= Antipodal equatorial intersection criterion for a great circle
{title2=$r^2=1+|c|^2$}
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.
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