Proof with Geometry.
unfold on_line_d in |- *...
intros...
decompose [and] H; clear H...
assert (Col P Q A)...
assert (Col P Q Y)...
assert (Col P A Y); eauto with Geom...
assert (A ** P + P ** Y = A ** Y)...
rewrite <- H7...
rewrite H5...
field...
Qed.

Theorem elim_length_ratio_on_line_1 :
forall (A C D P Q Y : Point),
on_line Y P Q ->
S A P Q = 0 ->
C <> D ->
A ** Y / C ** D = (A ** P / P ** Q + P**Y/P**Q) / (C ** D / P ** Q).