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

Miscellaneous Exercises. 159
26. Any great circle is cut in involution by the sides and diagonals of a
spherical quadrilateral.
27. If two diagonals of a spherical quadrilateral be quadrants, the third
is a quadrant.
28. Show that the method given in "Sequel," p. 121, for describing a
circle touching three circles, may be extended to the sphere.
29. Inscribe in a spherical triangle or in a small circle a triangle whose
sides shall pass through three given points.
30. Prove that if CC' be the symmedian drawn from the angle C of a
spherical triangle,
tan C' = 2 jcos2c - COs2 i (a + b) cos2 (a b) (492)
cot a sin b + cot b sin a
31. If ABCD be a cyclic quadrilateral, and P any point in the circumcircle, prove that
sinAPB. sin CPD sin A AB. sin ~ CD
sinAPC. sin BPD sin 'AC. sin B) (4
32. If three great circles having two points common intersect the sides of
a spherical triangle in angles ai, a2, a3; 81, $2, J3: 71, 72, 73, respectively,
prove that
COs al, COS a2, COS a3
cos B1i C, s 2, cs 3 = 0. (494)
cs 71, COS 72, COS 73
33. Given the base of a spherical triangle and the two bisectors of the
vertical angle, solve the triangle.
34. If two sides of a spherical triangle be given in position, and a point
in the base fixed, if the base be bisected at the fixed point, prove that the
area is either a maximum or a minimum.
35. If the sines of the perpendiculars let fall from a point on the sides of
a spherical polygon, each multiplied by a given constant, be given, the
locus of the point is a circle.
36. O, S are two points on the surface of the sphere; O is fixed, and S
suffers a small displacement along OS proportional to sin OS; prove that the
displacement estimated in the directions of two great circles at right angles
to each other, passing through S, are proportional to the cosines of the
distances of their poles from 0.