\(x^2+y^2+z^2=xy+yz+zx\)
<=> \(2x^2+2y^2+2z^2-2xy-2yz-2zx=0\)
<=> \(\left(x^2-2xy+y^2\right)+\left(y^2-2yz+z^2\right)+\left(z^2-2zx+x^2\right)=0\)
<=> \(\left(x-y\right)^2+\left(y-z\right)^2+\left(z-x\right)^2=0\)
<=> \(\hept{\begin{cases}x-y=0\\y-z=0\\z-x=0\end{cases}}\) <=> \(\hept{\begin{cases}x=y\\y=z\\z=x\end{cases}}\)<=> \(x=y=z\) (đpcm)
\(x^2+y^2+z^2=xy+xz+yz\)
\(\Rightarrow2x^2+2y^2+2z^2=2xy+2xz+2yz\)
\(\Rightarrow2x^2+2y^2+2z^2-2xy-2xz-2yz=0\)
\(\Rightarrow\left(x^2-2xy+y^2\right)+\left(y^2-2yz+z^2\right)+\left(x^2-2xz+z^2\right)=0\)
\(\Rightarrow\left(x-y\right)^2+\left(y-z\right)^2+\left(x-z\right)^2=0\)(1)
Ta có: \(\hept{\begin{cases}\left(x-y\right)^2\ge0\forall x;y\\\left(y-z\right)^2\ge0\forall y;z\\\left(x-z\right)^2\ge0\forall x;z\end{cases}\Rightarrow\left(x-y\right)^2+\left(y-z\right)^2+\left(x-z\right)^2\ge0\forall x;y;z}\) (2)
Từ (1) và (2) \(\Rightarrow\hept{\begin{cases}x-y=0\\y-z=0\\x-z=0\end{cases}\Rightarrow\hept{\begin{cases}x=y\\y=z\\x=z\end{cases}\Rightarrow}x=y=z}\)
Chúc bạn học tốt.
