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Betweenness Proposition List

    A purpose of the Hilbert Betweenness Axioms (Hilbert's Order Axioms) is to give meaning to the undefined term between; seeing as between is undefined, we have only the axioms to guarantee that the points in this geometry behave in a way that is consistent with our interpretation of "betweenness". This topic points out that there are exactly two sides to a line in the Euclidean plane (Half-Planes Proposition), and that lines are not circular (Betweenness Property), and how to decompose a line into its parts (Linear Decomposition and Line Separation Propositions). The interior of an angle and the Crossbar Proposition are also detailed. Some of these results were taken for granted by Euclid.

Proposition (Half-Planes) Every line bounds exactly two half-planes and these half-planes have no point in common.

Proposition (Three-Point Property) Given If is a point not on the line determined by and then
    (i) and are on the same side of and
    (ii) and are on opposite sides of

Proposition (Betweenness Property) Given points and if either
    (i) and or
    (ii) and
then and are distinct and collinear. Further,
    (iii) if and then
    (iv) if and then
    (v) if and then and
    (vi) if and then

Proposition (Linear Decomposition) For any two points and ,
    (i)  
    (ii)   and
    (iii) if is a line containing and not then every point of the ray (except ) is on the same side of as
Further, given
    (iv) and and
    (v) and

Proposition (Line Seperation Property) If and is the line through and then for every point lying on lies either on ray or on the opposite ray

Proposition (The Pasch Theorem)  If and are distinct noncollinear points and is any line intersecting in a point between and then also intersects  either or If does not lie on then does not intersect both and

Proposition (Interior Angle Property) Given an angle
    (i) If the point is lying on line then is in the interior of if and only if
Further, if is in the interior of then
    (ii) so is every other point on ray except
    (iii) no point on the opposite ray to is in the interior of and
    (iv) if then is in the interior of

Proposition (Crossbar Theorem) If is between and then intersects segment

Proposition (Cutting Triangle Theorem) Given any triangle
    (i) If a ray emanting from an exterior point of intersects side in a point between and then also intersects side or side
    (ii) If a ray emantes from an interior point of then it intersects one of the sides, and if it does not pass through a vertex, it intersects only one side.
    (iii) There does not exist a line in the interior of

Proposition (Extension Property) Given any two distinct points and , there exist a point lying on such that

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