Question

prove a=b=c if
${a}^{2} + {b}^{2} + {c}^{2} = ab + bc + ac$

Answer 1

a² + b² + c² = ab + bc + ac

a² + b² + c² -ab - bc -ca = 0

Multiply by 2 .

2 ( a² + b² + c² -ab - bc -ca) = 0

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

Each term is 0 .

(a-b)² = 0
a = b.

(b-c)²=0
b = c

(c-a)² = 0
c = a.

a = b = c.

Well, I am also not sure of this solution. But here, we can say if a = b = c then a² + b² + c² = ab + bc + ac