How find this minimum Help me!
Let $x,y,z\ge0$ such that: $xy+yz+zx=1$.
Find the minimum value of: $A=\dfrac{1}{x^2+y^2}+\dfrac{1}{y^2+z^2}+\dfrac{1}{z^2+x^2}+\dfrac{5}{2}(x+1)(y+1)(z+1)$

I found minimum value of $A$ is $\dfrac{25}{2}$ iff $(x,y,z)=(1,1,0)$ or any permutation. But I can't solve that.

first to prove:
$\dfrac{1}{x^2+y^2}+\dfrac{1}{y^2+z^2}+\dfrac{1}{z^2+x^2} \ge \dfrac{5(1+xyz)}{2}$ ...(1)
$A \ge \dfrac{25}{2} \implies x+y+z+2xyz\ge 2 $...(2)
(2) is easy to prove. But (1) is hard. I have a ugly solution and I think to wait some days to post it if there is no better way. 
