ILL 1986 Problem 17

We call a tetrahedron right-faced if each of its faces is a right-angled triangle.

(a) Prove that every orthogonal parallelepiped can be partitioned into six right-faced tetrahedra.

(b) Prove that a tetrahedron with vertices $A_1,A_2,A_3,A_4$ is right-faced if and only if there exist four distinct real numbers $c_1, c_2, c_3$ , and $c_4$ such that the edges $A_jA_k$ have lengths $A_jA_k=\sqrt{|c_j-c_k|}$ for $1\leq j < k \leq 4.$