Show that the lines `(x-a+d)/(alpha-delta)=(y-a)/alpha=(z-a-d)/(alpha+delta)`and `(x-b+c)/(beta-gamma)=(y-b)/beta=(z-b-c)/(beta+gamma)`are coplanar.

Explanation

Given lines are ` (x-a+d)/(alpha-delta)= (y-a)/(alpha)= (z-a-d)/(alpha+delta), (x-b-c)/(beta-gamma)= (y-b)/(beta)=(z-b-c)/(beta+gamma)`
Now
`" "|{:(a-d-b+c,,a-b,,a+d-b-c),(alpha-delta,,alpha,,alpha+delta),(beta-gamma,,beta,,beta+gamma):}|`
`" "=|{:(-d+c,,a-b,,d-c),(-delta,,alpha,,delta),(-gamma,,beta,,gamma):}|=0`
`" " ("Applying " C_(1)to C_(1)-C_(2) and C_(3)to C_(3)-C_(2))`
Hence, the lines are coplaner.
Equation of plane containing these lines is
`" "|{:(x-a+d,,y-a,,z-a-d),(alpha-delta,,alpha,,alpha+delta),(beta-gamma,,beta,,beta+gamma):}|=0`
`" "|{:(x-2y+z,,y-a,,z-a-d),(0,,alpha,,alpha+delta),(0,,beta,,beta+gamma):}|=0`
`" "("Applying "C_(1)to C_(1)+C_(3)-2C_(2))`
or `" "(x-2y+z)(alphagamma-betadelta)=0`
or `" "x-2y +z=0 " "[because (alphagamma ne betadelta)]`