Minimum value of an expression in two variables $$\sqrt{x^2-7\sqrt{2}x+49} + \sqrt{x^2+y^2-\sqrt{2}xy} + \sqrt{50+y^2-10y}$$
$x$ and $y$ are positive real numbers, what is the minimum value?
I have tried finding the minimum value of the first expression and the third expression, with the first expression being $\frac{7}{\sqrt{2}}$ while the second expression being $5$, I do not know how to move on from this (note that the answer is an integer from $0$ to $999$).

Let $OABCD$ be a pentagon such that $OA=5\sqrt2$, $OB=y$, $OC=x$, $OD=7$, $\measuredangle AOB=45^{\circ}$, $\measuredangle BOC=45^{\circ}$ and  $\measuredangle COD=45^{\circ}$.
Thus, $$\sqrt{x^2-7\sqrt{2}x+49} + \sqrt{x^2+y^2-\sqrt{2}xy} + \sqrt{50+y^2-10y}=$$
$$=DC+CB+BA\geq AD=\sqrt{(5\sqrt2)^2+7^2-2\cdot5\sqrt2\cdot7\cos135^{\circ}}=13.$$
The equality occurs, when $\{B,C\}\subset AD,$ which says that $13$ is a minimal value. 
Done!
