What is the theorem of varignons

User: Therber

Varignon parallelogram

This article is a contribution to the didactic seminar "MMS / Computer in the MU of the upper secondary level".

→ MMS / Computer_im_MU_der_Secundarstufe_II

The aim is to help the students to develop a mathematical proof for the Varignon theorem by representing the statement geometrically and by intuitively experiencing the laws associated with it during the construction. The possibility of using software to see continuous transitions makes it possible to develop an understanding of the geometric behavior and to make considerations from this that create the proof.

Dynamic 3D software offers a great advantage: while any square (without coordinates - i.e. only as an image) in space does not yet provide a clear statement about its position and aspect ratios, the brain understands which positional relationships as soon as the structure is rotated in space exist because it merges the many images of the "film" into one overall piece of information and creates a spatial image of the body that is not - or at least not unmistakably - created on a two-dimensional 3D drawing.

Possible use as a teaching task:

  • Geometric representation of the Varignon theorem.
  • joint finding of a proof (e.g. geometrical, vectorial, physical)
  • Geometric representation of the evidence
  • Discussion of the scope of the evidence: strength of evidence, historical development, etc.

Movement to the Varignon parallelogram

"Each square forms a parallelogram with its sides."


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Geometric proof:

Sharing the routes and with the same proportions (here 1/1) through the points and , generates one to the diagonal according to the ray theorem parallel route . The same goes for the route which in turn parallel to must run. The routes are obtained in the same way and . Since the lines were created by halving the sides of the square, they also form a closed square, which always forms a parallelogram due to the opposing parallel sides.

(Without having proved the ray theorem, one can also prove it. Consider the triangle and the height m. By parallel shifting the straight line through , at half the height of m, you get the intersection points and , which due to the constant gradient of the routes and cut them in half. So the proof is to be continued as above.)


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Vector proof:

If one considers the sides of the inner quadrilateral as vectors, then it has to be shown that two opposite sides have the same direction and are of equal length.

For the position vectors of the corner points of the inner quadrilateral we get:

.

This results in the associated pages:

.

So it's the side . So is a parallelogram.


Proof by the center of gravity theorem (physical interpretation):

If you weight every corner of a quadrilateral equally (e.g. one kg point mass), the center of gravity of the system lies middle between and . The one for the system middle between and (red system). The focal points obtained in this way apply to the entire system common center of gravity in the middle of their connecting route. The same must apply to the blue system, with a focus on and . The straight lines connecting the centers of gravity and are the diagonals of the inner square. Because they bisect, the square is point-symmetrical, i.e. a parallelogram.[1]


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This phenomenon does not only occur in the case of flat rectangles, but in general in space.

The proof for this is already available, because neither in the vectorial nor in the geometric proof, the two-dimensionality is involved at any point.

This is immediately visible in the vector calculation.

In the case of the geometric proof, this can also be seen quickly as soon as you have it in front of your eyes (e.g. supported by dynamic 3-D software, or even faster drawn by hand) and see that the parallel shift in space does not change the parallelism. The square can also be divided into two flat triangles in space. If you connect the opposite corners of the outer square, you get two lines, to which the opposite sides of the inner square run parallel: form a parallelogram.

Since the proof of the center of gravity is the inner square with only two connecting lines and is defined and because of the same overall center of gravity these have an intersection point that also bisects the straight line, the square can only lie in one plane and be point-symmetrical, which makes it a parallelogram.

The proof must also be given by means of a cross product. The cross product of two vectors is the normal male vector, which describes the parallelogram spanned by the vectors in terms of its size (= length of the vector) and its position in space (orthogonal to the vector). Are the normal vectors the same, the three parallelograms can only lie in one plane, be of the same size and, due to the bisecting property of the parallelogram diagonals, result in only one parallelogram. The evidence is cluttered, long, and offers no further insight, so it is inappropriate for teaching. What is interesting, however, is that it offers a further approach that intuitively came to my mind as a necessary requirement for the figure before working on the simple proof variant, which was also easy to construct and check using dynamic geometry software (which of course does not yet represent a proof) .


To the secondary school curriculum

According to the guidelines for upper secondary school in North Rhine-Westphalia, mathematics lessons should be experienced as a means of clarifying complex issues and thereby demonstrate the cultural and civilizational significance of mathematics. The abstraction of a physical force problem on a mathematical model that enables its calculation, fulfills both the requirement for the elementaryization of complex circumstances, as well as for the cultural integration, considering that the evidence has grown historically from the description via the mathematical model to the analysis of the center of gravity, or the geometric description by means of the theorem of rays, up to vector geometry in Euclidean space. The vector calculation, which was discovered very late in the history of mathematics and had an immense influence on the calculation possibilities in the sciences, opened a gateway to mathematics and science, which has a profound influence on our civilization.

The Ministry of Schools (NRW) summarizes the requirements for high school teaching in a few embroidery words. The proof of the Varignon parallelogram carried out in the classroom corresponds to the points:

  • Acquisition of fundamental
  • Models of geometry / linear algebra,
  • exemplary insights into the historical genesis of mathematics,
  • Establish connections between mathematical and non-mathematical culture,
  • Understanding the elementary nature of complex issues,
  • heuristic work through the dynamization by software.

