Prove $\sum \frac{1}{(x+2y{)}^{2}}\ge \frac{1}{xy+yz+zx}$

Let $x,y,z>0$ Prove that:

$\frac{1}{(x+2y{)}^{2}}+\frac{1}{(y+2z{)}^{2}}+\frac{1}{(z+2x{)}^{2}}\ge \frac{1}{xy+yz+zx}$

I tried to apply Cauchy - Schwarz's inequality but I couldn't prove this inequality!

Let $x,y,z>0$ Prove that:

$\frac{1}{(x+2y{)}^{2}}+\frac{1}{(y+2z{)}^{2}}+\frac{1}{(z+2x{)}^{2}}\ge \frac{1}{xy+yz+zx}$

I tried to apply Cauchy - Schwarz's inequality but I couldn't prove this inequality!