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

Formulae Relative to E. 91
10. The cosines of the arcs joining the middle points of the sides of a
spherical triangle are proportional to the cosines of half the sides.
11. Solve a spherical triangle, being given a, b ~ e, and E.
12. If s denote the semiperimeter of a spherical triangle, A, a,, A, A
its area, and the areas of its colunar triangles; prove that
tan2 - = tan a A. cot A. cot b A. cot - ac. (383)
13. Prove sin E = cot R tan a tan b tan 2 c. (384)
14.,, sin s = sin a cos B cos C sin A. (385)
15. If (a + b + c) = 7r, prove cos A + cos B + cos C 1, cos A. (385)
B C
16. In the same case, prove that cos A = tan - tan -. (386)
2 2
sin21A + sin2 B +- sin2 C!- 1
17. Prove coss=.+ (387)
2sin A sin i B sin ( C
8., sin sinEcos(A-E) cos o(B-E)cos2C-/) (388)
2 4 sin ~A sin B sin ~ C
b c
19. If E = r, prove that cosA = - cot - cot -. (389)
= 2z%' 2
o2 cos2B+ cos21 C - cos2.
20. Prove cos (s - a) = 2 cos 2-c 2 (390)
2 sin A cos B cos ( C
21. If I be incentre of a spherical triangle, prove that
A B C
Cos2 - os2 - - cos2 -
2 2 2
cos BIC = B C. (NEUBEaG.) (391)
1B C
2 cos - cos -
2 2
22. If Ia, Ib, Ic be the incentres of the triangles colunar to ABC, prove
that
A4 B C
cos2- - sin2 -sin2
cos BIaC = - - (Ibid.) (392)
2 sin - sin -
2 2
23. The angle BI,, C corresponds in the supplemental triangle to the are
joining the middle points of two sides. (Ibid.)
24. If Ia be the incentre of the colunar triangle A'BC, from Ia let fall
perpendiculars IaD, IE, IF on the sides BC, CA, AB, respectively;
then the angle BIa C = ~ FIE = FIaA. The triangle FIaA gives
cos FI A = cos AF. sin FAIa = cos s. sin. A.
Hence cosBI C = co s. sin ~ A.