Dihedral Angles of 4-Ball Tetrahedra

A tetrahedron is a 4-ball tetrahedron if there are four externally tangent spheres centered at the vertices of the tetrahedron. It is known that a tetrahedron being a 4-ball tetrahedron is equivalent to (1) three pairs of the sum of opposing edge lengths are the same, and to (2) there is a sphere tangent to each edge of the tetrahedron. We will prove that a tetrahedron is a 4-ball tetrahedron if, and only if three pairs of the sums of opposing dihedral angles are the same.