(** Euclid Book III Prop 6 If two circles touch one another, then they do not have the same center. *)

(** Euclid Book III Prop 10 A circle does not cut a circle at more than two points. *)
Lemma cop_onc2__oreq : forall A B C D P Q,
InterCCAt A B C D P Q -> Coplanar A C P Q ->
forall Z, OnCircle Z A B -> OnCircle Z C D -> Coplanar A C P Z -> Z=P \/ Z=Q.