(***************************************************************************)
(* 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.

Theorem elim_py_on_line_d_middle :
forall A B P Q Y r,
on_line_d Y P Q r ->
Py A Y B = - Py A P B * r + Py A P B + r * r * Py Q P Q + r * Py A Q B- r * Py Q P Q .