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

24 Connecting Sides and Angles of a Spherical Triangle.
14. Prove
cos2 ~a + cos a' cos a cos a' cosa = cos2 b+ cos2 b'+ 2 cos b cos b' cos
= cos2 ~ C + cos2 c' + 2 cos c cos a c' cos y.
15. Given the base of a spherical triangle and the sum of the cosines of
the sides, find the locus of the vertex.
16. In a spherical quadrilateral the arcs joining the middle points of
opposite sides, and the arc joining the middle points of the diagonals, are
concurrent. (NEUBERG.)
17. If D be any point in the side BC of a spherical triangle, prove that
- cos AD sin BC= cos AB sin DC+ cos AC sin BD. (42)
The theorem of this exercise may be called STEWART'S Theorem. It is
a generalization of a theorem due to that Geometer.-Sequel to Euclid,
Prop. ix., p. 24.
18. If ABC be an arc of a great circle, and AA', BB', CC', arcs perpendicular to any other great circle, prove that
sin BC sin AA' + sin CA sin BB' + sin AB sin CC' = 0. (43)
19. Prove n = /(1 - cos2- cob - cosc + 2 cos a cos b cos c).
(44)
20. If c be the diametral side of a diametral triangle, prove
c~ a b
sin2 = sin2 + sin2. (45)
2 2 2
Case II.-Two Sides and the Angles opposite to them.
29. The sines of the sides of a spherical triangle areproportional
to the sines of their opposite angles.
DEM.-From equations (16), (19) we get, by multiplieation,
2sin i cosi = 2 /sins sin (s - a) sin(s-b) sin (s-c)
2 sin b A cos c A =s
sin b sin e
or sin = si si n c; (46)
a sa 2n
sin a sin a sin b sin