisectors of ∠ABC\angle ABC and ∠ADC\angle ADC meet ACAC at EE and FF respectively, and meet each other at point PP. Let MM be the midpoint of ACAC and let ω\omega be the circumcircle of triangle BPDBPD. Segments BMBM and DMDM intersect ω\omega again at XX and YY respectively. Denote by QQ the intersection point of lines XEXE and YFYF. Prove that PQ⊥ACPQ \perp AC.
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