HMMT 十一月 2013 · 团队赛 · 第 5 题

英文解析

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 Band G . Given that AD = F G = 4 and BF = , find the length of CG .1
   √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
   √
   which rearranges to 2(5 x − 4) = 5 x ( x + 4). Squaring and noting x > gives (5 x − 16)(15 x − 4) =4
   0 = ⇒ x = .165
   ′5