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

(* Specifications: *)
Lemma tangentcc__neq: forall A B C D,
A<>B ->
TangentCC A B C D ->
A<>C.
Lemma interccat__neq : forall A B C D P Q, InterCCAt A B C D P Q -> A <> C.
(** Prop 17 construction of the tangent to a circle at a given point *)
Lemma tangent_construction : forall O P X, segment_circle -> OutCircle X O P
-> exists Y, Tangent X Y O P.
Lemma interccat__ncol : forall A B C D P Q,
InterCCAt A B C D P Q -> ~ Col A C P.
(** 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.