Proof with Geometry.
intros.
unfold parallel, on_line in *.
basic_simpl.
decompose [and] H.
replace (S P Y Q) with (- S Y P Q)...
Qed.

Lemma invariant_par_on_line_d_1_length_ratio_3 :
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 -> parallel P Y P Q.