How do I use Cauchy-Schwarz inequality? I wouldn't have posted if I hadn't searched in every site about Cauchy-Schwarz or bcs or Bunyakovsky inequality. But the only thing I can find is the statement and a proof.
In many answers in this SE I found referrals to CBS. but I cannot understand which vectors I can use to get the result. For example
$$\frac{a^4}{a^2+ab} + \frac{b^4}{b^2 + bc} + \frac{c^4}{c^2 + ac} \ge \frac{(a^2+b^2+c^2)^2}{a^2 + b^2 + c^2 + ab + ac + bc}$$
I have also noted that many times the inequality looks like
$$\frac{a^2}{b} + \frac{c^2}{d} + \frac{e^2}{f} \ge \frac{(a+c+e)^2}{b+d+f}$$
Is something like this valid? If yes, how can I end with this, using BCS (or any other inequality)?

In case $b+d+f>0$, multiplying $(b+d+f)$ to both sides will give $(\frac{a^2}{b}+\frac{c^2}{d}+\frac{e^2}{f})(b+d+f)\geq (a+c+e)^2$, and it is a direct corollary from Cauchy-Schwarz inequality.  
