If a² + b² + c² -ab – bc – ca = 0 then Prove that a = b = c

Answer: The given expression is: 

a²  + b² + c² – ab – bc – ca = 0

Now, if we multiply both sides with 2, we get 

2a²  + 2b² + 2c² – 2ab – 2bc – 2ca = 0

So, 

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

This implies, 

(a – b)² + (b – c)² + (a – c)² = 0

Now, the square of two numbers can never be less than 0. So, 

a – b = 0; b – c = 0; a – c = 0

So, we can also write this as

a = b; b = c; a = c, or, 

a = b = c (HENCE PROVED)