HMMT 十一月 2010 · 冲刺赛 · 第 15 题

英文解析

15. [ 8 ] Distinct points A, B, C, D are given such that triangles ABC and ABD are equilateral and bothare of side length 10. Point E lies inside triangle ABC such that EA = 8 and EB = 3, and point Flies inside triangle ABD such that F D = 8 and F B = 3. What is the area of quadrilateral AEF D ?
    √
    91 3
    ∼
    Answer: Since AEB DF B , we have ∠ EBA = ∠ F BD . Thus, ∠ EBF = ∠ EBA + ∠ ABF =
    4=
    °
    ∠ F BD + ∠ ABF = ∠ ABD = 60 . Since EB = BF = 3, this means that EBF is an equilateral triangleof side length 3. Now we have [ AEF D ] = [ AEBD ] − [ EBF ] − [ F BD ] = [ AEB ] + [ ABD ] − [ EBF ] −
    √ √
    3 91 3
    2 2
    [ F BD ] = [ ABD ] − [ EBF ] = (10 − 3 ) = .
    4 4