HMMT 十一月 2014 · THM 赛 · 第 9 题
HMMT November 2014 — THM Round — Problem 9
题目详情
英文原题
- In equilateral triangle ABC with side length 2, let the parabola with focus A and directrix BC intersectsides AB and AC at A and A , respectively. Similarly, let the parabola with focus B and directrix
1 2
CA intersect sides BC and BA at B and B , respectively. Finally, let the parabola with focus C and
1 2
directrix AB intersect sides CA and CB at C and C , respectively.
1 2
Find the perimeter of the triangle formed by lines A A , B B , C C .
1 2 1 2 1 2 k
解析
英文解析
- In equilateral triangle ABC with side length 2, let the parabola with focus A and directrix BC intersectsides AB and AC at A and A , respectively. Similarly, let the parabola with focus B and directrix
1 2
CA intersect sides BC and BA at B and B , respectively. Finally, let the parabola with focus C and
1 2
directrix AB intersect sides CA and CB at C and C , respectively.
1 2
Find the perimeter of the triangle formed by lines A A , B B , C C .
1 2 1 2 1 2
√
Answer: 66 − 36 3 Since everything is equilateral it’s easy to find the side length of the wantedtriangle. By symmetry, it’s just AA + 2 A B = 3 AA − AB . Using the definition of a parabola,
1 1 2 1
√
√
AA = A B so some calculation gives a side length of 2(11 − 6 3), thus the perimeter claimed.3
1 1
1 2 22
This can be seen rigorously by showing that five points in the plane determine at most one quadratic equation ax + by +
cxy + dx + ey + f = 0, up to scaling.
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