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

142 Polyhedra.
9. If ABCD be a tetrahedron, and if we denote by AB the angle between
the faces ABC, ABD, prove that
~4-~2.C/2 sn2 AtD- A
1 AB2. CD. sin2 (AB. CD) = ABC2 + ABD2 - 2ABC. ABD cos AB,
where ABC denotes the area of the triangle ABCe (455)
Project the triangle BCD into B'C'D' on a plane perpendicular to AB;
we have then A
C'D' = CD sin (AB CD),
and C'D'2 = B'C'2 + B'D'2 - 2B'C'. B'D'. cos C'B'D';
D
C'
BFig. 56
Fig. 56.
then, multiplying by AB2, and remembering that B'C', B'D' are equal to
the altitudes of the triangles ABC, ABD, the proposition is proved.
10. If MD be any point in the edge CD, prove that
A
ABM2. CD2 = ABC2. MD2+ ABD2. CM2 + 2ABC. ABD. CM. MD cosAB.
(456)
Draw M'Pparallel to BD, and we have
B'M'2 = B'P2 + M'P2 + 2B'P. PM' cos AB;
B'P CHM PM ' 7MD
also FB') = C' B'C' = - &c.
11. If M be the middle point of CD,
AB2 + ABD2 +. ABD cos A. (457)
4ABM2= ABC2 + ABD2 + 2ABC..ABD. cos AB. (457)