Proof with Geometry.
intros.
unfold on_line_d in *.
DecompAndAll.
RewriteVar lambda H3...
clear H3.
assert (1 - P ** Y / P ** Q = Y ** Q / P ** Q)...
assert (Y ** P + P ** Q = Y ** Q)...
rewrite <- H1...
assert (P ** Y = - Y ** P)...
rewrite H3; field...
rewrite H1...
replace (S A B Y) with (S Y A B)...
replace (S A B Q) with (S Q A B)...
replace (S A B P) with (S P A B)...
apply l2_9...
Qed.

(* Specifications: *)
Theorem elim_area_on_line :
forall A B P Q Y : Point,
on_line Y P Q ->
S A B Y = P ** Y / P ** Q * S A B Q + (1 - P ** Y / P ** Q) * S A B P.