If a² + b² + c² - ab - bc - ca = 0 then a : b : c is -

a² + b² + c² - ab - bc - ca = 0 .....(i)
Multiple equation (i) by 2 we get
⇒ 2a² + 2b² + 2c² - 2ab - 2bc - 2ca = 0
⇒ (a² + b² - 2ab) + (b² + c² - 2bc) + (c² + a² - 2ca) = 0 [ (a + b)² = a² + b² + 2ab]
⇒ (a - b)² + ((b - c)² + (c - a)² = 0 [if x² + y² + z² = 0 then x = 0, y = 0, z = 0]
∴ a - b = 0
⇒ a = b
b - c = 0
⇒ b = c
c - a = 0
⇒ c = a
∴ a : b : c = 1 : 1 : 1