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• The sum and difference of the areas of Napoleon triangles

  99-108

  Sándor Kiss

  Views:

  9

  The sum of the areas of the Napoleon triangles is the average of the areas of the three outward equilateral triangles on the sides of triangle ABC, and the differerence of these areas is the area of triangle ABC. In this paper we examine how to change these properties if we build on the sides of the triangle ABC, outwards and inwards, three similar triangles.

• Ein ungewöhnlicher Weg zu Jakob Steiners Umellipse eines Dreiecks und zur Steiner–Hypozykloide

  49-65

  Oswald Giering

  Views:

  21

  In real projective geometry of triangles two problems of collinear points are discussed. The problems differ only from the running through the vertices of a given triangle ABC. Resolving the problems we find two cubic curves kS and kT . Affine specialization leads to the circumscribed Steiner ellipse about the triangle ABC and shows us this ellipse in more general surroundings. Euclidean specialization leads to Steiners three-cusped hypocycloid.

• Besondere Punkte der Euler-Geraden

  145-157

  Oswald Giering

  Views:

  23

  In the following article the concepts "Euler line of a triangle" and "radical centre of three circles" are connected. In this way we could find some relations between special points of a triangle (orthocentre H, centre of gravity S, circumcentreM, midpoint F of the nine-point circle) and the radical centres of special triples of circles.

• Decomposition of triangles into isosceles triangles II: complete solution of the problem by using a computer

  275-300

  József Kosztolányi

  Zoltán Kovács

  Erzsébet Nagy

  Views:

  29

  We solve an open decomposition problem in elementary geometry using pure mathematics and a computer programme, utilizing a computer algebra system.

• Decomposition of triangles into isosceles triangles I: let the students ask bravely

  163-184

  József Kosztolányi

  Zoltán Kovács

  Erzsébet Nagy

  Views:

  26

  We report about working up an open geometric problem as a mathematical research with pupils of a mathematics camp. This paper shows the didactic aims and the methods we worked with, the didactic results. The second part of this paper gives a general solution of the problem, using pure mathematics and a computer programme.

• Über ähnliche Aufsatzdreiecke einer Strecke

  337-348

  Oswald Giering

  Views:

  28

  In this article we investigate (with methods of school geometry) a figure (PQ,ABC) consisting of three given similar triangles PQA, PQB, PQC with side PQ in common (Figure 1). We combine other triangles with this figure such as triangle ABC which is proved to be similar to the given triangles. The incircles of three additional triangles adjacent to triangle ABC will be determined.

• Charakteristische Dreieckpunkte in der projektiv-erweiterten hyperbolischen Ebene

  299-315

  Ana Sliepčević

  Ivanka Babić

  Views:

  9

  Some basic planimetric constructions regarding segments, angles and triangles are shown in the Cayley-Klein model of the hyperbolic plane. Relationship with the situation in the Euclidean plane is given. H-triangles are classified considering the location of their vertices and sides with respect to the absolute. There are 28 types of triangles. It is shown that there exist 12 pairs of dual triangles, while 4 types of triangles are dual to themselves. For every type of triangle the existence and number of the characteristic points are determined. Few examples of triangles with construction of their characteristic points, incircles and circumcircles are given.

• On the past of a famous theorem: the predecessors of a theorem of Pythagoras

  255-267

  Lajos Klukovits

  Views:

  40

  The well-known Theorem of Pythagoras asserts a relation among the sides of any right-angled triangle. It can be found any secondary school textbook. An interesting question whether this result due to the Pythagoreans from the VIth century BC, or it was known in earlier civilizations. The first answer is a vague yes. According to the legends the Egyptian rope-stretchers used a triangle with sides 3,4,5 units to create right angle. But are there real evidences that this result was known earlier? We will argue that in almost all river-valley civilizations it was known and used.

• Über die Verwendung von Maple für die Simulation von Mechanismen

  21-39

  Engin Can

  Views:

  25

  Maple is used to do numerical computation, plot graphs and do exact symbolic manipulations and word processing. This paper demonstrates how Maple can be used for the simulation of mechanisms. This offers the possibility for students to become familiar with this particular section of mathematical modelling. The mechanism under consideration is a so-called F-mechanisms, i.e., a planar parallel 3-RRR robot with three synchronously driven cranks. It turns out that at this example it is not possible to find the poses of the moving triangle exactly by graphical methods with traditional instruments only. Hence, numerical methods are essential for the analysis of motions which can be performed by an F-mechanism.

• Analysis of a problem in plane geometry discussed in an 11th grade group study session

  181-193

  Richárd Rakamazi

  Views:

  27

  The main aim of this paper is to show those strategies and proof methods we try to teach in secondary maths education through an interesting geometric problem: Find a relation for the sides of a triangle where an angle is the double of another angle. Is the converse also true? Is it possible to generalize the problem? We try to answer these questions while discussing the upcoming difficulties in detail and presenting more possible solutions. Hopefully the paper can be successfully used in study group sessions and problem solving seminars in secondary schools.

• Combinatorics teaching experiment

  27-44

  Károly Csabay

  József Daragó

  Views:

  36

  Teaching combinatorics has got its conventional method. One has to see: the combinatorical formations won't be follow each other by a heuristic way. The formulas kept by pupils seem to come from "deus ex machina". We try to offer now an alternative way to approach combinatorical concepts from a nontraditional direction and point of view.

• On a special class of generalized conics with infinitely many focal points

  87-99

  Ábris Nagy

  Zsolt Rábai

  Csaba Vincze

  Views:

  8

  Let a continuous, piecewise smooth curve in the Euclidean space be given. We are going to investigate the surfaces formed by the vertices of generalized cones with such a curve as the common directrix and the same area. The basic geometric idea in the background is when the curve runs through the sides of a non-void triangle ABC. Then the sum of the areas of some triangles is constant for any point of such a surface. By the help of a growth condition we prove that these are convex compact surfaces in the space provided that the points A, B and C are not collinear. The next step is to introduce the general concept of awnings spanned by a curve. As an important example awnings spanned by a circle will be considered. Estimations for the volume of the convex hull will be also given.