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 $ac$.

However line segment is not an undefined term, we are not allowed to call upon it at this stage.

We offer an ostensive definition:


 * Betweenness.png

In the picture, point $c$ is between the two points $a, b$, and we write:
 * $\mathsf{B}acb$

However, point $d$ is not between the two points $a$ and $c$, and we write:
 * $\neg \left({\mathsf{B}adc}\right)$

In Euclidean 2-Space
Define the following coordinates in the $xy$-plane:

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

Then:


 * Betweenness(Analytic Def'n).png


 * $\mathsf{B}abc \dashv \vdash \left({x_1-y_1}\right) \cdot \left({y_2-z_2}\right) = \left({x_2-y_2}\right) \cdot \left({y_1-z_1}\right) \land$


 * $\left({0 \le \left({x_1-y_1}\right) \cdot \left({y_1-z_1}\right)}\right) \land \left({0 \le \left({x_2-y_2}\right) \cdot \left({y_2-z_2}\right)}\right)$