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

Incircles. 77
ExERCISES.-XXII.
1. Prove tan r = sin a sin ~ B sin ~ C sec A, (306)
2.,, tan r = N 2 cos 1A cos B cos C. (307)
3.,, cot r= 2N {sin(A- E) +sin (B-E) +sin (C-BE)-sinE}.
(308)
4.,, if a + b + c = r, prove tan r = tan 2 A tan B tan ~ C. (309)
5.,, tan r tana ra. tan rb tan rc = n2. (310)
6.,, tan ra = sin a cos 1 B cos ~ C sec A. (311)
7.,, cotra = 2- {sin E+ sin(B - E) + sin(C- E) -sin (A - E)}.
(312)
8.,, cot r - cot ra = {sin (A - E) - sin E}. (313)
9. Prove that the centre of the incircle is the orthocentre of the triangle
formed by the excentres.
10. Prove cot ra + cot rb + cot rc = (cot ~ A + cot 2 B + cot ~ C) - sin s.
(314)
11. Prove that the common tangents of the escribed circles taken in
pairs are a + b, b + c, c + a, respectively.
12. If Oa, Oh, Oc be the centres of the escribed circles, prove that
cos ra COS rb cos re
cos 00a: cos 00b: cos 00c: cos ra::. (315)
cos(s-a) cos(s-b) cos(s-c)
13. Prove that cot r 4 cot ra + cot rb + cot rc
= { sinE+sin(A-E)+sin(B-E) +sin(C-.E)}. (316)
14. Prove
sin(s-a) sin(s-b) sin(s-c)
sin2. 0: sin2 BO: sin2 CO:: in sic (317)
sin a sin b sin c
15. Prove that the cosines of the angles of the triangle 0,, Ob, Oc are
respectively equal to
cos s. sin 2 A, cos s. sin -B, cos s. sin C.