Prove that $\sum \frac {xy}{\sqrt{ (x^2+z^2)(y^2+z^2)}}\leq \frac {3}{2} $ Prove that $$\sum \frac {xy}{\sqrt{ (x^2+z^2)(y^2+z^2)}}\leq \frac {3}{2} $$
I tried to apply Cauchy Schwarz but then I obtain Nesbitt's inequality with the wrong sign.

By AM-GM
$$\sum_{cyc}\frac{xy}{\sqrt{(x^2+z^2)(y^2+z^2)}}\leq\frac{1}{2}\sum_{cyc}\left(\frac{x^2}{x^2+z^2}+\frac{y^2}{y^2+z^2}\right)=$$
$$=\frac{1}{2}\sum_{cyc}\left(\frac{x^2}{x^2+z^2}+\frac{z^2}{z^2+x^2}\right)=\frac{3}{2}.$$
