Prove this inequality $\frac{1}{xy+z}+\frac{1}{yz+x}+\frac{1}{zx+y}\le\frac{1}{2}$

Question

Let $x,y,z>0$ and such $xy+yz+xz\ge 2(x+y+z)$,show that $$\dfrac{1}{xy+z}+\dfrac{1}{yz+x}+\dfrac{1}{zx+y}\le\dfrac{1}{2}$$

Answer 1

It's wrong! Try $z\rightarrow0^+$ and $x=y\rightarrow4$.