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

Lemma circumcenter_perp : forall A B C A' G, A<>B -> B<>C -> A<>C -> G <> A' -> is_circumcenter G A B C -> Midpoint A' B C -> Perp_bisect G A' B C.