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

72 Various Applications.
EXERCISES.-XXI.
1-2. If the medians AA', BB', CC' of the triangle ABC intersect in G,
prove1~. sin GA - sin GA' = 2 cos 2 a. (294)
sin AA' sin BB' sin CC'
2. sin A'G sin B'G sin C'G (295)
3. If through a fixed point P we draw any two transversals PAB, PA'B'
to a fixed angle XSY, the locus of the points of intersection of the arcs
AB', A'B is a great circle called the polar of P, with respect to the angle
XSY.
4. If two spherical triangles ABC, A'B'C' are such that the arcs
AA', BB', CC' are concurrent, the pairs of corresponding sides AB, A'B';
BC, B'C'; CA, C'A' intersect on the same great circle.
This may be proved by transversals (see Sequel to Euclid, p. 131), or by
considering the tetrahedrons (O - ABC)(O - A'B'C') cut by the same plane,
which gives two rectilineal triangles in perspective.
5. If we take on the three sides of a triangle from their middle points
arcs equal to a quadrant, the six points thus obtained are on the same
great circle.
6. If the arcs AP, BP, CP meet the sides BC, CA, AB in A', B', C';
and if A", B", C" be the symmetriques* of A', B', C', with respect to the
middle points of the sides, then AA", BB", CC" meet in the same point P',
called the isotomic conjugate of P, with respect to the triangle.
7. Prove that the three arcs AD, BE, CF, each bisecting the area of
a spherical triangle ABC, are concurrent.-(STEINER.)
From the given conditions the spherical excess of each of the triangles
BAD, CAD is E. Hence
sin AD - |si s in (B-E) Isin E. sin (C- E)
2 s sin BAD. sin ADB ~ sin CAD. sin ADC. sin BAD: sin CAD::sin (B- - E): sin (C - ),
from which and two similar proportions the proposition follows.
* For shortness, we say that the extremities of an arc of a great circle
are symmetriques, with respect to the middle of that arc.