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(* 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. *)

Theorem elim_py_on_foot_left_right :
forall B P U V Y : Point,
on_foot Y P U V ->
Py Y B Y =
(Py P U V * Py P U V * Py V B V + Py P U V * Py P V U * Py V B U +
Py P U V * Py P V U * Py U B V + Py P V U * Py P V U * Py U B U) / (Py U V U *  Py U V U).