HMMT 二月 2016 · 冲刺赛 · 第 14 题

英文解析

14. [ 9 ] Let ABC be a triangle such that AB = 13, BC = 14, CA = 15 and let E , F be the feet of thealtitudes from B and C , respectively. Let the circumcircle of triangle AEF be ω . We draw three lines,
    tangent to the circumcircle of triangle AEF at A , E , and F . Compute the area of the triangle thesethree lines determine.
    Proposed by: Christopher Shao
    Answer:462
    Note that AEF ∼ ABC . Let the vertices of the triangle whose area we wish to compute be P, Q, R ,5
    opposite A, E, F respectively. Since H, O are isogonal conjugates, line AH passes through the circumcenter of AEF , so QR ‖ BC .
    Let M be the midpoint of BC . We claim that M = P . This can be seen by angle chasing at E, F tofind that ∠ P F B = ∠ ABC , ∠ P EC = ∠ ACB , and noting that M is the circumcenter of BF EC . So,
    the height from P to QR is the height from A to BC , and thus if K is the area of ABC , the area wewant is K .QR
    BC tan BBC
    Heron’s formula gives K = 84, and similar triangles QAF, M BF and RAE, M CE give QA = ,
    2 tan A
    BC tan C tan B +tan C tan B tan C − 1 11QR
    RA = , so that = = = ,
    2 tan A BC 2 tan A 2 10
    since the height from A to BC is 12. So our answer is .462
    ( ) ( ) ( )5
    π 2 π 3 π