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# If a,b and C are real numbers and a2+b2+c2-ab-bc-ca =0 then show that a=b=c?

Asked by Vikas 20th March 2017, 5:58 PM

Consider,

a2 + b2 + c2 – ab – bc – ca = 0

Multiply both sides with 2, we get
2( a2 + b2 + c2 – ab – bc – ca) = 0
⇒ 2a+ 2b2 + 2c2 – 2ab – 2bc – 2ca = 0

⇒ a2 + a+ b2 + b+ c2 + c2– 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 squares is zero, it means 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

Answered by Expert 20th March 2017, 6:13 PM
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