Question

${a}^{2} + {b}^{2} + {c}^{2} - ab \: - bc \: - ca \: = 0 \: then \: show \: that \: a = b = c \: \\ \\ \\ \\ solve \: this \: question \: \\ \\ thanks$

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Answer 1

To show:- a=b=c

Consider, a2 + b2 + c2 – ab – bc – ca = 0

Multiply both sides with 2, we get

=>2( a2 + b2 + c2 – ab – bc – ca) = 0

=> 2a2 + 2b2 + 2c2 – 2ab – 2bc – 2ca = 0

=> (a2 – 2ab + b2) + (b2 – 2bc + c2) + (c2 – 2ca + a2) = 0

=> (a –b)2 + (b – c)2 + (c – a)2 = 0

Since the sum of square is zero then each term should be zero

=> (a –b)2 = 0,  (b – c)2 = 0, (c – a)2 = 0

=> (a –b) = 0,  (b – c) = 0, (c – a) = 0

=> a = b,  b = c, c = a

∴ a = b = c