Proof with Geometry.
intros.
unfold inter_ll in *.
DecompAndAll.
assert (P <> Q);eauto with Geom.
assert (S Y A B = P ** Y / P ** Q * S Q A B + Y ** Q / P ** Q * S P A B)...
apply l2_9...
unfold parallel in H3.
assert (P ** Y / P ** Q = S P U V / S4 P U Q V)...
assert (Q ** Y / P ** Q = S Q U V / S4 P U Q V)...
replace (Q**Y) with (- Y**Q) in H5.
2:symmetry;Geometry.
uniformize_signed_areas.
IsoleVar (Y ** Q) H5...
rewrite H1.
rewrite H4.
rewrite H5.
field...
Qed.

(* Specifications: *)
Theorem elim_area_on_parallel_d :
forall (A B P Q R Y : Point) (lambda : F),
on_parallel_d Y R P Q lambda ->
S A B Y = S A B R + lambda * S4 A P B Q.