(** If two circles intersect at a point which is not on the line joining the center, then they intersect on any half-plane delimited by that line. *)

Lemma circle_circle_os : forall A B C D I P,
OnCircle I A B -> OnCircle I C D -> ~ Col A C I -> ~ Col A C P ->
exists Z, OnCircle Z A B /\ OnCircle Z C D /\ OS A C P Z.