Proof with Geometry.
intros.
unfold not;intro.
assert (parallel C D P Q)...
assert (parallel A Y P Q).
eapply parallel_transitivity.
apply H1.
Geometry.
Geometry.
unfold inter_ll in *.
DecompAndAll.
assert (P<>Q).
unfold not;intro;subst P.
intuition.
assert (U<>V).
unfold not;intro;subst U.
intuition.
assert (Col A P Q).
eapply par_col_col_3.
2:apply H7.
Geometry.
assert (Col U A Y).
eapply col_trans_1 with (B:=V)...
assert (parallel P Q A U).
eapply col_par_par.
apply H2.
Geometry.
Geometry.
assert (U<>A).
unfold not;intro.
subst U.
assert (parallel P Q A V).
eapply col_par_par.
apply H2.
Geometry.
Geometry.
intuition.
assert (parallel P Q U V).
eapply col_par_par.
apply H14.
Geometry.
Geometry.
intuition.
Qed.

(* Specifications: *)
Lemma elim_length_ratio_inter_ll_2 :
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 ->
A ** Y / C ** D = S A P Q / S4 C P D Q.