section_path: Circle of Apollonius/Cross-ratios
section_name: Cross-ratios
paragraph_text: Cross-ratios
  
  A closely related property of circles involves the geometry of the cross-ratio of points in the complex plane. If A, B, and C are as above, then the circle of Apollonius for these three points is the collection of points P for which the absolute value of the cross-ratio is equal to one: | | = 1. {\displaystyle {\bigl |}{\bigr |}=1.}
  
  Stated another way, P is a point on the circle of Apollonius if and only if the cross-ratio is on the unit circle in the complex plane.