Pasch's Theorem
Contents |
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:
- If $b$ is between $a$ and $c$;
- and $c$ is between $b$ and $d$
then:
- $b$ is between $a$ and $d$.
Proof
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Also see
Source of Name
This entry was named for Moritz Pasch.
This intuitively obvious statement in geometry is bizarrely not provable from Euclid's axioms.
Pasch founded the discipline of Ordered Geometry, which is an axiomatic system which specifically defines the concept of betweenness, and hence can be viewed as a branch of order theory.
