If $y$ is mean proportional between $x$ and $z$ , prove that
$x y z (x + y + z)^{3} = (x y + y z + z x)^{3}$

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Solution

We have,
$y$ is the mean proportional between $x + z$ then,
$\begin{matrix} y^{2} = x z \Rightarrow y = \sqrt{x y} \end{matrix}$

To prove $x y z {(x + y + z)}^{3} = {(x y + y z + z x)}^{3}$

R.H.S
$\begin{matrix} {(x y + y z + z x)}^{3} = {(x y + y z + z x)}^{3} = y^{3} {(x + y + z)}^{3} = y^{2} . y {(x + y + z)}^{3} = z x . y {(x + y + z)}^{3} = x y z {(x + y = z)}^{3} \end{matrix}$

L.H.S
Hence, proved.

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