HMMT 二月 2011 · 冲刺赛 · 第 28 题

英文解析

28. [ 16 ] Let ABC be a triangle, and let points P and Q lie on BC such that P is closer to B than Q is.
    Suppose that the radii of the incircles of triangles ABP , AP Q , and AQC are all equal to 1, and that the radii of the corresponding excircles opposite A are 3, 6, and 5, respectively. If the radius of theincircle of triangle ABC is , find the radius of the excircle of triangle ABC opposite A .3
    Answer: 1352
    Let t denote the radius of the excircle of triangle 4 ABC .
    Lemma: Let 4 ABC be a triangle, and let r and r be the inradius and exradius opposite A . Thenr B CA = tan tan .
    r 2 2
    Guts Round a
    Proof. Let I and J denote the incenter and the excenter with respect to A . Let D and E be the footof the perpendicular from I and J to BC , respectively. Thenr = ID = BI sin b
    °2
    180 − B Br = JE = BJ sin = BJ cos
    2 2A
    BI = BJ tan ∠ AJB = BY tan .C
    The last equation followed from 2
    °
    180 − B A C
    °
    ∠ AJB = 180 − ∠ ABJ − ∠ JAB = − = .
    2 2 2
    Hencesin br BI B C = · = tan · tan .2
    r BJ 2 2Bcos
    2A
    ∠ AP Q ∠ AQP ∠ AQC
    ∠ AP B
    Noting tan tan = tan tan = 1 and applying the lemma to 4 ABC , 4 ABP ,
    2 2 2 2
    4 AP Q , and 4 AQC give
    3 / 2 ∠ ABC ∠ ACB = tan · tant 2 2
    ( ) ( ) ( )
    ∠ ABC ∠ AP B ∠ AP Q ∠ AQP ∠ AQC ∠ ACB = tan · tan · tan · tan · tan · tan
    2 2 2 2 2 2
    1 1 1 = · ·
    3 6 5
    Therefore, t = 135.