Question

If y is the mean proportional between x and z. Prove that xyz(x+y+z)^3=(xy+yz+zx)^3

Answer 1

y2=xz(mean proportional) let a be mean proportional of (x2+y2) and (y2+z2) so,,,, a2=(x2+y2) (y2+x2) a=√(x2+y2) (y2+z2) √x2y2+2y4+y2z2. √2y4+y2(x2+y2)

Answer 2

y is mean proportion of x and z
then,
y² = xz

LHS =xyz( x + y + z)³

=y(xz)( x + y + z)³

=y.y²( x + y + z)³

={ y( x + y + z)}³

=(xy + yz +zx )³ = RHS