If a, b are rational numbers such that a² + b² + c² – ab – bc – ca = 0, prove that a = b = c.

a² + b² + c² – ab – bc – ca = 0

Multiplying both sides by 2 we get

2 (a² + b² + c² – ab – bc – ca ) = 0

⇒ (a² + b² - 2ab) + ( b² + c² - 2bc) + (c² + a² -2ac) = 0

The individual terms inside the brackets can be expressed as a whole square

⇒ (a – b)² + (b – c)² + (c – a)² = 0

Since a, b, c are rational and none of the term is equal to zero so each of the terms inside the bracket must individually be equal to zero

⇒ a – b = 0

⇒ a = b

⇒ b – c = 0

⇒ b = c

⇒ c – a = 0

⇒ c = a

So together we can say that

a = b = c