HMMT 二月 2017 · 几何 · 第 10 题

英文解析

10. Let ABCD be a quadrilateral with an inscribed circle ω . Let I be the center of ω let IA = 12,
    IB = 16, IC = 14, and ID = 11. Let M be the midpoint of segment AC . Compute , where N is i mthe midpoint of segment BD .IN
    Proposed by: Sam Korsky
    Answer:21
    Let points W, X, Y, Z be the tangency points between ω and lines AB, BC, CD, DA respectively. Now 22
    ′ ′ ′ ′
    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 ratior 2
    ′ ′
    where r is the radius of ω . Similarly figure IB P D is similar to figure IDM B with similitude
    IA · ICr 2
    ratio . Therefore
    IB · ID
    2 2
    r r
    IP = · IM = · IN
    IA · IC IB · IDwhich yields
    IM IA · IC 12 · 14 21 = = =
    IN IB · ID 16 · 11 22