[color=#999999]This activity belongs to the [i]GeoGebra book[/i] [url=https://www.geogebra.org/m/sw2cat9w]GeoGebra Principia[/url].[/color][br][br][br]Let's return to our familiar Euclidean metric. So far, we have used algebra to facilitate the observation of loci. Let's now see an example of the reciprocal process: using geometry to facilitate the observation of algebraic structures.[br][br]Normally, we think of algebraic structures (groups, rings, fields...) as something inherent to certain numerical structures, like integers or real numbers.[br][br]However, we can easily create equivalent geometric structures, with the advantage that we can visualize each arithmetic operation as a geometric construction.[br][br]If we fix a point O in the plane, we can consider the (Euclidean) distance from the rest of the points to O. [b]We will denote OP as the distance from O to P.[/b][br][br]The points equidistant from O form a CIRCLE.[br][br]By fixing another point I different from O, we establish a DIRECTION, an ORIENTATION O→I and a LINE r.[br] [br]We will take the distance OI as the UNIT. Additionally, two points on the line limit a semicircle. A point P is on the line r if it satisfies any of these equalities:[br] [br] OI = OP + PI (P is between O and I)[br] OP = OI + IP (I is between O and P)[br] PI = PO + OI (O is between P and I)[br][list][*]Point reflection (symmetry): If A is on the line, there exists only another point A' on it at the same distance from O as A.[br][/*][*]Perpendicular bisector: Given two distinct points A and B, we can find all the points that are equidistant from them.[br][/*][*]Midpoint: Intersecting the perpendicular bisector with the line r, we obtain the midpoint M[sub]AB[/sub].[br][/*][*]Perpendicular: The perpendicular bisector allows us to draw perpendicular lines (simply draw the circle with center P through any point on r).[br][/*][*]Parallel: With two perpendicular lines we obtain a line parallel to r through P.[br][/*][*]Inversion (reflection with respect to the circle) [url=https://en.wikipedia.org/wiki/Inversive_geometry][img]https://www.geogebra.org/resource/scjbyz2p/0tuzuVw455vxurEw/material-scjbyz2p.png[/img][/url]: With the circle and the perpendicular line we can construct the inversion of A, A[sup]–1[/sup]. [/*][/list]

[color=#999999]Author of the construction of GeoGebra: [url=https://www.geogebra.org/u/rafael]Rafael Losada[/url].[/color]