unfold Same_dir.
induction (eq_dec_points A B).
left.
tauto.
right.
exists B.
split.
apply out_trivial.
auto.
apply eqv_refl.
Qed.

Lemma same_dir_out : forall A B C, Same_dir A B A C -> Out A B C \/ A = B /\ A = C.
intros.