Definition:Between (Geometry)

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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:

Betweenness.png

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:

\(\displaystyle a\) \(=\) \(\displaystyle \left({x_1, x_2}\right)\) $\quad$ $\quad$
\(\displaystyle b\) \(=\) \(\displaystyle \left({y_1, y_2}\right)\) $\quad$ $\quad$
\(\displaystyle c\) \(=\) \(\displaystyle \left({z_1, z_2}\right)\) $\quad$ $\quad$

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

Let:

\(\displaystyle \Delta x_1\) \(=\) \(\displaystyle x_3 - x_2\) $\quad$ $\quad$
\(\displaystyle \Delta x_2\) \(=\) \(\displaystyle x_2 - x_1\) $\quad$ $\quad$
\(\displaystyle \Delta y_1\) \(=\) \(\displaystyle y_2 - y_1\) $\quad$ $\quad$
\(\displaystyle \Delta y_2\) \(=\) \(\displaystyle y_3 - y_2\) $\quad$ $\quad$

Then:

Betweenness(Analytic Def'n).png
$\mathsf{B}abc \dashv \vdash \left({\Delta x_1 \Delta y_1 = \Delta x_2 \Delta y_2}\right) \land$
$\left({0 \le \Delta x_1 \Delta y_1 \land 0 \le \Delta x_2 \Delta y_2}\right)$


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

\(\displaystyle \Delta x_1 \Delta y_1\) \(=\) \(\displaystyle \Delta x_2 \Delta y_2\) $\quad$ $\quad$
\(\displaystyle \implies \ \ \) \(\displaystyle \frac {\Delta y_1}{\Delta x_2}\) \(=\) \(\displaystyle \frac {\Delta y_2}{\Delta x_1}\) $\quad$ $\quad$

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


Sources