HMMT 十一月 2013 · 团队赛 · 第 5 题

HMMT November 2013 — Team Round — Problem 5

[ 4 ] In triangle ABC , ∠ BAC = 60 . Let ω be a circle tangent to segment AB at point D and segment AC at point E . Suppose ω intersects segment BC at points F and G such that F lies in between B 1 and G . Given that AD = F G = 4 and BF = , find the length of CG . 2 ′

解析

[ 4 ] In triangle ABC , ∠ BAC = 60 . Let ω be a circle tangent to segment AB at point D and segment AC at point E . Suppose ω intersects segment BC at points F and G such that F lies in between B 1 and G . Given that AD = F G = 4 and BF = , find the length of CG . 2 √ 16 3 Answer: Let x = CG . First, by power of a point, BD = BF ( BF + F G ) = , and 5 2 √ CE = x ( x + 4). By the law of cosines, we have ( ) 2 √ √ 9 11 11 2 2 ( x + ) = + (4 + x ( x + 4)) − (4 + x ( x + 4)) , 2 2 2 √ 4 which rearranges to 2(5 x − 4) = 5 x ( x + 4). Squaring and noting x > gives (5 x − 16)(15 x − 4) = 5 16 0 = ⇒ x = . 5 ′