Question
If $a^2 + b^2 + c^2 - ab - bc - ca = 0$, than.
✓
Answer
Given: $a^2 + b^2 + c^2 - ab -bc - ca = 0$
$\Rightarrow 2(a^2 + b^2 + c^2 - ab - bc - ca) = 0$
$\Rightarrow (2a^2 + 2b^2 + 2c^2 - 2ab - 2bc - 2ca) = 0$
$\Rightarrow ({a^2 + b^2 -2ab} + {b^2 + c^2 - 2bc} + {c^2 + a^2 - 2ca}) = 0$
$\Rightarrow (a - b)^2 + (b - c)^2 + (c - a)^2 = 0$
Now, since the sum of all squares is zero.
$\Rightarrow a - b = 0 \Rightarrow a = b$
$\Rightarrow b - c = 0 \Rightarrow b = c$
$\Rightarrow c - a = 0 \Rightarrow c = a$
$\Rightarrow a = b = c$
$\Rightarrow 2(a^2 + b^2 + c^2 - ab - bc - ca) = 0$
$\Rightarrow (2a^2 + 2b^2 + 2c^2 - 2ab - 2bc - 2ca) = 0$
$\Rightarrow ({a^2 + b^2 -2ab} + {b^2 + c^2 - 2bc} + {c^2 + a^2 - 2ca}) = 0$
$\Rightarrow (a - b)^2 + (b - c)^2 + (c - a)^2 = 0$
Now, since the sum of all squares is zero.
$\Rightarrow a - b = 0 \Rightarrow a = b$
$\Rightarrow b - c = 0 \Rightarrow b = c$
$\Rightarrow c - a = 0 \Rightarrow c = a$
$\Rightarrow a = b = c$
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