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.

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.