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If a2 plus b2 plus c2 - ab - bc - ca equals 0 then prove a equals b equals c?
Sambhavrulz
a2 + b2 + c2 - ab - bc - ca = 0
=> 2a2 + 2b2 + 2c2 - 2ab - 2bc - 2ca = 0
Rearranging,
a2 - 2ab + b2 + b2 - 2bc + c2 + c2 - 2ca + a2 = 0
=> (a2 - 2ab + b2) + (b2 - 2bc + c2) + (c2 - 2ca + a2) = 0
or (a - b)2 + (b - c)2 + (c - a)2 = 0
so a - b = 0, b - c = 0 and c - a = 0 (since each square is >=0)
that is, a = b = c
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