Proof with Geometry.
intros...
uniformize_dir_seg.
field...
Qed.
Hint Resolve dirseg_3 dirseg_4: Geom.

(* Specifications: *)
Theorem dirseg_simpl_1 :
forall A B, A<>B -> A**B / A**B = 1.
Theorem dirseg_simpl_2 :
forall A B, A<>B -> B**A / A**B = -(1).
Theorem dirsur_1 :
forall A B C D E F : Point,
~ Col D E F -> S A B C / S D E F = S A C B / S D F E.