Each point/antipode pair determines a great circle of points of distance pi/2 radians away from both, e.g., the north pole/south pole pair corresponds to the equator.
Generically, n great circles divide the sphere into n^2-n+2 regions (for n>0). Each region corresponds to a choice of point or antipode such that all chosen points are within pi/2 radians of every point in the region, hence contained within all hemispheres centered in that region.
This is analogous with what happens for the circle, and it's just an interpretation of what happens in
this more general proof.