Antipodal equatorial intersection criterion for a great circle ($r^2=1+|c{|}^2$)

A planar circle other than the unit circle lifts to a great circle exactly when it meets the unit circle at two opposite points. A great circle other than the equator intersects the equator in an antipodal pair. Conversely the plane containing a lifted circle and an antipodal pair contains their midpoint, the sphere's centre. Algebraically, adding the equations $|z-c{|}^2=r^2$ and $|-z-c{|}^2=r^2$ with $|z|=1$ gives the displayed condition.