What is the manipulation used to transform $\sum_{cyc} \dfrac{a-bc}{a+bc}$ into $\sum_{cyc}\dfrac{bc}{a+bc}$?

I'm trying to prove the inequality shown in this post, but I'm confused on how OP concluded that, $$\dfrac{a-bc}{a+bc}+\dfrac{b-ac}{b+ac}+\dfrac{c-ab}{c+ab}\le \dfrac32$$

Implies, $$\dfrac{bc}{a+bc}+\dfrac{ca}{b+ca}+\dfrac{ab}{c+ab}\ge \dfrac34$$.

I tried doing it on my own and only yielded the opposite, $\dfrac{a}{a+bc}+\dfrac{b}{b+ca}+\dfrac{c}{c+ab}\ge \dfrac34$. Any kind of hint is appreciated.