If y is the mean proportional between x and z prove that

xyz(x+y+z)^3=(xy+yz+zx)^3

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Solution

Y is the mean proportional between x and z

so, y/z = x/y

y^2 =xz

now,

LHS =xyz(x + y + z)^3

=y(xz){x + y + z }^3

=y^3(x + y + z)^3
=((y)(x+y+z))^3
=(xy+y^2+yz)^3
=(xy+xz+yz)^3
(Note:y^2=xz)
=(xy + yz + zx)^3 = RHS

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