Prove that : a^3+b^3+c^3-3a b c=1/2(a+b+...

Prove that : `a^3+b^3+c^3-3a b c=1/2(a+b+c){(a-b)^2+(b-c)^2+(c-a)^2}`

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For verifying the given, we need to prove that the left hand side (L.H.S) of the equation is equal to right hand side (R.H.S).
Therefore R.H.S.
=`((a + b + c) ((a−b)² + (b − c)² + (c-a)²))`......(1)
By using the formula `(a - b)²= a² + b² - 2ab`
Then, as per the above formula the equation ( 1) becomes,
`(1/2)(a+b+c) 2(a²- 2ab + b²) + (b² - 2bc + c²) + (c² -2ca + a²)`
=` (1/2)(a + b + c)2(a² + b² + c² — ab — bc — ca)`
=` (a + b + c) (a² + b² +c²-ab-bc-ca)`
=`a³ + ab + ac²-ba² - abc-ca²+ba² + b³ + bc²-ab² -bc²-abc+ca²+ cb²+ c³-abc-bc²-c²a`
After simplification,
=`a³ + b³ + c³-3abc`
L.H.S=R.H.S
Hence, proved.

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