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

106 Small Circles on the Sphere.
10. If two points be the inverses of two other points, the four points are
concyclic.
11. If through the centre of similitude S (see fig., ~ 96) another great
circle be drawn, intersecting the circles Y, Z in the points g, y'; v', v, corresponding to the points M, M', N', N, the systems of points M, ', j, v';
M',.N, z', V; M, N', N ',,; M', N, l, v', are each concyclic, and the
planes of the four circles pass through a common point.
12. If a variable circle on the sphere touch two fixed circles, the sine of
its radius has a constant ratio to the sine of the perpendicular drawn from
its spherical centre to the radical axis of the fixed circles.
13. If a variable circle touch two fixed circles, the ratio of the sines of
half the tangents drawn to it from the limiting points is constant.
14. If a variable circle touch two fixed circles of a coaxal system, it cuts
any circle of the system at a constant angle.
15. The inverse of a coaxal system is a coaxal system.
16. The inverse of a system of great circles passing through two common
points is a coaxal system.
17. The inverse of a system of small circles having a common spherical
centre is a coaxal system.
SECTION III.-POLES AND POLARS.
100. LEMMAS. —If the segment AB be harmonically divided in
the points C, D; and E the middle point of AB; then
1~. tan2EB = tan EC. tan FD. (399)
20. cot AB = (cot A C + cot AD). (400)
^E B
Fig. 38.
For, by definition,
sin CA sin DA
sin CB sin.DB;
sin CA - sin CB sin DA - sin DB
sin CA + sin CB sin DA + sin DB'