The pictured tetrahedron has four identical rectangular (i.e., right-angled or pythagorean) triangles as faces, with

AB=CD=p,
AC=BD=q,
AD=BC=r,
and p²+q²=r².

What's its volume, as a function of p, q and r?

D, if it exists, is the intersection of four different spheres:

* a sphere with radius r and certer at A
* a sphere with radios q and center at B
* a sphere with radius p and cneter at C
* a sphere with diameter r and center at the midpoint of BC

It's easy to see that if D is the symmetric of A with regard to the midpoint of BC, then D satisfies all four coinditions, but it should be proved that no otherp oint also does.

Posted by e.g. on 2004-04-26 12:25:38