Cho a2 + b2 + c2 = ab + bc + ca. Chứng minh a = b = c .

Ta có: a² + b² +c² = ab + bc + ca

<=> 2(a² + b² +c²) = 2(ab + bc + ca)

<=> 2a² + 2b² + 2c² = 2ab + 2bc + 2ca

<=> (a² − 2ab + b²)+ (b² − 2bc +c²) + (c² − 2ca + a²) = 0

<=> (a − b)² + (b − c)² + (c − a)² = 0

Mà (a − b)² ≥ 0; (b − c)² ≥ 0; (c − a)² ≥ 0 nên suy ra

a−b2=0b−c2=0c−a2=0⇔a=bb=cc=a⇔a=b=c (đpcm)