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

Lexell's Theorem. 93
- A C'B' - r = r A - B'- C'. Hence B' + C'- A is constant;
and by the lemma the locus of A is the circumcircle of the
triangle B'C'A.
Or thus: SERRET'S PRooF.-Let, as before, B', C' be the
antipodes of B, C; let E' be the spherical excess of B'C'A,
and R' its circumradius; then we have (~ 75),
tan R' = tan a a ' sin (A - E') = tan a a ' sin E.
Hence since a and E are given in magnitude, R' is given in
magnitude, and the circumcircle of B'C'A is evidently given in
position, and is the locus required.
89. Steiner's Theorem.-The great circles through angular
points of a spherical triangle ABC, and which bisect its area, are
concurrent. Let the circles bisecting the area meet the opposite
sides in the points a, /, y, respectively; also, let A', B', C' be
the antipodes of A, B, C. Now the areas of the triangles ABa,
AB,3 are equal, each being half of ABC. Hence, by Lexell's
theorem, the points A', B', a, / are concyclic. Similarly, each
of the systems of points B', C', /, y; C', A', y, a, is concyclic.
A
B C
Fig. 33.
Let the point common to the planes of these three small circles
be P, then the lines of intersection of these planes two by two
pass through P. Hence, if O be the centre of the sphere, the
planes OB'f3B, OC'7yC, OA'aA have a common line of intersection, namely, the line OP. Hence the proposition is proved.