Axiom:Inner Transitivity of Betweenness

Axiom

Let $\mathsf{B}$ be the relation of betweenness.

This axiom asserts that:

$\forall a, b, c, d: \mathsf{B}abd \land \mathsf{B}bcd \implies \mathsf{B}abc$

where $a, b, c, d$ are points.

Intuition

Let $abd$ and $bcd$ be line segments.

Suppose they are arranged to form one straight line segment.

Then the points $a,b,c$ are collinear.

Note that this axiom still holds in the degenerate cases where the points are not distinct.

For example, if we are dealing with exactly three points, this axiom could be interpreted as "three points on a line are collinear".

Also see

• Outer Transitivity of Betweenness

Sources

• June 1999:  Alfred Tarski and Steven Givant: Tarski's System of Geometry (The Bulletin of Symbolic Logic Vol. 5, no. 2: 175 – 214) : Page 185 : Axiom $15$