minimum value of $3$ square root function of variables $x,y$ Finding minimum value of $$\sqrt{x^2+49-7\sqrt{2}x}+\sqrt{x^2+y^2-\sqrt{2}xy}+\sqrt{y^2+25-5\sqrt{2}y}, x,y \in \mathbb{R}$$
Attempt: $$\sqrt{(x+7)^2-14x-7\sqrt{2}x}+\sqrt{(x+y)^2-2xy-\sqrt{2}xy}+\sqrt{(y+5)^2-10y-5\sqrt{2}y}$$
could some help me how to solve it , thanks

We can assume that  $x$ and $y$ are non-negatives.
Consider three triangles $\Delta ABC$, $\Delta CBD$ and $\Delta DBE$, 
where $\measuredangle ABC=\measuredangle CBD=\measuredangle DBC=45^{\circ}$, $AB=5$, $BE=7$, $CB=y$ and $DB=x$.
Thus, $AC+CD+DE\geq AE=\sqrt{25+35\sqrt2+49}=\sqrt{74+35\sqrt2}.$
The equality occurs, when $\{C,D\}\subset AE$, which is possible of course.
