(***************************************************************************)
(* Formalization of the Chou, Gao and Zhang's decision procedure.          *)
(* Julien Narboux (Julien@narboux.fr)                                      *)
(* LIX/INRIA FUTURS 2004-2006                                              *)
(* University of Strasbourg 2008-2009                                      *)
(***************************************************************************)
Require Export euclidean_constructions.
Require Export area_elimination_lemmas.
Require Export ratios_elimination_lemmas.

(* Specifications: *)
(* lemme 3_2 chou *)
Lemma l_24_a : forall A B P D,
Col A D B -> perp A B P D ->
A<>B -> B<>D ->
A**D / D**B = Py P A B / Py P B A.
Lemma perp4_perp : forall A B P,
perp A B P A -> Py P A B = 0.
Lemma perp_not_eq_not_perp : forall A B P D, perp A B P D ->
Col A D B ->
A <> B ->  B <> D -> A <> D ->
Py P A B <> 0.
(* lemma 3_2 chou *)
Lemma l_24_b : forall A B P D,
Col A D B -> perp A B P D ->
A<>B ->
A**D / A**B = Py P A B / (2 * A**B * A**B).
(* lemma 3_2 chou *)
Lemma l_24_c : forall A B P D,
Col A D B -> perp A B P D ->
A<>B ->
D**B / A**B = Py P B A / (2 * A**B * A**B).
Lemma l_24_c_on_foot : forall P U V Y,
on_foot Y P U V ->
U**Y / U**V = Py P U V / Py U V U.
Lemma per_area: forall A B C,
per A B C ->
2 * 2 * S A B C * S A B C = A**B * A**B * B**C * B**C.
Lemma per_col_eq : forall A B C,
per A B C -> Col A B C -> A = B \/ B = C.
Lemma perp_col_perp : forall P Q B C,
Q<>B ->
per P Q B -> Col Q B C -> per P Q C.
Lemma l_3_4 : forall A B C P,
Col A B C -> Py P A C <> 0 ->
Py P A B / Py P A C = A**B/ A**C.