TITLE: Why is "lies between" a primitive notion in Hilbert's Foundations of Geometry?
QUESTION [10 upvotes]: I read this question: Hilbert's Foundations of Geometry Axiom II, 1 : Why is this relevant? and its answer by Eric Wofsey. In this answer, it is stated that "lies between" a primitive notion in Hilbert's Foundations of Geometry. 
Why is it a primitive notion? Can't we define that a point $B$ is between points $A$ and $C$ if $AB < AC$ and $BC < AC$? Or maybe that a point $B$ is between points $A$ and $C$ if $AC=AB+BC$?

REPLY [2 votes]: Why is [the relation "is between"] a primitive notion? Can't we define that a point $B$ is between points $A$ and $C$ if $AB < AC$ and $BC < AC$? Or maybe that a point $B$ is between points $A$ and $C$ if $AC=AB+BC$? 

This assumes that a distance function is given whose values can be ordered.  Without the between-ness axiom the geometry could be coordinatized by a non-orderable field such as complex numbers, and an ordering of distances would break the correspondence between points on a line segment and distances.  And if distances do correspond to points on a line, without an ordering of the line there is no trichotomy and there does not always exist a permutation of any $3$ points on a line for which one is "between" the others.