HMMT 二月 2003 · 几何 · 第 10 题

英文解析

10. Convex quadrilateral M AT H is given with HM/M T = 3 / 4, and AT M = M AT =
    °
    AHM = 60 . N is the midpoint of M A , and O is a point on T H such that lines 6
    M T, AH, N O are concurrent. Find the ratio HO/OT .
    Solution: 9 / 16
    6 6
    Triangle M AT is equilateral, so HM/AT = HM/M T = 3 / 4. Also, AHM = AT M ,
    so the quadrilateral is cyclic. Now, let P be the intersection of M T, AH, N O . Extend
    M H and N O to intersect at point Q . Then by Menelaus’s theorem, applied to triangle
    AHM and line QN P , we have 4
    HQ M N AP
    · · = 1 ,
    QM N A P Hwhile applying the same theorem to triangle T HM and line QP O gives
    HQ M P T O
    · · = 1 .
    QM P T OH
    Combining gives HO/OT = ( M P/P T ) · ( AN/N M ) · ( HP/P A ) = ( M P/P A ) · ( HP/P T )
    (because AN/N M = 1). But since M AT H is cyclic, 4 AP T ∼ 4 M P H , so M P/P A =
    HP/P T = HM/AT = 3 / 4, and the answer is (3 / 4) = 9 / 16. (See figure.)2
    OHTQ
    A MP
    5N