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_all : forall A B C A' B' C' G, A<>B -> B<>C -> A<>C -> G <> A' -> G <> B' -> G <> C' -> is_circumcenter G A B C -> Midpoint A' B C -> Midpoint B' A C -> Midpoint C' A B -> Perp_bisect G A' B C /\ Perp_bisect G B' A C /\ Perp_bisect G C' A B.