HMMT 十一月 2013 · 团队赛 · 第 7 题

HMMT November 2013 — Team Round — Problem 7

[ 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 E and F . If EF = 4 and AB = 8, determine | AE − F D | . Periodicity x + x +1 k − 1 k − 2

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 E and 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 . 5 5 √ √ 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