Proof with Geometry.
intros.
unfold parallel, on_line_d in *.
basic_simpl.
decompose [and] H.
replace (S P Y Q) with (- S Y P Q)...
Qed.
(* TODO update tactic to take care of C<>D *)

Lemma non_zero_denom_on_line_d_2_length_ratio :
forall (A C D P Q Y : Point) (lambda : F),
on_line_d Y P Q lambda ->
S A P Q <> 0 ->
parallel A Y C D ->
C<>D ->
S4 C P D Q <> 0.