Three, (18 points) Solve the system of equations:
$$
\left\{\begin{array}{l}
2(x+y+z)-5 \sqrt{x+y+z+5}=2, \\
\frac{x}{3}=\frac{y}{4}=\frac{z}{5} .
\end{array}\right.
$$
Three, let $\sqrt{x+y+z+5}=t \geqslant 0$. Then $x+y+z=t^{2}-5$.
Substitute into equation (1) to get
$$
2\left(t^{2}-5\right)-5 t=2 \text {, }
$$

which simplifies to $2 t^{2}-5 t-12=0$.
Solving this, we get $t=4$ or $t=-\frac{3}{2}$ (discard).
From $\sqrt{x+y+z+5}=4 \Rightarrow x+y+z=11$.
From equation (2) and the property of proportions, we get
$$
\frac{x}{3}=\frac{y}{4}=\frac{z}{5}=\frac{x+y+z}{12}=\frac{11}{12} \text {. }
$$

Solving this, we get $x=\frac{11}{4}, y=\frac{11}{3}, z=\frac{55}{12}$.