If a, b, c are positive numbers then prove that $\frac{a^2+b^2}{a+b}+\frac{b^2+c^2}{b+c}+\frac{c^2+a^2}{c+a}\geq a+b+c$. 
If a, b, c are positive numbers then prove that $\frac{a^2+b^2}{a+b}+\frac{b^2+c^2}{b+c}+\frac{c^2+a^2}{c+a}\geq a+b+c$.

I tried using AM-GM inequality, $$\frac{a+b}{2}\geq \sqrt{ab}$$ $$a^2+b^2+2ab\geq 4ab$$ $$ a^2+b^2\geq 2ab$$
Dividing both sides by $a+b$, $$\frac{a^2+b^2}{a+b}\geq \frac{2ab}{a+b}$$
Similarly, $$\frac{b^2+c^2}{b+c}\geq \frac{2bc}{b+c}$$ $$\frac{c^2+a^2}{c+a}\geq \frac{2ca}{c+a}$$
Adding all, we get, $$\frac{a^2+b^2}{a+b}+\frac{b^2+c^2}{b+c}+\frac{c^2+a^2}{c+a}\geq \frac{2ab}{a+b}+\frac{2bc}{b+c}+\frac{2ca}{c+a}$$
Now, I have no idea what to do. Can we simplify RHS to get $a+b+c$ or is there a different method?

Hint: Use $\dfrac{x^2+y^2}{x+y} \ge \dfrac{x+y}{2}$.
