Require Export GeoCoq.Highschool.circumcenter.
Section Concyclic.
Context `{TE:Tarski_euclidean}.
Definition Concyclic A B C D := Coplanar A B C D /\ exists O, Cong O A O B /\ Cong O A O C /\ Cong O A O D.

(* Specifications: *)
Lemma concyclic_aux : forall A B C D, Concyclic A B C D ->
exists O, Cong O A O B /\ Cong O A O C /\ Cong O A O D /\ Coplanar A B C O.
Lemma concyclic_trans : forall A B C D E,
~ Col A B C ->
Concyclic A B C D  -> Concyclic A B C E -> Concyclic A B D E.
Lemma concyclic_perm_1: forall A B C D,
Concyclic A B C D -> Concyclic A B D C.
Lemma concyclic_perm_2 : forall A B C D,
Concyclic A B C D -> Concyclic A C B D.
Lemma concyclic_perm_3 : forall A B C D,
Concyclic A B C D -> Concyclic A C D B.
Lemma concyclic_perm_4 : forall A B C D,
Concyclic A B C D -> Concyclic A D B C.
Lemma concyclic_perm_5 : forall A B C D,
Concyclic A B C D -> Concyclic A D C B.
Lemma concyclic_perm_6 : forall A B C D,
Concyclic A B C D -> Concyclic B A C D.
Lemma concyclic_perm_7 : forall A B C D,
Concyclic A B C D -> Concyclic B A D C.
Lemma concyclic_perm_8 : forall A B C D,
Concyclic A B C D -> Concyclic B C A D.
Lemma concyclic_perm_9 : forall A B C D,
Concyclic A B C D -> Concyclic B C D A.
Lemma concyclic_perm_10 : forall A B C D,
Concyclic A B C D -> Concyclic B D A C.
Lemma concyclic_perm_11 : forall A B C D,
Concyclic A B C D -> Concyclic B D C A.
Lemma concyclic_perm_12 : forall A B C D,
Concyclic A B C D -> Concyclic C A B D.
Lemma concyclic_perm_13 : forall A B C D,
Concyclic A B C D -> Concyclic C A D B.
Lemma concyclic_perm_14 : forall A B C D,
Concyclic A B C D -> Concyclic C B A D.
Lemma concyclic_perm_15 : forall A B C D,
Concyclic A B C D -> Concyclic C B D A.
Lemma concyclic_perm_16 : forall A B C D,
Concyclic A B C D -> Concyclic C D A B.
Lemma concyclic_perm_17 : forall A B C D,
Concyclic A B C D -> Concyclic C D B A.
Lemma concyclic_perm_18 : forall A B C D,
Concyclic A B C D -> Concyclic D A B C.
Lemma concyclic_perm_19 : forall A B C D,
Concyclic A B C D -> Concyclic D A C B.
Lemma concyclic_perm_20 : forall A B C D,
Concyclic A B C D -> Concyclic D B A C.
Lemma concyclic_perm_21 : forall A B C D,
Concyclic A B C D -> Concyclic D B C A.
Lemma concyclic_perm_22 : forall A B C D,
Concyclic A B C D -> Concyclic D C A B.
Lemma concyclic_perm_23 : forall A B C D,
Concyclic A B C D -> Concyclic D C B A.
Lemma concyclic_1123 : forall A B C,
~ Col A B C ->
Concyclic A A B C.