Proof with Geometry.
intros.
unfold Det3.
replace (S A B Y) with (S A B O + S A O Y + S O B Y)...
replace (S A O Y) with (- S O A Y).
2:symmetry;Geometry.
replace (S O A Y) with (1/ (S O U V) * (S O A V * S O U Y + S O A U * S O Y V)).
2:symmetry;eapply freepoint_elimination_aux...
replace (S O B Y) with (1/ (S O U V) * (S O B V * S O U Y + S O B U * S O Y V)).
2:symmetry;eapply freepoint_elimination_aux...
replace (S A B O) with (S O A B).
2:Geometry.
replace (S O A B) with (1/ (S O U V) * (S O A V * S O U B + S O A U * S O B V)).
2:symmetry;eapply freepoint_elimination_aux...
uniformize_signed_areas.
field...
Qed.

Lemma free_points_ratio_elimination_1 : forall O U V A B C D : Point,
parallel A B C D ->
C<>D ->
S O U V <> 0 ->
S U A B <> 0 ->
A**B / C**D = (S O U A * S O V B + S O U A * S O U V - S O V A * S O U B -
S O U B * S O U V) /
(S O U A * S O V D - S O U A * S O V C + S O V A * S O U C -
S O V A * S O U D + S O U V * S O U C - S O U V * S O U D).