(* set (X_C := S O U C). set (X_A := S O U A). set (X_B := S O U B). set (X_D := S O U D). set (Y_C := S O V C). set (Y_A := S O V A). set (Y_B := S O V B). set (Y_D := S O V D). *)

(* Specifications: *)
Lemma free_points_ratio_elimination_6 : 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 ->
A**B / C**D = (S O U A * S O V B - S O V A * S O U B) /
(S O U C * S O V D - S O V C * S O U D).
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 : 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 ->
A**B / C**D =
(S O U V * (S O U B - S O U A) - S O U A * S O V B + S O U B * S O V A) /
(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).
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 free_points_ratio_elimination_8 : forall O U V A B C D: Point,
parallel A B C D ->
C<>D ->
~ Col O U V ->
~ Col V A C ->
Col A C D ->
A**B / C**D =
( S O U V * (S O V B - S O V A) + S O V A * S O U B - S O V B * S O U A) /
( S O U V * (S O V D - S O V C) + S O V C * S O U D - S O V D * S O U C).