$\begin{array}{l}x y+x z+y z=0 \\ \frac{x+y}{z}+\frac{x+z}{y}+\frac{y+z}{x}=?\end{array}$

Step 1 ：Given that $xy+xz+yz=0$, we can rewrite the expression as $rac{x+y}{z}+rac{x+z}{y}+rac{y+z}{x}$

Step 2 ：By multiplying each term by $x$, $y$, and $z$ respectively, we get $rac{x^2+xy}{xz}+rac{xy+y^2}{yz}+rac{xz+z^2}{xz}$

Step 3 ：Rearranging the terms, we get $rac{x^2}{xz}+rac{xy}{xz}+rac{xy}{yz}+rac{y^2}{yz}+rac{xz}{xz}+rac{z^2}{xz}$

Step 4 ：Simplifying each term, we get $rac{x}{z}+rac{y}{z}+rac{x}{y}+rac{z}{y}+x+z$

Step 5 ：Since $xy+xz+yz=0$, we can substitute $y$ and $z$ in terms of $x$ to get $rac{x}{-x-x}+rac{-x-x}{x}+rac{x}{-x}+rac{-x-x}{-x}+x+(-x-x)$

Step 6 ：Simplifying the above expression, we get $-1-2-1-2+1-2$

Step 7 ：So, $rac{x+y}{z}+rac{x+z}{y}+rac{y+z}{x} = \boxed{-7}$