Section CLOCKWISE.
Parameter Clockwise : Point -> Point -> Point -> Prop.
Definition HalfPlane (A B : Point) : Figure := Clockwise A B.
Definition EquiOriented (A B C D: Point) : Prop  :=
SubFigure (HalfPlane A B) (HalfPlane C D).
Definition EquiDirected := fun A B C D : Point =>
EquiOriented A B C D \/ EquiOriented A B D C \/
EquiOriented B A C D \/ EquiOriented B A D C \/
EquiOriented C D A B \/ EquiOriented C D B A \/
EquiOriented D C A B \/ EquiOriented D C B A.
Definition HalfLine (A B : Point) : Figure  :=
fun C : Point => EquiOriented A B A C.
Definition Collinear (A B C : Point) := ~Clockwise A B C /\ ~Clockwise B A C.
Definition Between (A B C : Point) :=
A <> B /\ EquiOriented A B B C.

Axiom FourCases : forall A B C,
Clockwise A B C \/ Clockwise B A C \/ HalfLine A B C \/ HalfLine B A C.