Prove that a2-b2-c2-a b-b c-c a is always non-negative for all values of a- b- and c-

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Prove that a2...

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Prove that a² + b² + c² −ab−bc−ca is always non-negative for all values of a, b, and c.

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a^{2} + b^{2} + c^{2} - a b - b c - c a

a^{2} + b^{2} + c^{2} - a b - b c - c a = \frac{1}{2} [{(b - c)}^{2} + {(c - a)}^{2} + {(a - b)}^{2}]

a + b + c

a + b + c = 0

\frac{a^{2}}{b c} + \frac{b^{2}}{c a} + \frac{c^{2}}{a b} = 3

∣ ∣ ∣ ∣ ∣ \begin{matrix} a^{2} + b^{2} + c^{2} & b c + c a + a b & b c + c a + a b b c + c a + a b & a^{2} + b^{2} + c^{2} & b c + c a + a b b c + c a + a b & b c + c a + a b & a^{2} + b^{2} + c^{2} \end{matrix} ∣ ∣ ∣ ∣ ∣

a^{2} + b^{2} + c^{2} \geq a b + b c + c a

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