(***************************************************************************)
(* Formalization of the Chou, Gao and Zhang's decision procedure.          *)
(* Julien Narboux (Julien@narboux.fr)                                      *)
(* LIX/INRIA FUTURS 2004-2006                                              *)
(* University of Strasbourg 2008                                           *)
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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_foot_spec_a :
forall Y P U V A C D : Point,
on_foot Y P U V ->
Col A U V ->
parallel A Y C Y ->
C <> Y ->
A <> Y ->
A**Y / C**Y = (Py P U V * Py4 P C A V + Py P V U * Py4 P C A U) /
(Py P U V * Py C V C + Py P V U * Py C U C - Py P U V * Py P V U).