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Right-faced tetrahedron

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August 29, 2010
geometry3D geometrytetrahedrongeometry unsolved

Problem Statement

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 A1,A2,A3,A4A_1,A_2,A_3,A_4 is right-faced if and only if there exist four distinct real numbers c1,c2,c3c_1, c_2, c_3, and c4c_4 such that the edges AjAkA_jA_k have lengths AjAk=∣cj−ck∣A_jA_k=\sqrt{|c_j-c_k|} for 1≤j<k≤4.1\leq j < k \leq 4.
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