HMMT 二月 2016 · 冲刺赛 · 第 10 题

英文解析

10. [ 7 ] Let ABC be a triangle with AB = 13, BC = 14, CA = 15. Let O be the circumcenter of ABC .
    Find the distance between the circumcenters of triangles AOB and AOC .
    Proposed by: Evan Chen
    Answer:91
    Let S, T be the intersections of the tangents to the circumcircle of ABC at A, C and at A, B respectively.6
    Note that ASCO is cyclic with diameter SO , so the circumcenter of AOC is the midpoint of OS , andsimilarly for the other side. So the length we want is ST . The circumradius R of ABC can be 1
    abc 652
    computed by Heron’s formula and K = , giving R = . A few applications of the Pythagorean
    4 R 8
    65 3991
    theorem and similar triangles gives AT = , AS = , so the answer is .
    6 2
    !6