\(a^2+b^2+c^2=ab+ac+bc\)
=> \(2a^2+2b^2+2c^2=2ab+2ac+2bc\)
=> \(2a^2+2b^2+2c^2-2ab-2ac-2bc=0\)
=> \(\left(a^2-2ab+b^2\right)+\left(a^2-2ac+c^2\right)+\left(b^2-2bc+c^2\right)=0\)
=> \(\left(a-b\right)^2+\left(a-c\right)^2+\left(b-c\right)^2=0\)
Vì \(\left(a-b\right)^2\ge0\) với mọi a, b ; \(\left(a-c\right)^2\ge0\) với mọi a, c ; \(\left(b-c\right)^2\ge0\) với mọi b, c.
Do đó \(\left(a-b\right)^2+\left(a-c\right)^2+\left(b-c\right)^2=0\) khi \(a-b=a-c=b-c=0\), suy ra a = b = c
\(a^2+b^2+c^2=ab+ac+bc\)
\(\Rightarrow2\left(a^2+b^2+c^2\right)=2\left(ab+ac+bc\right)\)
\(\Rightarrow a^2+a^2+b^2+b^2+c^2+c^2-2ab-2ac-2bc=0\)
\(\Rightarrow\left(a^2-2ab+b^2\right)+\left(a^2-2ac+c^2\right)+\left(b^2-2bc+c^2\right)=0\)
\(\Rightarrow\left(a-b\right)^2+\left(a-c\right)^2+\left(b-c\right)^2=0\)
Vì \(\left(a-b\right)^2\ge0\forall a,b\)
\(\left(a-c\right)^2\ge0\forall a,c\)
\(\left(b-c\right)^2\ge0\forall b,c\)
Do đó \(\left(a-b\right)^2+\left(a-c\right)^2+\left(b-c\right)^2\ge0\forall a,b,c\)
Dấu "=" xảy ra \(\Leftrightarrow a=b,a=c,b=c\)
\(\Rightarrow a=b=c\)
