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.