Proof with Geometry.
intros.
unfold not;intro.
assert (parallel C D P Q)...
clear H3.
unfold on_parallel_d in *.
DecompAndAll.
assert (parallel A Y P Q).
eapply parallel_transitivity.
apply H1.
Geometry.
Geometry.
assert (parallel A Y Y R).
eapply parallel_transitivity.
apply H3.
Geometry.
Geometry.
assert (Col A Y R)...
unfold parallel, S4 in *.
basic_simpl.
Geometry.
assert (Col A R Y)...
Qed.

(* Specifications: *)
Lemma lambda_zero_on_parallel_d : forall Y R P Q lambda,
on_parallel_d Y R P Q lambda ->
(R=Y <-> lambda = 0).