unfold on_line in |- *.
intuition.
Qed.

(* Specifications: *)
Theorem non_zero_denom_on_line_1_length_ratio_seg :
forall (Y P Q : Point), on_line Y P Q -> P**Q <> 0.
Theorem elim_length_ratio_on_line_d_1 :
forall (A C D P Q Y : Point) (lambda : F),
on_line_d Y P Q lambda ->
S A P Q = 0 ->
C <> D ->
A ** Y / C ** D = (A ** P / P ** Q + lambda) / (C ** D / P ** Q).