Inequality with condition $x+y+z=xy+yz+zx$ I'm trying to prove the following inequality:
For $x,y,z\in\mathbb{R}$ with $x+y+z=xy+yz+zx$, prove that
$$
\frac{x}{x^2+1}+\frac{y}{y^2+1}+\frac{z}{z^2+1}\ge-\frac{1}{2}
$$
My approach:
After slight manipulation the inequality is equivalent to:
$$
\sum_{cyc}\frac{(x+1)^2}{x^2+1}\ge 2
$$
Now, applying CS is legitimate and it reduces the inequality to proving:
$$
s^2-10s-3\le0
$$
with $s=x+y+z=xy+yz+zx$, but I'm not quite sure if this is still true. Could anybody give me a hint in the right direction? Any help is highly appreciated.

It's just $$(x-yz)^2+(y-zx)^2+(z-xy)^2+(x+y+z+1)^2+(x+y+z+xyz)^2\geq0.$$
Done!
