Proof with Geometry.
intros.
assert (S4 C P D Q <>0).
eapply non_zero_denom_inter_ll_2_length_ratio;eauto.
unfold inter_ll in *.
DecompAndAll.
cases_equality A Y.
subst Y.
rewrite H5.
basic_simpl.
field.
Geometry.
apply (elim_length_ratio_on_line_2 A C D)...
unfold on_line.
split...
unfold not;intro;subst Q...
unfold not;intro.
assert (P<>Q).
unfold not;intro;subst P...
assert (Col P A Y).
eapply col_trans_1 with (B:=Q)...
assert (parallel C D A P).
eapply col_par_par.
apply H.
Geometry.
Geometry.
assert (P<>A).
unfold not;intro.
subst P.
clear H6 H10 H11.
assert (~ parallel C D A Q)...
assert (parallel C D A Q).
eapply col_par_par.
apply H.
Geometry.
Geometry.
intuition.
assert (parallel C D P Q).
eapply col_par_par.
apply H12.
Geometry.
Geometry.
assert (~ parallel C D P Q)...
Qed.

Lemma non_zero_denom_on_parallel_d_2_length_ratio :
forall (A C D P Q R Y : Point) (l : F),
on_parallel_d Y R P Q l ->
parallel A Y C D ->
C <> D ->
S A R Y <> 0 ->
S4 C P D Q <> 0.