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!!)

It will take me forever to post the diagram so here is a description.
Draw the circle with centre $(0,0)$ and radius $\sqrt c\,$.  Locate the point $(x,y)$ on this circle: by assumption, $x$ and $y$ are integers.  Draw the tangent to this circle starting at $(x,y)$ and having length $\sqrt c\,$.  This will give a point distant $\sqrt{2c}$ from the origin (because we have a right angled isosceles triangle), and the point will have integer coordinates because it is obtained from $(x,y)$ by a displacement of $(y,-x)$ or $(-y,x)$, depending which way we drew the tangent.
Update: see another answer for the picture.  Thanks Oleg!
