# Definition:Between (Geometry)

## Definition

**Betweenness** is one of the undefined terms in Tarski's Geometry.

Intuitively, a point $b$ is between two others $a$ and $c$ if it lies on the line segment $a c$.

However as *line segment* has not yet been defined, we are not allowed to call upon it at this stage.

We offer an ostensive definition:

In the picture, point $b$ is **between** the two points $a, c$, and we write:

- $\mathsf{B}abc$

However, point $d$ is *not* **between** the two points $a$ and $c$, and we write:

- $\neg \left({\mathsf B a d c}\right)$

### In Euclidean 2-Space

Define the following coordinates in the $xy$-plane:

\(\ds a\) | \(=\) | \(\ds \left({x_1, x_2}\right)\) | ||||||||||||

\(\ds b\) | \(=\) | \(\ds \left({y_1, y_2}\right)\) | ||||||||||||

\(\ds c\) | \(=\) | \(\ds \left({z_1, z_2}\right)\) |

where $a, b, c \in \R^2$.

Let:

\(\ds \Delta x_1\) | \(=\) | \(\ds x_3 - x_2\) | ||||||||||||

\(\ds \Delta x_2\) | \(=\) | \(\ds x_2 - x_1\) | ||||||||||||

\(\ds \Delta y_1\) | \(=\) | \(\ds y_2 - y_1\) | ||||||||||||

\(\ds \Delta y_2\) | \(=\) | \(\ds y_3 - y_2\) |

Then:

- $\mathsf{B}abc \dashv \vdash \left({\Delta x_1 \Delta y_1 = \Delta x_2 \Delta y_2}\right) \land$

- $\left({0 \le \Delta x_1 \Delta y_1 \land 0 \le \Delta x_2 \Delta y_2}\right)$

As a justification of this definition, consider the case where $\Delta x_1, \Delta x_2 \ne 0$.

\(\ds \Delta x_1 \Delta y_1\) | \(=\) | \(\ds \Delta x_2 \Delta y_2\) | ||||||||||||

\(\ds \implies \ \ \) | \(\ds \frac {\Delta y_1}{\Delta x_2}\) | \(=\) | \(\ds \frac {\Delta y_2}{\Delta x_1}\) |

Hence, the right triangles with hypotenuses $ab$ and $bc$ are similar.

Furthermore, the hypotenuses are parallel, because they have the same slope.

They are similarly oriented because $\Delta x$ is by construction parallel to the $x$-axis, $\Delta y$ to the $y$-axis.

They are touching because there are only three points under consideration.

Lastly, the inequalities assure that vertex $b$ lies between the two triangles, because otherwise the inequalities wouldn't hold.

## In Euclidean $n$-Space

In the Euclidean $n$-dimensional case, consider three points $A$,$B$ and $C$. We say that $B$ is between $A$ and $C$ if:

- $\norm{\vec{AB}} < \norm{\vec{AC}}$

and:

- $\vec{AB} \cdot\vec{AC} = \norm {\vec{AB}} * \norm{\vec{AC}}$

For intuition behind definition see Definition:Between (Geometry)/N-dimensional Euclidean space Intuition

## Sources

- June 1999: Alfred Tarski and Steven Givant:
*Tarski's System of Geometry*(*Bull. Symb. Log.***Vol. 5**,*no. 2*: pp. 175 – 214) : p. $201$