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

Incircles. 75
n' 1
Hence (297)
Hence n ^/ 1 +_ 2(1 + cos a + cos b + cos c)
18. If tangents be drawn at A, B, C to the circumcircle of the triangle
ABC, forming a triangle A'B'C', the arcs AA', BB', CC', are concurrent.
The point of concurrence, K, is called the LEMOINE point of the Triangle.
19. Prove that the normal co-ordinates of K are
sin (A - ), sin (B -E), sin (C - E).
20. The triangular co-ordinates of the orthocentre are tan A, tan B, tan C,
and the normal co-ordinates, sec A, sec B, sec C.
DEF. XXIV.-The isogonal conjugate of 0, the intersection of
the medians, is called the SYMMEDIAN point.
21. Prove that the Symmedian point of a spherical triangle does not coincide with its Lemoine point. Its normal co-ordinates are sin a, sin b, sin c.
22. The normal co-ordinates of the pole of the circumcircle are
cos (A-E), cos (B-E), cos(C-E).
23. If M be any point of the sphere, and the arcs MA, MB, MC meet
the sides BC, CA, AB in A', B', C', if O be the pole of the circumcircle,
sin MA4' sin'MB' sin MC' cos MO
sin A' sin BB' sin CC' cos (STN.) (298)
24. If two equianharmonic pencils have a common ray, the intersection
of three corresponding pairs of rays lie on a great circle. Compare Sequel
to Euclid, Prop. v., p. 131.
SECTION II.-INCIRCLES.
74. To find the radius of the incircle of a spherical triangle
ABC. A
F E
D C
Fig. 27.
Sol.-Bisect the angles A, B by the arcs A 0, B 0. O is the