Pasch's Theorem

Theorem
Let $a, b, c, d$ be points on a line.

Let $\left({a, b, c}\right)$ denote that $b$ lies between $a$ and $c$.

Then $\left({a, b, c}\right)$ and $\left({b, c, d}\right)$ together imply that $\left({a, b, d}\right)$.

That is: then:
 * If $b$ is between $a$ and $c$;
 * and $c$ is between $b$ and $d$
 * $b$ is between $a$ and $d$.

Also see

 * Outer Transitivity of Betweenness

This intuitively obvious statement in geometry is bizarrely not provable from Euclid's axioms.

Pasch founded the discipline of ordered geometry, which is a branch of geometry centered around the concept of betweenness, and hence can be viewed as a branch of order theory.