Proof with Geometry.
intros.
unfold not;intro.
assert (parallel C D U V)...
assert (parallel A Y U V).
eapply parallel_transitivity;eauto.
unfold inter_ll in *.
DecompAndAll.
assert (Col A U V).
eapply par_col_col_3.
2:apply H8.
Geometry.
intuition.
Qed.

(* Specifications: *)
Lemma elim_length_ratio_inter_ll_1 :
forall A C D U V P Q Y : Point,
inter_ll Y P Q U V ->
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.