Proof with Geometry.
intros.
unfold on_parallel_d in *.
DecompAndAll.
assert (parallel P Q Y A).
eapply col_par_par.
assert (Y<>R)...
apply H.
Geometry.
Geometry.
eapply parallel_transitivity.
apply H2.
Geometry.
Geometry.
Qed.

(* Specifications: *)
Theorem elim_length_ratio_on_parallel_d_1 :
forall (A C D P Q R Y : Point) (lambda : F),
on_parallel_d Y R P Q lambda ->
S A R Y = 0 ->
C <> D ->
A ** Y / C ** D = (A ** R / P ** Q + lambda) / (C ** D / P ** Q).