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Quadrilateral APBQ

Source: USAMO 2015 Problem 2, JMO Problem 3

April 28, 2015
USA(J)MOUSAMOgeometrycyclic quadrilateralcomplex bashfixed locus

Problem Statement

Quadrilateral APBQAPBQ is inscribed in circle ω\omega with ∠P=∠Q=90∘\angle P = \angle Q = 90^{\circ} and AP=AQ<BPAP = AQ < BP. Let XX be a variable point on segment PQ‾\overline{PQ}. Line AXAX meets ω\omega again at SS (other than AA). Point TT lies on arc AQBAQB of ω\omega such that XT‾\overline{XT} is perpendicular to AX‾\overline{AX}. Let MM denote the midpoint of chord ST‾\overline{ST}. As XX varies on segment PQ‾\overline{PQ}, show that MM moves along a circle.
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