Preliminary Constructions
This activity belongs to the GeoGebra book GeoGebra Principia.
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.
Normally, we think of algebraic structures (groups, rings, fields...) as something inherent to certain numerical structures, like integers or real numbers.
However, we can easily create equivalent geometric structures, with the advantage that we can visualize each arithmetic operation as a geometric construction.
If we fix a point O in the plane, we can consider the (Euclidean) distance from the rest of the points to O. We will denote OP as the distance from O to P.
The points equidistant from O form a CIRCLE.
By fixing another point I different from O, we establish a DIRECTION, an ORIENTATION O→I and a LINE r.
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:
OI = OP + PI (P is between O and I)
OP = OI + IP (I is between O and P)
PI = PO + OI (O is between P and I)
- 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.
- Perpendicular bisector: Given two distinct points A and B, we can find all the points that are equidistant from them.
- Midpoint: Intersecting the perpendicular bisector with the line r, we obtain the midpoint MAB.
- Perpendicular: The perpendicular bisector allows us to draw perpendicular lines (simply draw the circle with center P through any point on r).
- Parallel: With two perpendicular lines we obtain a line parallel to r through P.
- Inversion (reflection with respect to the circle)
: With the circle and the perpendicular line we can construct the inversion of A, A–1.
Author of the construction of GeoGebra: Rafael Losada.