In the following, by working on this topic, two of the central ideas of mathematics teaching are dealt with in particular. The seven central ideas of mathematics lessons are characterized by the property of fulfilling a dual function, namely the interlocking of the disciplines of mathematics to make visible, as well as the connection to non-mathematical disciplines. It is only through the latter that mathematics loses the notion, which is often attached to it, of having an unnecessarily high end in itself. The pupil then sees it more as an instrument, both to describe existing problems and to develop new problems, or better, new possibilities.


Idea of ​​spatial structuring

If one interprets the mathematical proposition physically, it is necessary to idealize concepts such as mass, extension, measurability and to combine them with concepts such as distance, point and weighting. The position of the masses or the alignment of forces can generally be related to one another, which promotes spatial imagination. Expanding this imagination through Euclidean geometry with vectors is also the goal of mathematics lessons and shows that there can and are still other descriptions of space. Accordingly, the geometric and vector proofs show a certain equivalence between mathematical procedures, which leads to the fundamental insight that mathematics serves as an instrument that can be used: you yourself set the framework, e.g. the type of coordinate system, the course of the Proof etc. firmly in order to reach the goal ... instead of working as a "servant" of given calculations, as it can easily appear to a student.


Idea of ​​mathematical modeling

This idea is closely linked to that of structuring. Varignon advanced statics in a remarkable way by modeling static conditions in a mathematical model. The sentence is an example from the field of modeling, of given forces, masses and their movements. In this way the required relationship between phenomena of the world and mathematics is achieved as a translation of the same. “In order to gain access to the idea of ​​mathematical modeling, it is necessary to get to know mathematical models. The simple one, astonishing before the evidence Sentence, conveys the process and also the range of mathematical models to the student. To lift the proof from the two-dimensional into the three-dimensional seems complicated at first, as long as one does not see the generally valid character of the ray theorem or the vector calculation. If the general validity is recognized, the possibility of using the sentence in general, in n-dimensional spaces with non-Cartesian coordinate systems, becomes clearer and the wide range of mathematics is shown.


Idea of ​​the functional connection

This idea does not come to the fore here very strongly. Nevertheless, the construction of a parallelogram using an arbitrary square is of course an illustration. A defined rectangle always depicts exactly one parallelogram with its center sides. Interestingly, the reverse is not the case, because a parallelogram with its corner points as the center of a square has an infinite number of associated squares. The concept of the functional connection is also touched on by the proof in this way.

History

Varignon's career

Pierre Varignon was born in France in 1654 and was a contemporary of Newton, Leipnitz, Rolle, Bernoulli, L'Hospital and other greats in mathematics. He was born into a poor working class family. His father earned so little as a bricklayer that he could not give Pierre financial support. Obviously ungrateful in this situation, he complains that he has not noticed anything from his family other than a sense of technology. But this one later turned out to be a source of his life's work. After a theological career as a priest, he happened upon Euclid's "Elements", which he began to read with great interest. This mathematical introduction brought him to Descartes' "Geometrié", whereupon he devoted himself to science. Varignon moved to Paris with a friend in 1686, where he sought contact with local scientists. Just one year later he published a project on mechanics, in which he dealt with a new type of connection between mechanics and Leibniz calculus. At the Academy of Sciences he was highly regarded and received a chair. He was always an advocate of Leibnitz's calculus and understood its immense value for science early on. Although he did not produce any groundbreaking mathematical knowledge himself, he was one of the most important mathematicians of that time. He developed the analytical dynamics with the help of Leibnitz's differential calculus, which he applied to Newton's mechanics in inertial systems. While greats like Huygens, who Newton held in high esteem, rejected the physical theory of the action of forces at a distance, he kept himself free from doubts and thus penetrated further and further into matter. With Rolle he had a vigorous exchange of blows over the infinitesimal calculus over several years, which only came to an end at the behest of the Academy. He achieved a lot in the field of mechanics and statics and published his “New Mechanics” in 1724, which was groundbreaking for the next 75 years.[2]


geometry

At the time of Varignon, the concept of vector did not yet exist. Varignon then played an important role in the discovery of the composition of forces. He develops the general concept of the parallelogram of forces and shows the complete compensation of forces whose "vector sum" results in zero. He developed a device that made it possible to find the common point of equilibrium for n occurring forces. In his work on forces in mechanics and statics he found the regularity of the general parallelogram formation in the interior of any quadrilateral: the centers of the adjacent corner points of a quadrilateral always form a parallelogram.

For proof

“The analogies between the vividly imaginable and the higher-dimensional come to light most clearly in linear terms and tasks. Therefore, the actual transition to n-dimensional geometry is closely connected with the development of linear algebra and the emergence of the concept of vector. ”It therefore seems to me to be appropriate to use the short proof in school both vectorially and physically (using the mathematical Center of gravity theorem). This can be done clearly with the help of geometry software. The range of this set is made particularly clear by dynamic geometry software, because after the construction has been made, any shape (including interlaced squares) can be generated in the blink of an eye and its effects can be checked. This does not yet provide the proof, but creates a deep understanding of the behavior of a square in space and thus opens the way to the search for evidence.[3]


[1] School of Mathematics and Statistics, University of St Andrews, Scotland, O'Connor and Robertson July 2007

[2] Scriba, Christoph J .; Schreiber, Peter (2010): 5000 years of geometry, Springer-Verlag Berlin Heidelberg, page 432

[3] Wittmann, Erich Ch. Elementary Geometry and Reality

http://www.uni-flensburg.de/mathe/zero/veranst/elemgeom/varignon/varignon.html#2.1
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