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Quadrilateral Inscribed in Semicircle

Source: 2013 USAJMO #5

May 1, 2013
trigonometrygeometryanalytic geometrytrapezoidsimilar trianglestrig identitiesratios

Problem Statement

Quadrilateral XABYXABY is inscribed in the semicircle ω\omega with diameter XYXY. Segments AYAY and BXBX meet at PP. Point ZZ is the foot of the perpendicular from PP to line XYXY. Point CC lies on ω\omega such that line XCXC is perpendicular to line AZAZ. Let QQ be the intersection of segments AYAY and XCXC. Prove that BYXP+CYXQ=AYAX.\dfrac{BY}{XP}+\dfrac{CY}{XQ}=\dfrac{AY}{AX}.
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