A treatise on spherical trigonometry, and its application to geodesy and astronomy, with numerous examples. By John Casey.

Parallelopipeds and Tetrahedra. 141
EXERCISES XXXIV.
1. If the four edges of a tetrahedron be tangents to a sphere, the sum of
each pair of opposite edges is constant.
For if ti, t2, t3, t4 be the tangents drawn to the sphere front the vertices
of the tetrahedron, it is evident that the sum = tl + t2 + t3 + t4.
2. If a, a' be two opposite edges of a tetrahedron, and d their shortest
distance, the volume
1 A
= aa'd sin (aa'). (448)
3. The four escribed spheres of an isosceles tetrahedron are equal, and
the radius of each is equal to the diameter of the inscribed sphere.
2t, t2 ta t4
4. Prove that the radius of the sphere in Ex. 1 = 23 t. (449)
5. If V be the volume of a tetrahedron, whose edges of a face are a, b, c,
and opposite edges a', b', c'; and V' the volume of a tetrahedron, whose
edges of a face are a', b', c', and opposite edges a, b, c; then
144 ( v2 - Y'2) = (a2 - a'2) (b2 - b'2) (c2 - c'2). (460)
(WOLSTENHOLME.)
6. If (a, a'), (b, b'), (c, c') be the three pairs of opposite edges of a tetrahedron, and denoting by the same letters the dihedral angles adjacent to
these edges, prove that if the altitudes cointersect1~. a2+a2 = b2+b'2=c2 + c2. (451)
2~. cos a cos a' = cos b cos b' = cos c cos c'. (452)
7. If the lines joining the summits of a tetrahedron to the points of contact of opposite faces with the inscribed sphere cointersect, prove that
cos ~ a cos a' cos b. cos b' = cos c. cos c'. (453)
8. If the spherical triangles which are equivalent to the two trihedrals
D - ABC, G - ABC (fig. Art. 127), be denoted by ABC, A'B'C', respectively, prove
tan a' inA (454)
/ sin b sin c cos a