Question

if a square+b square+c square-ab-bc-ca=0,than prove that a=b=c​

Answer 1

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