if a2+b2+c2-ab-bc-ca=0 then show that a=b=c .

Hello student,
Please find the answer to your question below
a² + b² + c² = ab + bc + ca
Multiplying both sides with "2", we have
2 ( a² + b² + c² ) = 2 ( ab + bc + ca)
2a² + 2b² + 2c² = 2ab + 2bc + 2ca
a² + a² + b² + b² + c² + c² - 2ab - 2bc - 2ca = 0
a² + b² - 2ab + b² + c² - 2bc + c² + a² - 2ca = 0
(a² + b² - 2ab) + (b² + c² - 2bc) + (c² + a² - 2ca) = 0
(a - b)² + (b - c)² + (c - a)² = 0
=>
(a - b)² = (b - c)² = (c - a)² = 0
=>
(a - b)² = 0 ---------- (1)
(b - c)² = 0 ---------- (2)
(c - a)² = 0 ---------- (3)
Simplifying Equ. (1), we have
(a - b)² = 0
Taking Square Root on both sides, we have
a - b = 0
a = b ---------- (4)
Simplifying Equ. (2), we have
(b - c)² = 0
Taking Square Root on both sides, we have
b - c = 0
b = c ---------- (5)
Simplifying Equ. (3), we have
(c - a)² = 0
Taking Square Root on both sides, we have
c - a = 0
c = a ---------- (6)
From Equation No. (4), (5) & (6) , it is proved that
a = b = c