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MP \cdot XY \ge 2 \cdot QX \cdot PY, related to a chord

Source: JBMO Shortlist 2018 G6

July 22, 2019
geometryinequalitiesgeometric inequalityChords

Problem Statement

Let XYXY be a chord of a circle Ω\Omega, with center OO, which is not a diameter. Let P,QP, Q be two distinct points inside the segment XYXY, where QQ lies between PP and XX. Let ℓ\ell the perpendicular line drawn from PP to the diameter which passes through QQ. Let MM be the intersection point of ℓ\ell and Ω\Omega, which is closer to PP. Prove that MP⋅XY≥2⋅QX⋅PY MP \cdot XY \ge 2 \cdot QX \cdot PY
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