HMMT 二月 2015 · 冲刺赛 · 第 29 题

英文解析

29. [ 20 ] Let ABC be a triangle whose incircle has center I and is tangent to BC , CA , AB , at D , E , F .
    ̂
    Denote by X the midpoint of major arc BAC of the circumcircle of ABC . Suppose P is a point online XI such that DP ⊥ EF .
    Given that AB = 14, AC = 15, and BC = 13, compute DP .
    √
    4 5
    Answer: Let H be the orthocenter of triangle DEF . We claim that P is the midpoint of DH .
    Indeed, consider an inversion at the incicrle of ABC , denoting the inverse of a point with an asterik.5
    ° ∗ ∗ °
    It maps ABC to the nine-point circle of △ DEF . According to ∠ IAX = 90 , we have ∠ A X I = 90 .
    ∗
    Hence line XI passes through the point diametrically opposite to A , which is the midpoint of DH ,
    as claimed.
    Guts
    The rest is a straightforward computation. The inradius of △ ABC is r = 4. The length of EF is given
    AF · r 16
    √
    by EF = 2 = . Then,
    5AI
    ( )
    ( )2
    1 1 64 16
    2 2 2 2
    DP = DH = 4 r − EF = 4 − = .
    2 4 5 5
    √
    4 5
    Hence DP = .
    Remark. This is also not too bad of a coordinate bash.5