Proof with assert_all.
intros.
assert_diffs.
Name G the intersection of the medians (A I)
which is a median since H0 and (B J)
which is a median since H1
of the non-flat triangle A B C H.
exists G; repeat split; try assumption.
Name D the symmetric of C wrt G.
assert_all.
show_distinct' A D.
permutation_intro_in_hyps.
assert (Par A D J G) by (apply (triangle_mid_par A D C G J H13 H14 H1)).
show_distinct' B G.
assert (Par G B A D)
by (perm_apply (par_col_par A D G J B)).
show_distinct' B D.
assert (Par B D I G) by (apply (triangle_mid_par B D C G I H104 H14 H0)).
show_distinct' A G.
assert (Par G A D B)
by (perm_apply (par_col_par B D G I A))...
show_distinct' D G.
assert (~ Col G A D)
by (intro; search_contradiction).
assert (Parallelogram G A D B)
by (apply (par_2_plg G A D B); finish).
Name Z the intersection of the diagonals (G D)
and (A B) of the parallelogram H17...
ColR.
Qed.

(* Specifications: *)
Lemma is_gravity_center_col : forall A B C G I,
is_gravity_center G A B C ->
Midpoint I A B ->
Col G I C.
Lemma is_gravity_center_diff_1 :
forall A B C G,
is_gravity_center G A B C ->
G<>A.
Lemma is_gravity_center_diff_2 :
forall A B C G,
is_gravity_center G A B C ->
G<>B.
Lemma is_gravity_center_diff_3 :
forall A B C G,
is_gravity_center G A B C ->
G<>C.