ELMO SL 2014 A9

Let $a$ , $b$ , $c$ be positive reals. Prove that $$\sqrt{\frac{a^2(bc+a^2)}{b^2+c^2}}+\sqrt{\frac{b^2(ca+b^2)}{c^2+a^2}}+\sqrt{\frac{c^2(ab+c^2)}{a^2+b^2}}\ge a+b+c.$$Proposed by Robin Park