HMMT 二月 2007 · 冲刺赛 · 第 32 题

英文解析

32. [ 18 ] Triangle ABC has AB = 4 , BC = 6 , and AC = 5 . Let O denote the circumcenter of ABC. Thecircle Γ is tangent to and surrounds the circumcircles of triangles AOB, BOC, and AOC . Determinethe diameter of Γ .
    √
    256 7
    Answer: . Denote by ω, Γ , Γ , and Γ the circumcenters of triangles ABC, BOC, COA, and
    1 2 3
    AOB , respectively. An inversion about ω interchanges Γ and line BC , Γ and line CA , and Γ and 17
    1 2 3
    line AB. This inversion also preserves tangency between generalized circles, so the image of Γ is a circletangent to AB , BC, and CA. It is the incircle of ABC because it is closer to O than these lines and
    ABC is acute.
    Now we run a few standard calculations. Where s, r, and R denote the semiperimeter, inradius, andcircumradius of ABC , respectively, we have the following:
    √
    √
    15 7
    [ ABC ] = s ( s − a )( s − b )( s − c ) = ;
    √4
    r = [ ABC ] /s = 7 / 2;
    abc 8
    R = = √
    4[ ABC ]
    OI = R ( R − 2 r ) = .287
    Let OI intersect the incircle of ABC at P and Q , with I between P and O. Then OP = r + OI and 7
    ′ ′
    OQ = r − OI, and P Q is a diameter. Under the inversion, P and Q map to P and Q respectively.
    ′ ′
    Because P, I, O, and Q are collinear in that order, P and Q are diametrically opposed on Γ. It followsthat the diameter of Γ is
    ( )
    2 2 2
    R R 1 1 2 r r
    ′ ′ ′ ′ 2
    P Q = OP + OQ = + = R + = .
    2 2
    OP OQ r + OI r − OI r − OI
    √
    256 7
    We plug in the values found above to arrive at .
    17