51. By Hölder's inequality, we have
$$\begin{array}{l} \left(\frac{x}{\sqrt{y^{2}+z^{2}}}+\frac{y}{\sqrt{z^{2}+x^{2}}}+\frac{z}{\sqrt{x^{2}+y^{2}}}\right)^{2}\left[x\left(y^{2}+z^{2}\right)+y\left(z^{2}+x^{2}\right)+z\left(x^{2}+y^{2}\right)\right] \geqslant \\ (x+y+z)^{3} \end{array}$$

It suffices to prove that
$$\begin{aligned} (x+y+z)^{3}> & 4\left[x\left(y^{2}+z^{2}\right)+y\left(z^{2}+x^{2}\right)+z\left(x^{2}+y^{2}\right)\right]= \\ & 4(x+y+z)(x y+y z+z x)-12 x y z \end{aligned}$$

By Schur's inequality, $(x+y+z)^{3}-4(x+y+z)(x y+y z+z x)+9 x y z \geqslant 0$, so
$$\begin{array}{l} (x+y+z)^{3} \geqslant 4(x+y+z)(x y+y z+z x)-9 x y z> \\ 4(x+y+z)(x y+y z+z x)-12 x y z \end{array}$$