If a2+b2+c2-ab-bc-ca=0,prove that a=b=c.

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.