induction(eq_dec_points A B).
subst B.
induction H.
apply False_ind.
apply plgs_sym in H.
apply H.
Col.
apply plgf_trivial_neq in H.
spliter.
subst D.
left.
tauto.
unfold Same_dir.
right.
exists C.
split.
apply out_trivial.
intro.
subst D.
induction H.
apply H.
Col.
apply plgf_sym in H.
apply plgf_trivial_neq in H.
tauto.
unfold EqV.
left.
assumption.
Qed.

Lemma same_dir_dec : forall A B C D,
Same_dir A B C D \/ ~ Same_dir A B C D.