---
tag: CE8GX6ADZS
kind: concept
name: Betweenness
msc: [51G05, 51N10]
sources:
  - url: https://en.wikipedia.org/wiki/Ordered_geometry
---

## Summary

Betweenness is the relation "the point $y$ lies between the points $x$ and $z$".
In an [[affine space|09XNAPTJQ7]] over an [[ordered field|0P3VCBDEE2]] it means that $y$ lies on the segment from $x$ to $z$.
In axiomatic geometry it is a primitive relation, and the axioms of order, such as [[Pasch's axiom|KJT98G16Y4]], describe it.
Segments, rays, half-planes and the inside of a triangle are all defined from it.

## Explanations

### L1 human

Walk along a straight road from $x$ to $z$.
Every point you pass on the way is between $x$ and $z$.
On the number line, $2$ is between $1$ and $5$, and $7$ is not.
This simple idea of order along a line is enough to define segments and sides of a line.