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Asquare plus b square plus c square -ab -bc -ca equals 0 then show that a equals b equals c?
a^2 + b^2 + c^2 - ab - bc - ca = 0=> 2a^2 + 2b^2 + 2c^2 - 2ab - 2bc - 2ca = 0
=> a^2 - 2ab + b^2 + b^2 - 2bc + c^2 + c^2 - 2ca + a^2 = 0
=> (a - b)^2 + (b - c)^2 + (c - a)^2 = 0
Each term on the left hand side is a square and so it is non-negative.
Since their sum is zero, each term must be zero.
Therefore:
a - b = 0 => a = b
b - c = 0 => b = c.
What else can I help you with?
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4
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