Pretty simple. Is there a definition in Euclidean geometry that says that one line segment is shorter than another? In number theory $(<)$ is defined such that $a<b$ if and only if there exists a number $c$ such that $a+c=b$. Is there an analog in Euclid's geometry?
Something like: given an $a,b\in \mathbb Q$ such that $a<b$ we can draw a line segment AB such that |AB|= $a$. Then it is possible to draw a point C on line AB such that |AC|= $b$ (this is analogous to "there exists a number $c$ such that $a+c=b$"). But I can't seem to restrict C from not being between A and B cause that would be similar to $a-c=b$.
My example may be way off. If that is too confusing just refer to the first paragraph.
This is maybe a little rough but you can say that $a<b$ if and only if a circle of radius b centered at an end point of a line segment of length a does not not intersect the line segment of length a, I think.