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

4 Spherical Geometry.
4. Either pole of a circle (great or small) on the sphere is
equally distant from every point in its circumference.
For (see fig., prop. n.), join PF. We have PF2 = PI2 + IF2;
but IF2 is constant = R2 - d, and PI2 = (R - d)2. IHence PF
is constant.
Cor. 1.- PF2 = 2R (R - d). (2)
Cor. 2.- P'F2= 2R (R + d). (3)
DEF. VII.-A great circle passing through the poles of another
circle (great or small) is called a secondary to that circle.
DEF. VIII.-The spherical radius of a small circle is the arc of
a secondary, intercepted between any point in the circumference
and the nearest pole. Thus the spherical radius of the small
circle D.EF (see fig., prop. in.) is the arc PD.
Cor. 1.-If OA be perpendicular to OP, the point A will
describe a great circle.
Cor. 2.-The spherical radius of a great circle is a quadrant.
This is evident; since P, P' are the poles of the great circle
ABC, and AP, CP are quadrants.
5. Only one great circle can be drawn through two points on the
surface of the sphere, unless they are diametrically opposite.
For only one plane can be drawn through tle centre and the
two points, unless they are collinear.
Cor. 1.-If two points 4, C be each 90~ distant from a third
point P, P is the pole of the great circle, determined by the
points A, C.
If O be the centre, the line OP is perpendicular to the lines
OA, OC, and therefore it is normal to their planes. Hence the
line PP' is the axis of the great circle in which the plane 0A C
cuts the sphere, and P, P' are its poles.
Cor. 2.-If the planes of two great circles be at right angles
to each other, their axes are perpendicular, and each passes
through the poles of the other.