HMMT 十一月 2015 · 冲刺赛 · 第 25 题

英文解析

25. [ 13 ] Let ABC be a triangle that satisfies AB = 13 , BC = 14 , AC = 15 . Given a point P in the plane,
    let P , P , P be the reflections of A, B, C across P . Call P good if the circumcircle of P P P
    A B C A B Cintersects the circumcircle of ABC at exactly 1 point. The locus of good points P encloses a region S .
    Find the area of S .
    Proposed by: Yang Liu
    4225
    Answer: π
    By the properties of reflection, the circumradius of P P P equals the circumradius of ABC . There-64
    A B Cfore, the circumcircle of P P P must be externally tangent to the circumcircle of ABC . Now it’s
    A B Ceasy to see that the midpoint of the 2 centers of ABC and P P P lies on the circumcircle of ABC.
    A B C
    So the locus of P is simply the circumcircle of ABC .
    abc 13 · 14 · 15 65
    Since [ ABC ] = , we find the circumradius is R = = , so the enclosed region has area
    4 R 84 · 4 8
    4225
    π .
    +64