Ltac cases_parallel A B C D := elim (parallel_dec A B C D);intros.Definition Det3 (x1 x2 x3 y1 y2 y3 z1 z2 z3 : F) : F :=
  x1 * (y2 * z3) - x1 * (y3 * z2) - x2 * (y1 * z3) + x2 * (y3 * z1) +
  x3 * (y1 * z2) - x3 * (y2 * z1).

(* Specifications: *)
Lemma freepoint_elimination_aux : forall O U V B Y:Point, ~ Col O U V -> S O B Y = 1/ (S O U V) * (S O B V * S O U Y + S O B U * S O Y V).
Lemma free_points_area_elimination : forall O U V A B Y : Point, S O U V <> 0 -> S A B Y = Det3 (S O U A) (S O V A) 1 (S O U B) (S O V B) 1 (S O U Y) (S O V Y) 1 / S O U V.
Lemma free_points_ratio_elimination_6_non_zero_denom: forall O U V A B C D: Point, parallel A B C D -> C<>D -> ~ Col O U V -> ~ Col O A C -> Col A C D -> S O U C * S O V D - S O V C * S O U D <> 0.
Lemma free_points_ratio_elimination_7_non_zero_denom : forall O U V A B C D: Point, parallel A B C D -> C<>D -> ~ Col O U V -> ~ Col U A C -> Col A C D -> (S O U V * (S O U D - S O U C) - S O U C * S O V D + S O U D * S O V C) <> 0.
Lemma aux: forall x, x<>0 -> -x<>0.