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

160 Miscellaneous Exercises.
37. If a chord PQ of a small circle whose spherical centre is C pass through
a fixed point O on the sphere, prove that
tan PCO. tan ~ OCQ is constant.
38. If through a given point O a great circle be drawn, cutting a small
circle in the points A, A', and on it a point X, taken so that cot OX = cot OA
+ cot OA', the locus of X is a great circle,
39. If arcs which intersect in a point O be drawn from the angles of a
triangle, meeting the opposite sides in the points dA', B', C', prove
tan 0'O tan B'O tan C'O
tan A'O + tan O + tan B'O + tan OB tan C'O + tan OC = (495
40. If a great circle, passing through a fixed point O, cut the sides of a
spherical polygon in the points A, B, C, &c.; and if X be a point, such
that cot OX = cot OA + cot OB + cot OC, &c., the locus of X is a great
circle.