Proof with Geometry.
intros.
unfold on_parallel_d in *.
DecompAndAll.
cut (parallel P Q R A)...
eapply col_par_par.
apply H1.
Geometry.
Geometry.
Qed.

Lemma invariant_par_on_parallel_d_1_length_ratio_2 :
forall (A C D P Q R Y : Point) (lambda : F),
on_parallel_d Y R P Q lambda ->
S A R Y = 0 ->
R<>Y ->
A<>Y ->
parallel A Y C D ->
parallel C D P Q.