Definition:Equidistance

Definition
Equidistance is one of the undefined terms in Tarski's Geometry.

As such, let the following ostensive definition suffice.



In the diagram, points $a,b$ are equidistant to points $c,d$, and we write $ab \equiv cd$.

However, points $a,c$ are not equidistant to points $a,b$, and we write $\neg \left({ac \equiv ab}\right)$ or $ac \not \equiv ab$.

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


 * $a = \left({x_1,y_1}\right)$


 * $b = \left({x_2,y_2}\right)$


 * $c = \left({x_3,y_3}\right)$


 * $d = \left({x_4,y_4}\right)$

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

Then:


 * $ab \equiv bc \dashv \vdash \left[{\left({x_1-y_1}\right)^2 + \left({x_2 - x_1}\right)^2 = \left({x_3-y_3}\right)^2 + \left({x_4-y_4}\right)^2}\right]$

Compare Distance Formula.