Hint Resolve inc112 onc212 onc_sym inc__outc onc__inc onc__outc
inc_outc__onc incs__inc outcs__outc : circle.
Ltac Circle := auto with circle.
Ltac treat_equalities :=
try treat_equalities_aux;
repeat
match goal with
| H:(Cong ?X3 ?X3 ?X1 ?X2) |- _ =>
apply cong_symmetry in H; apply cong_identity in H;
smart_subst X2;clean_reap_hyps
| H:(Cong ?X1 ?X2 ?X3 ?X3) |- _ =>
apply cong_identity in H;smart_subst X2;clean_reap_hyps
| H:(Bet ?X1 ?X2 ?X1) |- _ =>
apply  between_identity in H;smart_subst X2;clean_reap_hyps
| H:(Midpoint ?X ?Y ?Y) |- _ => apply l7_3 in H; smart_subst Y;clean_reap_hyps
| H : Bet ?A ?B ?C, H2 : Bet ?B ?A ?C |- _ =>
let T := fresh in not_exist_hyp (A=B); assert (T : A=B) by (apply (between_equality A B C); finish);
smart_subst A;clean_reap_hyps
| H : Bet ?A ?B ?C, H2 : Bet ?A ?C ?B |- _ =>
let T := fresh in not_exist_hyp (B=C); assert (T : B=C) by (apply (between_equality_2 A B C); finish);
smart_subst B;clean_reap_hyps
| H : Bet ?A ?B ?C, H2 : Bet ?C ?A ?B |- _ =>
let T := fresh in not_exist_hyp (A=B); assert (T : A=B) by (apply (between_equality A B C); finish);
smart_subst A;clean_reap_hyps
| H : Bet ?A ?B ?C, H2 : Bet ?B ?C ?A |- _ =>
let T := fresh in not_exist_hyp (B=C); assert (T : B=C) by (apply (between_equality_2 A B C); finish);
smart_subst A;clean_reap_hyps
| H:(Le ?X1 ?X2 ?X3 ?X3) |- _ =>
apply le_zero in H;smart_subst X2;clean_reap_hyps
| H : Midpoint ?P ?A ?P1, H2 : Midpoint ?P ?A ?P2 |- _ =>
let T := fresh in not_exist_hyp (P1=P2);
assert (T := symmetric_point_uniqueness A P P1 P2 H H2);
smart_subst P1;clean_reap_hyps
| H : Midpoint ?A ?P ?X, H2 : Midpoint ?A ?Q ?X |- _ =>
let T := fresh in not_exist_hyp (P=Q); assert (T := l7_9 P Q A X H H2);
smart_subst P;clean_reap_hyps
| H : Midpoint ?A ?P ?X, H2 : Midpoint ?A ?X ?Q |- _ =>
let T := fresh in not_exist_hyp (P=Q); assert (T := l7_9_bis P Q A X H H2);
smart_subst P;clean_reap_hyps
| H : Midpoint ?M ?A ?A |- _ =>
let T := fresh in not_exist_hyp (M=A); assert (T : l7_3 M A H);
smart_subst M;clean_reap_hyps
| H : Midpoint ?A ?P ?P', H2 : Midpoint ?B ?P ?P' |- _ =>
let T := fresh in not_exist_hyp (A=B); assert (T := l7_17 P P' A B H H2);
smart_subst A;clean_reap_hyps
| H : Midpoint ?A ?P ?P', H2 : Midpoint ?B ?P' ?P |- _ =>
let T := fresh in not_exist_hyp (A=B); assert (T := l7_17_bis P P' A B H H2);
smart_subst A;clean_reap_hyps
| H : Midpoint ?A ?B ?A |- _ =>
let T := fresh in not_exist_hyp (A=B); assert (T := is_midpoint_id_2 A B H);
smart_subst A;clean_reap_hyps
| H : Midpoint ?A ?A ?B |- _ =>
let T := fresh in not_exist_hyp (A=B); assert (T := is_midpoint_id A B H);
smart_subst A;clean_reap_hyps
| H : OnCircle ?A ?A ?B |- _ =>
apply cong_reverse_identity in H;smart_subst B;clean_reap_hyps
| H : OnCircle ?B ?A ?A |- _ =>
apply cong_identity in H;smart_subst B;clean_reap_hyps
| H : InCircle ?B ?A ?A |- _ =>
apply le_zero in H;smart_subst B;clean_reap_hyps
| H : OutCircle ?A ?A ?B |- _ =>
apply le_zero in H;smart_subst B;clean_reap_hyps
end.
Section Circle_2.
Context `{TnEQD:Tarski_neutral_dimensionless_with_decidable_point_equality}.
(** If a point is strictly inside a segment of a disk, it is strictly inside the circle. *)

Lemma bet_inc2__incs : forall O P U V X,
X <> U -> X <> V -> Bet U X V ->
InCircle U O P -> InCircle V O P ->
InCircleS X O P.