Proof with Geometry.
unfold on_inter_parallel_parallel in |- *.
unfold parallel in |- *.
intros.
assert (S4 G C H D = - S4 C G D H)...
rewrite H1 in H0.
intuition.
Qed.

(* Specifications: *)
Theorem elim_area_on_line_d :
forall (A B P Q Y : Point) (lambda : F),
on_line_d Y P Q lambda -> S A B Y = lambda * S A B Q + (1 - lambda) * S A B P.