HMMT 十一月 2014 · 冲刺赛 · 第 27 题

HMMT November 2014 — Guts Round — Problem 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . HMMT NOVEMBER 2014, 15 NOVEMBER 2014 — GUTS ROUND Organization Team Team ID# − 1 3 2

解析

[ 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 AC in 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 EF cyc cyc ∏ = BC (1 + sin A ) cyc Plugging in, squaring the result, and multiplying by K = 30 gives the answer. ABC Guts Round − 1 3 2