CBSE Class 9 Answered

Consider,

a² + b² + c² – ab – bc – ca = 0

Multiply both sides with 2, we get
2( a² + b² + c² – ab – bc – ca) = 0
⇒ 2a² + 2b² + 2c² – 2ab – 2bc – 2ca = 0

⇒ a² + a² + b² + b² + c² + c²– 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 squares is zero, it means 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