Proof with Geometry.
intros.
unfold on_parallel_d in *.
DecompAndAll.
split;intro.
subst Y.
basic_simpl.
IsoleVar (lambda) H3.
rewrite H3.
field...
Geometry.
subst lambda.
basic_simpl...
Qed.

Lemma invariant_par_on_parallel_d_1_length_ratio :
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 ->
parallel A Y C D ->
parallel A R P Q.