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

70 Various Applications.
Reciprocally.-If the points A', B', C' satisfy either of these
relations, they lie on a great circle.
Cor. 1.-The great circle which bisects two sides of a triangle meets the third side at the distance of 90~ from its
middle point.
Cor. 2.-The feet of two internal bisectors, and the foot of
the third external bisector, of the angles of a triangle lie on
the same great circle.
71. If the arcs which join the vertices of a triangle ABC to
the same point O of the sphere meet the opposite sides in the
points A', B', C', then,
1~. (ab, CC') (bc, AA') (ca, BB') = - 1. (291)
2~. (AB, C') (B C, A') ( CA, B') = - 1. (292)
1~. This follows from applying the theorem, ~ 29, to the
three triangles A OB, BOC, COA, and considering that if the
point O be inside the triangle ABC, the three ratios of section
(ab, CC), &c., are negative; and if 0 be outside, two are positive and one negative.
2~ follows from 1~ by equation (55).
Reciprocally, if the points A', B', C' satisfy either of the equalities (291), (292), the arcs AA', BB', CC' are concurrent.
Cor. 1.-The three medians AA', BB', CC', are concurrent.
Cor. 2.-The three altitudes, ha, hb, h, of a spherical triangle
are concurrent.
Cor. 3.-The homologous sides of two supplemental triangles
intersect in points situated on the same great circle, having as
pole the common orthocentre of the two triangles.
Cor. 4.-The arcs which join the vertices of a spherical triangle to the points of contact of opposite sides with inscribed
circle, or with any of the escribed, meet in the same point.
(The GERiGONNE point of the Triangle.)