\(\left(y-z\right)^2+\left(z-x\right)^2+\left(x-y\right)^2=\left(y+z-2x\right)^2+\left(z+x^2-2y\right)^2+\left(y+x-2z\right)^2\\ \Leftrightarrow\left(y^2-2yz+z^2\right)+\left(z^2-2xz+x^2\right)+\left(x^2-2xy+y^2\right)=\left[\left(y+z\right)^2-2\left(y+z\right)\cdot2x+\left(2x\right)^2\right]+\left[\left(x+z\right)^2-2\left(x+z\right)\cdot2y+\left(2y\right)^2\right]+\left[\left(x+y\right)^2-2\left(x+y\right)\cdot2z+\left(2z\right)^2\right]\\ \Leftrightarrow2\left(x^2+y^2+z^2\right)-2\left(xy+yz+xz\right)=y^2+2yz+z^2-4xy-4xz+4x^2+x^2+2xz+z^2-4xy-4yz+4y^2+x^2+2xy+y^2-4xz-4yz+4z^2\\ \Leftrightarrow2\left(x^2+y^2+z^2\right)-2\left(xy+yz+xz\right)=6\left(x^2+y^2+z^2\right)-2\left(xy+yz+xz\right)\\ \Leftrightarrow2\left(x^2+y^2+z^2\right)=6\left(x^2+y^2+z^2\right)\\ \Leftrightarrow4\left(x^2+y^2+z^2\right)=0\\ \Leftrightarrow x^2+y^2+z^2=0\\ \Leftrightarrow\left\{{}\begin{matrix}x=0\\y=0\\z=0\end{matrix}\right.\Leftrightarrow x=y=z\)
