Require Export C6_Parallele_Prop.
Section SECANT_LINES_PROPERTIES.

(* Specifications: *)
Lemma  StrictlyParallel : forall A B C D,
EquiOriented A B C D ->
~Collinear A B C ->
~Collinear A B D.
Lemma NotClockwiseTwoCases : forall A B C : Point,
~Clockwise A B C ->
Clockwise B A C \/ Collinear A B C.
Lemma FourPointsSecantLine : forall A B C D,
Clockwise A B C ->
~Clockwise A B D ->
~EquiDirected A B C D.
Lemma NotEquidirectedDistinctAB : forall A B C D,
~EquiDirected A B C D->
A<>B.
Lemma NotEquidirectedDistinctCD : forall A B C D,
~EquiDirected A B C D->
C<>D.
Lemma LinesIntersectionPoint : forall A B C D,
~EquiDirected A B C D->
{I : Point | Collinear A B I /\ Collinear C D I}.
Lemma UniqueIntersectionLines : forall A B C D I J : Point,
A <> B ->
C <> D ->
~EquiDirected A B C D ->
Collinear A B I ->
Collinear A B J ->
Collinear C D I ->
Collinear C D J ->
I = J.
Lemma FourPointsIntersectionPoint : forall A B C D,
Clockwise A B C ->
Clockwise B A D ->
{I : Point | Collinear A B I /\ Between C I D}.
Lemma PaschABC  : forall A B C M N,
Clockwise A B C ->
Between A M B  ->
~Collinear A M N ->
~Collinear B M N ->
~Collinear C M N ->
(exists P : Point, Collinear M N P /\ Between A P C) \/
(exists P : Point, Collinear M N P /\ Between B P C).