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

18 Spherical Geometry.
DEF. XVI.-A spherical triangle ABC is said to be diametrical when its
circumcentre is the middle point D of one of its sides AB. This side is called
the DIAMETRICAL SIDE.
7. In a diametrical triangle, the angle opposite the diametrical side is
equal to the sum of 'the two remaining angles, and is greater than a right
angle.
8. Two of the colunar triangles of a diametrical triangle are also diametrical triangles, and the spherical excess of the third colunar triangle is equal
to two right angles.
9. If the opposite sides of a spherical quadrilateral be equal, the diagonals
bisect each other, and the opposite angles are equal.
10. If in a spherical quadrilateral ABCD the angle A = C and B = D;
then the side AB = CD, and BC = AD.
Produce the sides AB, CD to meet in E and F; then triangles EBC,
FAD have the three angles of one respectively equal to the three angles
of the other.
DEF. XVII.-If the diagonals of a spherical quadrilateral bisect each
other, it is called a SPHERIOAL PARALLELOGRAM.
11. If the four sides of a spherical quadrilateral be equal, the diagonals
are perpendicular to each other, and they bisect its angles. Such a figure is
called a SPHERICAL LOZENGE.
12. If the four angles of a spherical quadrilateral be equal, the diagonals
are equal.
13. In two supplemental triangles ABC, A'B'C', the arcs A', BB', CC'
are perpendiculars to the corresponding sides of the two triangles, and the
corresponding altitudes of the two triangles are supplemental.
14. The poles of the small circle inscribed in a spherical triangle are also
the poles of the small circle circumscribed to its supplemental triangle, and
the spherical radii of both circles are complementary.
15. If two small circles on a sphere touch each other, the angle between
their planes is equal to the sum or the difference of their spherical radii.
16. The angle of intersection of a great circle and a small circle is greater
than the inclination of their planes.
17. The length of a degree on a parallel of latitude is equal to the length
of a degree of the equator multiplied by cos lat.
For if r be the radius of the equator, and r' the radius of the parallel,
then, degree on parallel divided by degree on equator = r'/r = cos lat.