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:
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:
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- $\mathsf{B}abc \dashv \vdash \left({\Delta x_1 \Delta y_1 = \Delta x_2 \Delta y_2}\right) \land$
- $\left({0 \le \Delta x_1 \Delta y_1 \land 0 \le \Delta x_2 \Delta y_2}\right)$
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.
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Work In Progress: I didn't mean prove it here, I meant raise a page where the definition is shown to be justified. (discuss) You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by completing it. When this has been done, the template should be removed from the code. |
In Euclidean $n$-Space
Sources
- June 1999: Alfred Tarski and Steven Givant: Tarski's System of Geometry (The Bulletin of Symbolic Logic Vol. 5, no. 2: 175 – 214) : Page 201
