HMMT 十一月 2013 · 团队赛 · 第 7 题
HMMT November 2013 — Team Round — Problem 7
题目详情
英文原题
- [ 7 ] In equilateral triangle ABC , a circle ω is drawn such that it is tangent to all three sides of the triangle. A line is drawn from A to point D on segment BC such that AD intersects ω at points Eand F . If EF = 4 and AB = 8, determine | AE − F D | .
Periodicityx + x +1
k − 1 k − 2
解析
英文解析
- [ 7 ] In equilateral triangle ABC , a circle ω is drawn such that it is tangent to all three sides of the triangle. A line is drawn from A to point D on segment BC such that AD intersects ω at points Eand F . If EF = 4 and AB = 8, determine | AE − F D | .
√
4 4 5
√
Answer: OR Without loss of generality, A, E, F, D lie in that order. Let x = AE , y = DF .
√ √55
2 2 2
By power of a point, x ( x + 4) = 4 = ⇒ x = 2 5 − 2, and y ( y + 4) = ( x + 4 + y ) − (4 3) = ⇒ y =
√
√
2 2
48 − ( x +4) 12 − (1+ 5)
4 4 5
√ √
= . It readily follows that x − y = = .
2( x +2) 5
5 5
x + x +1
k − 1 k − 2