Require Export GeoCoq.Axioms.continuity_axioms.
Require Export GeoCoq.Meta_theory.Continuity.elementary_continuity_props.
Require Export GeoCoq.Meta_theory.Parallel_postulates.parallel_postulates.
Section Tarski_neutral_to_Euclid_neutral.
Context `{TnEQD:Tarski_neutral_dimensionless_with_decidable_point_equality}.
Definition Tcircle : Type := Tpoint*Tpoint*Tpoint %type.
Definition OnCirc P (C:Tcircle) :=
match C with
(X,A,B) => tarski_axioms.Cong X P A B
end.
Definition CI (J:Tcircle) A C D := J=(A,C,D) /\ C<>D.
Definition InCirc P (J:Tcircle) :=
match J with
(C,A,B) =>
exists X Y, Definitions.BetS X C Y /\ tarski_axioms.Cong C Y A B /\
tarski_axioms.Cong C X A B /\ Definitions.BetS X P Y
end.
Definition OutCirc P (J:Tcircle) :=
match J with
(C,A,B) =>
exists X, Definitions.BetS C X P /\ tarski_axioms.Cong C X A B
end.

Lemma inside : forall A B C J P,
CI J C A B /\ InCirc P J <->
exists X Y, CI J C A B /\
Definitions.BetS X C Y /\
tarski_axioms.Cong C Y A B /\
tarski_axioms.Cong C X A B /\
Definitions.BetS X P Y.