(***************************************************************************)
(* Formalization of the Chou, Gao and Zhang's decision procedure.          *)
(* Julien Narboux (Julien@narboux.fr)                                      *)
(* LIX/INRIA FUTURS 2004-2006                                              *)
(* University of Strasbourg 2008                                           *)
(***************************************************************************)
Require Export pythagoras_difference_lemmas.
Require Export ratios_elimination_lemmas.
Proof with Geometry.
intros.
assert (on_line_d Y P Q (P ** Y / P ** Q)).
apply on_line_to_on_line_d...
apply elim_py_on_line_d_right...
Qed.
(* shows that we need A<>B in elim_py_on_line_d_middle Lemma essai : forall A P Q Y : Point, forall r:F, r<> 0 -> on_line_d Y P Q r -> False. *)
(* elimination of area for euclidean constructions *)
(* elimination of ratios for euclidean constructions *)
(* the condition A<>Y is required *)

Theorem elim_ratio_on_perp_d_a_aux :
forall Y P Q A C D : Point, forall r: F,
on_perp_d Y P Q r->
Col A P Y  ->
parallel A Y C D ->
C<>D ->
A<>Y ->
A<>P ->
A**Y / C**D = (S A P Q -r/(2+2) * Py P Q P)/(S4 C P D Q).