Proof with Geometry.
intros.
unfold on_parallel_d in H.
DecompAndAll.
assert (A ** Y + Y ** R = A ** R)...
assert (Y ** R = - R ** Y)...
rewrite H3 in H.
RewriteVar (A ** Y) H.
RewriteVar lambda H5...
field...
Qed.

(* Specifications: *)
Lemma elim_length_ratio_on_parallel_d_2 :
forall (A C D P Q R Y : Point) (lambda : F),
on_parallel_d Y R P Q lambda ->
~ Col A R Y ->
C <> D ->
parallel A Y C D ->
A ** Y / C ** D = S4 A P R Q / S4 C P D Q.