Definition is_circumcenter G A B C := Cong A G B G /\ Cong B G C G /\ Coplanar G A B C.

Lemma is_circumcenter_cases : forall A B C G, is_circumcenter G A B C \/ is_circumcenter G A C B \/ is_circumcenter G B A C \/ is_circumcenter G B C A \/ is_circumcenter G C A B \/ is_circumcenter G C B A -> is_circumcenter G A B C.