Visual explanation of the following statement: Can somebody fill me in on a visual explanation for the following:
If there exist integers $x, y$ such that $x^2 + y^2 = c$, then there also exist integers $w, z$ such that $w^2 + z^2 = 2c$
I know why it is true (ex. take $w = x-y, z = x+y$), but I would think there is a visual explanation hiding somewhere because of squared terms (we can make squares!!)

If you imagine an integer-sized coordinate grid (as in Oleg567's answer) overlaid with another coordinate grid that has been shrunk by a factor of $\sqrt2$ and rotated by $45^{\circ}$ about the origin, the original unit vectors still have integer coordinates. 
So the point $(x,y)$ also has integer coordinates in the new system, being a linear combination of unit vectors. Its distance to the origin is now $\sqrt2$ times the old distance, because we did shrink the unit vectors.
