Question

Chứng minh đẳng thức, bất đẳng thức: \(\left(a+b+c\right)^2+a^2+b^2+c^2=\left(a+b\right)^2+\left(b+c\right)^2+\left(c+a\right)^2\)

Answer 1

\(\left(a+b+c\right)^2+a^2+b^2+c^2=\left(a+b\right)^2+\left(b+c\right)^2+\left(c+a\right)^2\)

VT : (a + b + c)² + a² + b² + c²

= a² + b² + c² + 2ab +2bc + 2ac + a² + b² + c²

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

= (a + b)² + (b + c)² + (a + c)² = VP

Vậy \(\left(a+b+c\right)^2+a^2+b^2+c^2=\left(a+b\right)^2+\left(b+c\right)^2+\left(c+a\right)^2\)(đpcm)