Proof with Geometry.
intros.
assert (on_line_d Y P Q (P**Y / P**Q)).
apply on_line_to_on_line_d...
eapply invariant_par_on_line_d_1_length_ratio_2;eauto.
Qed.

Lemma invariant_par_on_line_1_length_ratio_3 :
forall (A C D P Q Y : Point),
on_line Y P Q -> S A P Q = 0 ->
parallel A Y C D -> parallel P Y P Q.