Require Export C4_Triangles_non_degeneres_egaux.
Section LINE_PROPERTIES.

(* Specifications: *)
Lemma ThreeCases : forall A B C,
Clockwise A B C \/ Clockwise B A C \/ Collinear A B C.
Lemma InOrOutLine : forall A : Point, forall F : Figure, forall D : Line F,
F A \/ ~F A.
Lemma CollinearHalfLine : forall A B C,
Collinear A B C ->
HalfLine A B C \/ HalfLine B A C.
Lemma ExclusiveTwoCases : forall A B C,
A <> C ->
Collinear A B C ->
HalfLine A B C \/
EquiOriented A B C A.
Lemma EquiOrientedSym : forall A B C,
A<>B -> EquiOriented A B C A -> EquiOriented C A A B.
Lemma EquiOrientedRev : forall A B C,
Between A B C -> EquiOriented B C A B.
Lemma BetweenTransC : forall A B C D : Point,
Between A B C ->
Between A C D ->
Between B C D.
Lemma BetweenTransA : forall A B C D : Point,
Between A B C ->
Between A C D ->
Between A B D.
Lemma NotFourBetween : forall A B C M P Q : Point,
Clockwise A M P ->
Between A M B ->
Between A P C ->
Between B Q C ->
Between M P Q ->
False.
Lemma ExclusiveCollinear : forall A B C,
A <> B ->
A <> C ->
Collinear A B C ->
HalfLine A B C \/ Between C A B.
Lemma CollinearTrans : forall A B C D,
A <> B ->
Collinear A B C ->
Collinear A B D ->
Collinear A C D.
Lemma MarkChange : forall A B C D,
A <> B ->
C <> D ->
Collinear A B C ->
Collinear A B D ->
Superimposed (Collinear A B) (Collinear C D).
Lemma LineAB : forall A B : Point, forall F : Figure, forall D : Line F,
A <> B ->
F A ->
F B ->
Superimposed F (Collinear A B).