HMMT 十一月 2014 · 冲刺赛 · 第 27 题

英文解析

27. [ 13 ] In triangle ABC , let the parabola with focus A and directrix BC intersect sides AB and AC at
    A and A , respectively. Similarly, let the parabola with focus B and directrix CA intersect sides BC
    1 2
    and BA at B and B , respectively. Finally, let the parabola with focus C and directrix AB intersect
    1 2
    sides CA and CB at C and C , respectively.
    1 2
    If triangle ABC has sides of length 5, 12, and 13, find the area of the triangle determined by lines
    A C , B A and C B .
    1 2 1 2 1 2
    6728
    Answer: By the definition of a parabola, we get AA = A B sin B and similarly for the
    1 1
    3375
    AB AC
    2 1
    other points. So = , giving B C ‖ BC , and similarly for the other sides. So DEF (WLOG,
    2 1
    AB ACin that order) is similar to ABC . It suffices to scale after finding the length of EF , which is
    B C − B F − EC
    2 1 2 1
    The parallel lines also give us B A F ∼ BAC and so forth, so expanding out the ratios from these
    2 1
    similarities in terms of sines eventually gives
    ∏ ∑
    2 sin A + sin A sin B − 1
    cyc cyc eF
    ∏
    BC (1 + sin A )=
    Plugging in, squaring the result, and multiplying by K = 30 gives the answer.cyc
    Guts Round aBC − 1 3 2