Proof with finish.
intros.
elim (intersection_two_medians_exist A B C I J H H0 H1); intros G HG; spliter.
exists G.
unfold unique.
assert_all.
repeat split...
intros G' HG'; spliter.
apply l6_21 with A I B J...
intro; search_contradiction.
show_distinct' B J...
Qed.
Definition is_gravity_center G A B C :=
~ Col A B C /\
exists I, exists J, Midpoint I B C /\ Midpoint J A C /\ Col G A I /\ Col G B J.

(* Specifications: *)
Lemma is_gravity_center_coplanar : forall A B C G,
is_gravity_center G A B C -> Coplanar G A B C.
Lemma is_gravity_center_exist_unique : forall A B C,
~ Col A B C ->
exists! G, is_gravity_center G A B C.