设A1A2A3.A7是圆内接正七边形,求证:1/(A1A2)等于1/(A1A3)+1/(A1A4)
问题描述：
设A1A2A3.A7是圆内接正七边形,求证:1/(A1A2)等于1/(A1A3)+1/(A1A4)
答
等同于证AC*AD=AB*(AC+AD)
设圆的半径为1
所以
AB=2sin(π/7)
AC=2sin(2π/7)
AD=2sin(3π/7)
AC*AD
=4sin(2π/7)sin(3π/7)
=(-2)(cos(5π/7)-cos(π/7))
AB*(AC+AD)
=4sin(π/7)*(sin(2π/7)+sin(3π/7))
=(-2)(cos(3π/7)-cos(π/7)+cos(4π/7)-cos(2π/7))
=(-2)(cos(π-(4π/7))-cos(π/7)+cos(4π/7)-cos(π-(5π/7)))
=(-2)(cos(5π/7)-cos(π/7))
=AC*AD
故命题得证