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

76 Various Applications.
incentre required; and the perpendiculars OD, OE, OF on the
sides are the angular radii.
DEM.-It is easy to see that OD, OE, OF are all equal; also
that AF = s - a. Now from the right-angled triangle OAF we
have, equation (110),
tan OAF= tan OF - sin AF;
or, denoting the radius by r,
tan A 2 = tan r sin (s - a). (299)
Hence, substituting for tan - A its value, equation (22), we get
tanr= sin(- ) sin(s - b) sin( - c) n (300)
sin s sin s
Cor. 1.-If in the expression for tan r we substitute for b, c
(7r - b), (7r - c), we get the expression for the in-radius of the
colunar triangle B CA' formed by producing the sides AB, A C.
Hence, denoting it by ra, we get
tan r, = -n (301)
sin (s - a)'
Similarly, tan rb =.n (302)
sin (s - b)'
and tan r, = in ( - (303)
sin (s - c)'
Cor. 2.-From the equations (299), (300) we get the following formulae for solving a spherical triangle when the three
sides are given:
ltan r= sin (s -a)+ Isin(s - b)+ Isin(s - c) - lsin s}.
(304)
ltan - l = tan r - l sin (s - a), &c. (305)
DEF. XXV.-The incircles of the colunar triangles are called
escribed circles.