If $x,y,z\in\mathbb{R}^+$, prove that $\sqrt{x^2-xz+z^2}+\sqrt{y^2-yz+z^2}\ge\sqrt{x^2+xy+y^2}.$ When I was doing Math Training, the coach gave a inequality problem.

If $x,y,z\in\mathbb{R}^+$, prove that $$\sqrt{x^2-xz+z^2}+\sqrt{y^2-yz+z^2}\ge\sqrt{x^2+xy+y^2}.$$

I tried to use the brute force method, and found out it was to hard and it involves lot of terms.
The coach later said that there is a beautiful way to solve this problem.
Can someone help me?

Hint: After one times squaring we get
$$2\sqrt{x^2-xz+z^2}\sqrt{y^2-yz+z^2}\geq xz+yz+xy-2z^2$$
Squaring again and factorizing we obtain
$$(xy-xz-yz)^2\geq 0$$ which is true.
