正三角形ABC的边长为12厘米,BD,DE,EF,FG四条线段把它的面积分成5等份,求AF,FD,DC,AG,EB的长.

Question

正三角形ABC的边长为12厘米,BD,DE,EF,FG四条线段把它的面积分成5等份,求AF,FD,
DC,AG,EB的长.

Answer 1

连结CE,S△CDE=S△BDE=S△ABC/5,（共DF底,且等高）,∴CD=AC/5=12/5,S△AFG=S△FGE=S△ABC/5,而S△AFG和S△FGE等高,∴AG=GE,S△DAE=3S△BDE,且△DAE和△BDE等高,∴AE=3BE,3BE+BE=12,BE=3,AE=9,AG=AE/2=9/2,S△DFE=S△AFE/2=S△ABC/5,∵△DEF和△AEF等高,∴DF=AF/2,AF=2DF,2DF+DF+CD=AC,3DF+12/5=12,DF=16/5,AF=32/5,AG=9/2,GE=9/2,BE=3,∴AF=32/5,DF=16/5,CD=12/5,AG=9/2,GE=9/2,BE=3.连结CE,S△CDE=S△BDE=S△ABC/5,（共DF底,且等高）,∴CD=AC/5=12/5,S△AFG=S△FGE=S△ABC/5,而S△AFG和S△FGE共高,∴AG=GE,S△DAE=3S△BDE,且△DAE和△BDE共高,∴AE=3BE,3BE+BE=12,BE=3,AE=9,AG=AE/2=9/2,S△DFE=S△AFE/2=S△ABC/5,∵△DEF和△AEF共高,∴DF=AF/2,AF=2DF,2DF+DF+CD=AC,3DF+12/5=12,DF=16/5,AF=32/5,AG=9/2,GE=9/2,BE=3,∴AF=32/5,DF=16/5,CD=12/5,AG=9/2,GE=9/2,BE=3.