(* TODO put in assert_diffs ? *)
(** We don't have ratio so we express that AG=2/3 AA' using midpoints. *)

(* Specifications: *)
Lemma is_gravity_center_third :
forall A B C G G' A',
is_gravity_center G A B C ->
Midpoint G' A G ->
Midpoint A' B C ->
Midpoint G A' G'.
Lemma is_gravity_center_third_reci :
forall A B C G A' A'',
Midpoint A' B C ->
Midpoint A'' A G ->
Midpoint G A' A'' ->
~ Col A B C ->
is_gravity_center G A B C.
Lemma is_gravity_center_perm : forall A B C G,
is_gravity_center G A B C ->
is_gravity_center G A B C /\ is_gravity_center G A C B /\
is_gravity_center G B A C /\ is_gravity_center G B C A /\
is_gravity_center G C A B /\ is_gravity_center G C B A.
Lemma is_gravity_center_perm_1 : forall A B C G,
is_gravity_center G A B C -> is_gravity_center G A C B.
Lemma is_gravity_center_perm_2 : forall A B C G,
is_gravity_center G A B C -> is_gravity_center G B A C.
Lemma is_gravity_center_perm_3 : forall A B C G,
is_gravity_center G A B C -> is_gravity_center G B C A.
Lemma is_gravity_center_perm_4 : forall A B C G,
is_gravity_center G A B C -> is_gravity_center G C A B.
Lemma is_gravity_center_perm_5 : forall A B C G,
is_gravity_center G A B C -> is_gravity_center G C B A.
Lemma is_gravity_center_cases : forall A B C G,
is_gravity_center G A B C \/
is_gravity_center G A C B \/
is_gravity_center G B A C \/
is_gravity_center G B C A \/
is_gravity_center G C A B \/
is_gravity_center G C B A ->
is_gravity_center G A B C.