[Panithi Vanasirikul]

$\sqrt{x^2+xy+y^2}+\sqrt{y^2+yz+z^2}+\sqrt{z^2+zx+x^2} $

$\ge \sqrt{\sqrt[3]{(x^2+xy+y^2)(y^2+yz+z^2)(z^2+zx+x^2)} }$ (AM-GM)
$= \sqrt{\sqrt[3]{(xy+y^2+x^2)(y^2+yz+z^2)(x^2+z^2+zx)}}$
$\ge \sqrt{xy+yz+zx}$ (Holder)