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_ts :
forall A B C D,
Same_dir A B C D ->
exists M, Bet A M D /\ Bet B M C.