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

Miscellaneous Exercises. 157
12. If x, tz denote the perpendiculars from the middle point of BC on the
internal and external bisectors of the angle A, prove that
2 sin À sin j = n. sin ~ (B + C) sec a. (490)
13. If there be any system of fixed points Ai, 42, A3, &c., and a corresponding system of multiples 11, 12, 13, &c., and P a point satisfying the
condition 2 (I cos AP) = constant, the locus of P is a circle.
DEM.-Let x, y, z denote the normal co-6rdinates of P with respect to a
fixed trirectangular triangle xi, yl, zl, &c., those of Ai, &c. Then (Art. 104)
we have (Ixi). x + (ly) y + (lzI) z = constant. Put (Ixl) =X, (yl) =Y,
(Izl) = Z; then, if O be a point whose normal co-ordinates are
X Y Z
y, R, 1, where 2 =X2+ y2+ 2),
we have 2 ( cos AP) = R cos OP = constant. Hence the locus of P is a
circle.
Cor.-If: (i cos AP = 0, either OP =-, and the locus is a great circle,
or. = 0, and then X, Y, Z must each separately vanish.
14. The sum of the cosines of the arcs, drawn from any point on the
surface of a sphere to all the summits of an inscribed regular polygon,
is equal to zero.
15. If O be the incentre of a spherical triangle ABC, prove that
cos OA sin(b- c) +cos OB sin(c-a) + cos OC sin(a-b)= 0. (491)
16. If the side AB of a spherical triangle be given in position and magnitude, and the side AC in magnitude, prove, if BC meet the great circle, of
which A is the pole in D, that the ratio cos BD: cos CD is constant.
17. The eight circles tangential to any three given circles on the sphere
may be divided into two tetrads, say X, Y, Z, W; X', Y', Z', W', of which
one is the inverse of the other, with respect to the circle, cutting the given
circles orthogonally.
18. Any three circles of either tetrad, and the non-corresponding circle
of the other tetrad, are touched by a fourth circle.-(HART.)
19. Any two circles of the first tetrad, and the two corresponding circles
of the second, have a fourth common tangential circle.-(Ibid.)