Problem - 3888
Given non-negative real numbers $x$, $y$ and $z$, prove
$$\sqrt{x^2+y^2-xy}+\sqrt{y^2 + z^2 - yz}\ge\sqrt{x^2+z^2+xz}$$
This problem can be solved geometrically by constructing the following graph.
By the law of cosines, we find
\begin{align*}
BD &= \sqrt{x^2+y^2-xy}\\
CD &=\sqrt{y^2+z^2 - yz}\\
BC &=\sqrt{x^2 + z^2 +xz}
\end{align*}
Noting $BD+CD\ge BC$ in $\triangle{BCD}$ leads to the conclusion immediately. The equality holds if and only if $BCD$ are collinear.