HMMT 十一月 2011 · 冲刺赛 · 第 33 题

英文解析

33. [ 17 ] Let ABC be a triangle with AB = 5, BC = 8, and CA = 7. Let Γ be a circle internally tangentto the circumcircle of ABC at A which is also tangent to segment BC . Γ intersects AB and AC atpoints D and E , respectively. Determine the length of segment DE .
    Answer:40
    Guts Round 9
    DEPA
    B Z X M CO
    First, note that a homothety h centered at A takes Γ to the circumcircle of ABC , D to B and E to C ,Ysince the two circles are tangent. As a result, we have DE ‖ BC . Now, let P be the center of Γ and
    O be the circumcenter of ABC : by the homothety h , we have DE/BC = AP/AO .
    − − →
    Let Γ be tangent to BC at X , and let ray AX meet the circumcircle of ABC at Y . Note that Y is theimage of X under h . Furthermore, h takes BC to the tangent line l to the circumcircle of ABC at Y ,
    ̂
    and since BC ‖ l , we must have that Y is the midpoint of arc BC . Therefore, AX bisects ∠ BAC .
    Now, let Z be the foot of the altitude from A to BC , and let M be the midpoint of BC , so that
    OM ⊥ BC . Note that AP/AO = ZX/ZM . Now, letting BC = a = 8, CA = b = 7, and AB = c = 5,
    we compute
    2 2 2
    c + a − b 5
    BZ = c cos B = =
    2 a 2
    by the Law of Cosines,
    ac 10
    BX = =
    b + c 3
    by the Angle Bisector Theorem, and
    BM = 4 .
    To finish,
    ( AP )( BC ) ( ZX )( BC ) (5 / 6)(8) 40
    DE = = = =
    AO ZM (3 / 2) 9
    .