HMMT 十一月 2010 · 冲刺赛 · 第 15 题

HMMT November 2010 — Guts Round — Problem 15

[ 8 ] Distinct points A, B, C, D are given such that triangles ABC and ABD are equilateral and both are of side length 10. Point E lies inside triangle ABC such that EA = 8 and EB = 3, and point F lies inside triangle ABD such that F D = 8 and F B = 3. What is the area of quadrilateral AEF D ?

解析

[ 8 ] Distinct points A, B, C, D are given such that triangles ABC and ABD are equilateral and both are of side length 10. Point E lies inside triangle ABC such that EA = 8 and EB = 3, and point F lies 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 triangle of 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