substitution - Showing an inequality using Cauchy-Schwarz

I managed to solve the following inequality using AM-GM:

$$\frac{a}{(a+1)(b+1)}+\frac{b}{(b+1)(c+1)}+\frac{c}{(c+1)(a+1)} \geq \frac{3}{4}$$
provided that $a,b,c >0$ and $abc=1$.

However it was hinted to me that this could also be solved with Cauchy-Schwarz inequality but I have not been able to find a solution using it and I'm really out of ideas.