I was reading about normalizing homogenous inequalities, by introducing constraints and conditions on the variables. For example, you can define the sum or product of the variables to be a constant.
Now I wonder exactly what types of conditions can be introduced WLOG? For instance, this is a problem I was trying to prove: $$ (a, b,c > 0) \implies \frac{a}{b+c} + \frac{b}{c+a} + \frac{c}{a+b} + \sqrt{\frac{ab+bc+ca}{a^2+b^2+c^2}}\ge\frac{5}{2} $$
Since this inequality is homogenous with degree 0, why can't I assume something like $$\frac{ab+bc+ca}{a^2+b^2+c^2} = \frac{25}{4}$$ But I can assume pretty much anything else like $a+b+c = 1$ or $ab + bc +ca = 5$ or $abc = 10$? I'm aware that the stated assumption is fundamentally wrong since it maxes out at 1, but if it was possible, would it be a valid condition in normalization? I often don't hear any other condition excluding sum, product and fixating a single variable.