Proof with Geometry.
unfold on_inter_line_parallel.
intros.
DecompAndAll.
eapply elim_length_ratio_on_line_2...
unfold on_line;split...
unfold not;intro;subst U...
Qed.

Lemma elim_length_ratio_on_inter_line_parallel_2 :
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 ->
Y <> A ->
Y <> R ->
parallel A Y C D ->
A ** Y / C ** D = S4 A P R Q / S4 C P D Q.