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).

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.