# Axiom:Euclid's Axiom

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## Axiom

### First Form

Let $\mathsf{B}$ be the relation of betweenness.

Let $=$ be the relation of equality.

This axiom asserts that:

- $\forall a, b, c, d, t: \exists x, y:$
- $\left({\mathsf{B}adt \land \mathsf{B}bdc \land \neg (a = d)}\right) \implies \left({\mathsf{B}abx \land \mathsf{B}acy \land \mathsf{B}xty}\right)$

where $a, b, c, d, t, x, y$ are points.

## Intuition

Draw an angle $bac$ such that $0^{\circ} \le \angle{bac} \le 180^{\circ}$

Let $t$ be some point in the interior of $\angle{bac}$.

Draw a ray starting at $a$ and passing through $t$.

Let $d$ be a point on ray $at$ such that $d$ is between $a$ and $t$.

Then there is some line $x y$ passing through $t$, where $x$ is a point on one side of angle $bac$ and $y$ is a point on the other.

### Second Form

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## Source of Name

This entry was named for Euclid.

## Also see

- Euclid's Fifth Postulate, an analogue of Euclid's Axiom in the context of Second-Order Logic.

## Sources

- June 1999: Alfred Tarski and Steven Givant:
*Tarski's System of Geometry*(*Bull. Symb. Log.***Vol. 5**,*no. 2*: pp. 175 – 214) : p. $183$ : Axiom $10$

Illustration courtesy of Steven Givant.