Definition:Equidistance

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Definition

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

Intuitively, two points $a, b$ are equidistant with respect to $c, d$ if the length of line segment $a b$ is the same as that of line segment $c d$.

However, at this point in the game, we have not yet defined line segment, or even distance.

As such, let the following ostensive definition suffice.

Equidistance.png

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

However, points $a, d$ are not equidistant to points $a, b$, and we write $\neg \paren {a d \equiv a b}$ or $a d \not \equiv a b$.


In Euclidean 2-Space

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

$a = \tuple {x_1, x_2}$
$b = \tuple {y_1, y_2}$
$c = \tuple {z_1, z_2}$
$d = \tuple {u_1, u_2}$

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

Equidistance(Analytic Def'n).png

$a b \equiv c d \dashv \vdash \paren {\paren {x_1 - y_1}^2 + \paren {x_2 - y_2}^2 = \paren {z_1 - u_1}^2 + \paren {z_2 - u_2}^2}$

Compare Distance Formula.


In Euclidean n-Space

Define the following coordinates in an Euclidean n-space:

$a = \tuple {x_1, x_2, \dotsc, x_n}$
$b = \tuple {y_1, y_2, \dotsc, y_n}$
$c = \tuple {z_1, z_2, \dotsc, z_n}$
$d = \tuple {w_1, w_2, \dotsc, w_n}$

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

Then:

$a b \equiv c d \dashv \vdash \paren {\paren {x_1 - y_1}^2 + \paren {x_2 - y_2}^2 + \dotsb + \paren {x_n - y_n}^2 = \paren {z_1 - w_1}^2 + \paren {z_2 - w_2}^2 + \cdots + \paren {z_n - w_n}^2}$

Compare Vector Length.


Sources