MathDB
Product of Lengths

Source: 2017 AMC 12A #24

February 8, 2017
AMCAMC 12AMC 12 Ageometry2017 AMC 12A2017 AMC

Problem Statement

Quadrilateral ABCDABCD is inscribed in circle OO and has sides AB=3AB = 3, BC=2BC = 2, CD=6CD = 6, and DA=8DA = 8. Let XX and YY be points on BD\overline{BD} such that \frac{DX}{BD} = \frac{1}{4}   \text{and}   \frac{BY}{BD} = \frac{11}{36}. Let EE be the intersection of intersection of line AXAX and the line through YY parallel to AD\overline{AD}. Let FF be the intersection of line CXCX and the line through EE parallel to AC\overline{AC}. Let GG be the point on circle OO other than CC that lies on line CXCX. What is XFXGXF \cdot XG?
<spanclass=latexbold>(A)</span>17<spanclass=latexbold>(B)</span>59523<spanclass=latexbold>(C)</span>911234<spanclass=latexbold>(D)</span>671023<spanclass=latexbold>(E)</span>18<span class='latex-bold'>(A) </span>17\qquad<span class='latex-bold'>(B) </span>\frac{59 - 5\sqrt{2}}{3}\qquad<span class='latex-bold'>(C) </span>\frac{91 - 12\sqrt{3}}{4}\qquad<span class='latex-bold'>(D) </span>\frac{67 - 10\sqrt{2}}{3}\qquad<span class='latex-bold'>(E) </span>18