Pasch's Theorem

Theorem

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

Let $\tuple {a, b, c}$ denote that $b$ lies between $a$ and $c$.

Then $\tuple {a, b, c}$ and $\tuple {b, c, d}$ together imply that $\tuple {a, b, d}$.

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

• Outer Transitivity of Betweenness

Source of Name

This entry was named for Moritz Pasch.

Historical Note

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.