If the line `x=y=z` intersect the line `s in Adotx+s in Bdoty+s in Cdotz=2d^2,s in2Adotx+s in2Bdoty+s in2Cdotz=d^2,` then find the value of `sinA/2dotsinB/2dotsinC/2w h e r eA ,B ,C` are the angles of a triangle.

Explanation

Any point on the line `x = y =z` is `(lamda, lamda, lamda)`
This point lies on the planes
`" "sinA*x+sinB*y+sinC*z=2d^(2)`
and `" "sin2A*x+sin2B*y+sin 2C*z=d^(2)`
`therefore" "sinA lamda+sin Blamda+sinC lamda = 2d^(2)`
and `" "sin2A lamda+sin 2B lamda +sin 2C lamda =d^(2)`
Eliminating `d^(2)`, we get
`" "sinA+sinB+sinC=2(sin2A+2sin2B+sin2C)`
or `" "4cos""(A)/(2) cos ""(B)/(2) cos""(C)/(2)=8sin A sin B sin C`
or `" "sin""(A)/(2)sin""(B)/(2)sin""(C)/(2)=(1)/(16)`