Generating Euler Bricks

Question

In a problem set assigned to my class, we were asked to show that if $(u,v,w)$ is a Pythagorean triple, then a cuboid with side lengths $u|4v^2-w^2|$, $v|4u^2-w^2|$, and $4uvw$ generated an Euler brick. I succeeded by replacing $w$ with $\sqrt{u^2+v^2}$, and then replacing $u$ and $v$ with $m^2-n^2$ and $2mn$ respectively, the well-known generator for Pythagorean triples. However, I am unsure how this would be derived. It seems like something easy to check but hard to get. I wasn't able to find any sort of derivation on the web, so does anyone have one?

Answer 1

The formula regarding the Euler Bricks is from Nicholas Saunderson. You can dig up the original derivation here.