unfold on_line in |- *.
intuition.
Qed.

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).