Prove that a²+b²+c²-ab-ca is always negative for all values of ab and c
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Let's say
X = a^2 + b^2 + c^2 -ab -bc -ca
So, 2X = 2(a^2 + b^2 + c^2 -ab -bc -ca)
2X = 2a^2 + 2b^2 + 2c^2 -2ab -2bc -2ca
2X = a^2 -2ab + b^2 + b^2 -2bc + c^2 + c^2 -2ca + a^2
2X = (a - b)^2 + (b - c)^2 + (c-a)^2
Here RHS has 3 terms having non-negative values, their summation always end up with positive number or zero i.e. non-negative.... :)
X = a^2 + b^2 + c^2 -ab -bc -ca
So, 2X = 2(a^2 + b^2 + c^2 -ab -bc -ca)
2X = 2a^2 + 2b^2 + 2c^2 -2ab -2bc -2ca
2X = a^2 -2ab + b^2 + b^2 -2bc + c^2 + c^2 -2ca + a^2
2X = (a - b)^2 + (b - c)^2 + (c-a)^2
Here RHS has 3 terms having non-negative values, their summation always end up with positive number or zero i.e. non-negative.... :)
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