(***************************************************************************)
(* 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 advanced_parallel_lemmas.
Definition Py A B C := A**B * A**B + B**C * B**C - A**C * A**C.
Definition Py4 A B C D := Py A B D - Py C B D.
Definition per A B C := Py A B C = 0.
Definition perp A B C D := Py4 A C B D = 0.
Definition on_foot (Y P U V : Point) := perp Y P U V /\ Col Y U V /\ U<>V.
(* We introduce three additionnal axioms *)
(* We use the square to get rid of sign issues *)

Lemma on_foot_area_paper : forall A B C F,
perp F A B C -> Col F B C ->
2 * 2 * S A B C * S A B C = A**F * A**F * B**C * B**C.