Có : a+b+c=6
\(\Rightarrow\) \(\left(a+b+c^{ }\right)^2=36\)
\(\Rightarrow a^2+b^2+c^2+2ab+2bc+2ac=36\)
\(\Rightarrow12+2\left(ab+bc+ca\right)=36\) ( vì \(a^2+b^2+c^2=12\))
\(\Rightarrow\) \(ab+bc+ca=12\)
\(\Rightarrow ab+bc+ca=a^2+b^2+c^2\) ( =12)
\(\Rightarrow a^2+b^2+c^2-ab-bc-ca=0\)
\(\Rightarrow2a^2+2b^2+2c^2+2ab+2bc+2ca=0\)
\(\Rightarrow\left(a-b\right)^2+\left(b-c\right)^2+\left(c-a\right)^2=0\)
Mà \(\left(a-b\right)^2\ge0\forall a,b;\left(b-c\right)^2\ge0\forall c,b;\left(c-a\right)^2\ge0\forall a,c\)
\(\Rightarrow\)\(\left\{{}\begin{matrix}a-b=0\\b-c=0\\c-a=0\end{matrix}\right.\Rightarrow\left\{{}\begin{matrix}a=b\\b=c\\c=a\end{matrix}\right.\Rightarrow a=b=c\)
Mặt khác : a+b=c=6(gt)
\(\Rightarrow a=b=c=2\left(đpcm\right)\)
\(\)
