HMMT 二月 2017 · 几何 · 第 10 题

HMMT February 2017 — Geometry — Problem 10

Let ABCD be a quadrilateral with an inscribed circle ω . Let I be the center of ω let IA = 12, IM IB = 16, IC = 14, and ID = 11. Let M be the midpoint of segment AC . Compute , where N is IN the midpoint of segment BD .

解析

Let ABCD be a quadrilateral with an inscribed circle ω . Let I be the center of ω let IA = 12, IM IB = 16, IC = 14, and ID = 11. Let M be the midpoint of segment AC . Compute , where N is IN the midpoint of segment BD . Proposed by: Sam Korsky 21 Answer: 22 Let points W, X, Y, Z be the tangency points between ω and lines AB, BC, CD, DA respectively. Now ′ ′ ′ ′ invert about ω . Then A , B , C , D are the midpoints of segments ZW, W X, XY, Y Z respectively. ′ ′ ′ ′ ′ ′ Thus by Varignon’s Theorem A B C D is a parallelogram. Then the midpoints of segments A C and ′ ′ ′ ′ B D coincide at a point P . Note that figure IA P C is similar to figure ICM A with similitude ratio 2 r ′ ′ where r is the radius of ω . Similarly figure IB P D is similar to figure IDM B with similitude IA · IC 2 r ratio . Therefore IB · ID 2 2 r r IP = · IM = · IN IA · IC IB · ID which yields IM IA · IC 12 · 14 21 = = = IN IB · ID 16 · 11 22