 Real cross-ratio criterion for a generalized circle

Normalize three distinct points of the Riemann sphere by a Möbius transformation $f$ sending them to $\infty ,0,1$. A fourth point lies on their generalized circle exactly when $f$ takes a real value there: the image of that generalized circle is the extended real line. Other standard cross-ratio ordering conventions differ by real fractional-linear transformations and preserve this reality criterion.