Proof with Geometry.
intros.
unfold on_inter_parallel_parallel in *.
DecompAndAll.
assert ((S A B Y) = (S A B W) + (W**Y/U**V) * (S4 A U B V)).
apply elim_area_on_parallel.
unfold on_parallel.
repeat split...
unfold not;intro;subst U...
assert (P<>Q).
unfold not;intro;subst P...
assert (U<>V).
unfold not;intro;subst U...
cases_equality Y W.
subst.
basic_simpl.
field...
assert (W ** Y / U ** V = (W ** R / P ** Q + R**Y / P**Q) / (U ** V / P ** Q)).
apply (elim_length_ratio_on_parallel_d_1 W U V)...
unfold  on_parallel_d;repeat split...
field...
subst...
rewrite H.
rewrite H8.
field...
Qed.

Lemma elim_length_ratio_on_inter_line_parallel_1 :
forall A C D U V R P Q Y : Point,
on_inter_line_parallel Y R U V P Q ->
S A U V <> 0->
C <> D ->
parallel A Y C D ->
A ** Y / C ** D = S A U V / S4 C U D V.