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

Preliminary Propositions and Definitions. 5
6. The locus of all the points of a sphere which are equidistant
from twofixed points A, B of the sphere, is the great circle, which
is perpendicular at its middle point to the arc of the great circle
AB.
DEM.-Let C be the middle point of the chord AB. At C
erect a plane P, perpendicular to the chord AB. P passes
through the centre of the sphere, and it is the locus of points
equally distant from A and B; therefore the points of the sphere,
where P intersects it, are the only points on it which are equidistant from the points A, B. Hence the proposition is proved.
7. Any two great circles of the sphere bisect each other.
DEir.-Let ABCD, AECF be the great circles; then
(Def. vi.) the plane of each passes through the centre of the
sphere. Hence the common section A C of these planes passes
H G
A 0
D
Fig. 3.
through the centre; but the common section of two planes is
a right line (Euc. XI., III.) Hence C is a diameter of the
sphere; therefore ABC, AEC are semicircles, and the proposition is proved.