Proof with Geometry.
intros.
assert (on_line_d Y P Q (P ** Y / P ** Q)).
apply on_line_to_on_line_d...
apply elim_area_on_line_d...
Qed.

Theorem elim_area_inter_ll :
forall A B P Q U V Y : Point,
inter_ll Y P Q U V ->
S A B Y = 1 / S4 P U Q V * (S P U V * S A B Q + S Q V U * S A B P).