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

Lexell's Theoren. 95
90. Keogh's Theorem.-The sine of half the spherical excess
is equal to twice the Staudtian of the triangle formed by joining
the middle points of the sides.-Nouv. Annales, 1857, p. 142.
DEM.-Let A',',, C' be the middle points of the sides. (See
figure, ~ 79.) Then we have, from the right-angled triangle
DAF,
cos DAF = sin D cos DF, equation (111);
that is, sin E = sin D sin B'A'.
But sin D = sine of the perpendicular from C' on B'A'. Hence,
if n' denote the Staudtian of A'B'C', we have
sin E = 2n'. (364)
91. To find the triangle of maximum area, two sides, b, c, being
given.
/A C
c
Fig. 34.
Sol.-The following is Steiner's geometrical solution:
Suppose the side AC to be fixed in position. Let A'C' be
the antipodes of A and C; through A', C' let a small circle be
described with pole P, such that the angle
PA'C'= - -;
then every triangle having A C as base, and vertex any point on