Are there any tetrahedra, with integer side lengths (A,B,C,D,E,F) AND three right angle-triangle faces (ABD, BCE and CDF)?
In algebraic terms, this amounts to the question: are there three integers; A, B and C; which fulfil the set of pythagorean equations:
A^2 + B^2 = D^2
B^2 + C^2 = E^2
A^2 + C^2 = F^2
Yes, such tetrahedra do exist. The associated rectangular prisms are known as Euler bricks. The smallest Euler brick (by volume) is given by $A = 44$, $B = 117$, and $C = 240$.