---
tag: KENSASDRWF
kind: definition
name: Euclidean geometry
msc: [51M05, 51M04]
defines: KRZD96GSVC
uses: [0XPR1E85W0, 0H3YY8CPKH, KM1FB8XZ2R]
sources:
  - url: https://en.wikipedia.org/wiki/Euclidean_geometry
---

## Definition

Let $n \ge 1$ and let $E^n$ be the [[Euclidean space|0XPR1E85W0]] of dimension $n$, that is, $\R^n$ with the distance $d(x, y) = \sqrt{\sum_{i=1}^n (x_i - y_i)^2}$.
The [[Euclidean group|0H3YY8CPKH]] is the group of all distance-preserving bijections of $E^n$.
*Euclidean geometry* of dimension $n$ is the study of the properties of subsets of $E^n$ that are invariant under the Euclidean group.
Synthetically, plane and solid Euclidean geometry is the theory given by [[Hilbert's axioms|KM1FB8XZ2R]] in dimension $2$ and $3$.

## Remarks

Example: the property "the three points $A$, $B$, $C$ form a right triangle with the right angle at $B$" is a property of Euclidean geometry.
Indeed, an isometry preserves distances, so it preserves the equation $d(A,B)^2 + d(B,C)^2 = d(A,C)^2$.
Non-example: the property "the segment $AB$ is horizontal" is not a property of Euclidean geometry, since a rotation moves a horizontal segment to a slanted one.
The two descriptions agree: every model of Hilbert's axioms of the plane is isomorphic to $\R^2$ with its usual lines, betweenness and congruence.