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Propostas de Atividade
Cabri
and the geometric contructions with compass alon
Abstract Mohr in 1672 and Mascheroni in 1797 proved the fundamental theorem according to which every ruler and compass construction can be made using only the compass. In this paper we describe how a Cabri based microworld can be built, in which only compass is available to the user. The analysis of Mascheroni’s work shows how his static contructions rely on implicitly made choices of points and they can not pass the dragging test. In order to have correctly draggable constructions we must use logical macros.
Theoretical and hystorical framework A geometric model can be defined as the set of tools available for geometric contructions and the set of elementary operations that can be performed using these tools. The Euclidean geometric model, based on unmarked ruler and compass, is well known; its graphical power and the wealth of possible constructions arose geometers’ interest in other geometric models with fewer resources and elementary operations. The most important and widely studied of these models is the compass model, which allows the use of compass only. The Danish geometer G. Mohr in 1672 discovered an important result about the compass model; but his book "Euclides Danicus" was lost and was rediscovered only three centuries later. In 1797 the Italian geometer Lorenzo Mascheroni (1750-1800) published in Pavia his "Geometria del compasso", stating (at least implicitely) the now called Mascheroni-Mohr theorem "All geometric contructions that can be made using ruler and compass can be made using only the compass"; in other words, the ruler and compass model and the compass model are equivalent.
The Mascheroni-Mohr Theorem The starting point of Mascheroni’s work is the following: every problem in elementary geometry can be reduced to determining points, since a line is given when two of its points are known and a circle is given when its center and a point are given. In the Euclidean model the possibility of using ruler and compass leads to an immediate solution of these three problems:
In the compass model (where a line is supposed to be given when a couple of its points are known) only the third problem is immediately solvable. The proof of the equivalence of the ruler and compass model and the compass model is reduced then to proving that the problems A and B can be solved using the compass alone. This proof is given by Mascheroni, as we describe below.
The "Geometria del compasso" in Cabri The study of the geometric contructions with compass alone can be made in Cabri in three steps:
Mascheroni solves the fundamental problems A and B using some simpler contructions:
Problem B is solved first, distinguishing two cases: when the line does not pass through the center of the circle and when it does. The constructions are shown in the figures below (the dotted lines are drawn just to make the figures more easily readable); looking at them we can see how the preceeding "lemmata" are used. For instance, in the second construction, given the circle of center O and point C and the line OL, we take a point A on the circle, we draw the simmetrical A’ with respect to the line OL and we bisect the arcs defined by A and A’, obtaining the points P and Q that solve the problem.
The contruction that solves Problem A is more lengthy; it uses the fourth proportional; it is shown in the figure below.
Dragging Mascheroni’s constructions The constructions made above following Mascheroni’s hints are correct "static" figures; but, in order to be correct Cabri figures and to become the starting point for the definition of macros, they must pass the dragging test. Unfortunately this does not happen. The analysis of the problem is quite lengthy, but the main point is the intersection of two circles that gives (in the non tangent case) two points. In paper and pencil contructions we first usually choose a suitable starting configuration (for instance, a suitable position for the points defining the lines in Problem A) and then, when required, we make a choice, implicit or explicit, between the two intersection points of the circles. As a side effect of this situation, we can notice that if we define a macro using these figures we can have uncorrect results when the macro is used. This can generate a cascade effect, if uncorrect macros are used in other constructions. A possible solution to these problems can be given using logical constructions, that allow us to make choices according to different geometrical situations. Without going into detail, the following logical macros can be defined:
As a conclusion, it is possible to define seven macro contructions:
The macro 1-4 are used in the definition of macros 5-7, that can be added to the menu bar of the "compass only Cabri", that looks like in figure.
Authors’adresses Paolo
Boieri Chiara Micheletti |
