Proof with Geometry.
intros.
assert (on_line_d Y P Q (P**Y/P**Q)).
apply on_line_to_on_line_d...
eapply non_zero_denom_on_line_d_2_length_ratio;eauto.
Qed.

Lemma elim_length_ratio_on_line_d_2 :
forall (A C D P Q Y : Point) (lambda : F),
on_line_d Y P Q lambda ->
S A P Q <> 0 ->
C <> D ->
parallel A Y C D ->
A ** Y / C ** D = S A P Q / S4 C P D Q.