Prove the following result by using suitable identities.
${(x - y)}^{2} + {(y - z)}^{2} + {(z - x)}^{2} = 2 (x^{2} + y^{2} + z^{2} - x y - y z - z x)$

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Solution

$L H S$
$= (x - y)^{2} +) (y - z)^{2} + (z - x)^{2}$
$= x^{2} - 2 x y + y^{2} + y^{2} - 2 y z + z^{2} + z^{2} - 2 z x + x^{2}$
$= 2 x^{2} + 2 y^{2} + 2 z^{2} - 2 x y - 2 y z - 2 z y$
$= 2 (x^{2} + y^{2} + z^{2} - x y - y z - z x) = R H S$ (Proved)

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