How to prove this inequality $\frac{a}{b+c} + \frac{b}{c+a} + \frac{c}{a+b} \ge 1.5$? Given $a$, $b$, and $c$ are three positive real numbers. How to prove that sum of $\frac{a}{b+c}$, $\frac{b}{c+a}$ and $\frac{c}{a+b}$ is greater than or equal to $1.5$?

by Cauchy Schwarz we have
$$\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}\geq \frac{a^2}{ab+ac}+\frac{b^2}{bc+ab}+\frac{c^2}{ac+bc}\geq \frac{(a+b+c)^2}{2(ab+bc+ca)}\geq \frac{3}{2}$$ if and only if $$a^2+b^2+c^2\geq ab+bc+ca$$ which is true.
