Given : a2+b2+c2−ab−bc−ca=0 Multiplying both sides by 2, we get : ⇒ 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 ∵ Sum of three positive sum is 0, then each term is equal to '0' ⇒ (a−b)=(b−c)=(c−a)=0 ⇒ a = b = c ∴ a:b:c=1:1:1