Proof with finish.
intros.
assert_bets.
elim (inner_pasch A B C J I)...
intro G;intros.
exists G.
spliter;assert_cols;split...
Qed.

(* Specifications: *)
Lemma intersection_two_medians_exist_unique :
forall A B C I J,
~Col A B C ->
Midpoint I B C -> Midpoint J A C ->
exists! G, Col G A I /\ Col G B J.