Victor Oxman

Two cevians intersecting on an angle bisector

Summary We prove the next generalization of the Steiner-Lehmus theorem: if two equal cevians intersect with each other on the angle bisector of the third triangle vertex, then the triangle is isosceles.

The concept of invariance in school mathematics

ABSTRACT In this paper, we highlight examples from school mathematics in which invariance did not receive the attention it deserves. We describe how problems related to invariance stimulated the interest of both teachers and students. In school mathematics, invariance is of particular relevance in teaching and learning geometry. When permitted change leaves some relationships or properties invariant, these properties prove to be inherently interesting to teachers and students.

The Properties of Special Points on the Brocard Circle in a Triangle

In this paper we investigate some properties related to the Brocard circle of a given triangle ∆ABC. Four of them correspond to the special point of a triangle that satisfies some conditions. Another one corresponds to the special case of ∆ABC for which the relation \( ({\text{AC}}^{2} + {\text{AB}}^{2} )/2 = {\text{BC}}^{2} \) holds. We analyzed the case where a symmedian is a tangent to the Brocard circle. We found some interesting properties related to the Brocard circle for right angle triangle and for isosceles triangle.

Proof Without Words: An Elegant Property of a Triangle Having an Angle of 60 Degrees

Summary In a triangle ABC in which angle A measures 60 degrees, the bisectors of angles B and C are used to construct a cyclic quadrilateral with two congruent sides.

Dynamic investigation of triangles inscribed in a circle, which tend to an equilateral triangle

ABSTRACT We present a geometrical investigation of the process of creating an infinite sequence of triangles inscribed in a circle, whose areas, perimeters and lengths of radii of the inscribed circles tend to a limit in a monotonous manner. First, using geometrical software, we investigate four theorems that represent interesting geometrical properties, after which we present formal proofs that rest on a combination between different fields of mathematics: trigonometry, algebra and geometry, and the use of the concept of standard deviation that is taken from statistics.

On teaching extrema triangle problems using dynamic investigation

ABSTRACT An important and interesting area in the study of triangle geometry is the related issue of extrema problems and inequalities. These problems play a significant role in the mathematics study program in high school. In tasks such as these, the difficulty level is high when one does not know in advance what the expected answer is. When one knows what to prove, the difficulty level is lower and most of the effort is aimed at attaining a proof of the expected answer. This can be done using dynamic geometric software. The possibility of making frequent changes to the geometric objects and the ability of dragging objects, contributes to the process of deducing properties, checking hypotheses and generalizing. In this paper, eight investigative tasks in Euclidean geometry are presented together with the applets developed for carrying out the dynamic investigation. Some of the tasks are well known, while others are almost unknown and are worthy of presentation as enrichment for those interested in the subject. The tasks were given to preservice teachers of mathematics as part of an advanced course for integrating technological tools in the teaching of mathematics.

Various solution methods, accompanied by dynamic investigation, for the same problem as a means for enriching the mathematical toolbox

ABSTRACT A geometrical task is presented with multiple solutions using different methods, in order to show the connection between various branches of mathematics and to highlight the importance of providing the students with an extensive ‘mathematical toolbox’. Investigation of the property that appears in the task was carried out using a computerized tool.

Dynamic investigation of loci with surprising outcomes and their mathematical explanations

The locus is a very important concept in Euclidean geometry since it serves as a tool for solving different problems, and allows geometric constructions to be carried out. The teaching of the subject of loci in various mathematics courses includes solution of different exercises in which the student is required to find the locus in accordance with the data of the question. The present paper offers a different view of the subject of loci, which brings about conceptual understanding of the subject with identification of conserved properties and suitable generalizations obtained through investigation that includes the use of dynamic geometric software (GeoGebra). General formulas were developed for the equation of the locus in two cases. In the article, there are links to geometric applets which allow one to demonstrate the loci formed in some cases.

Proof without Words: An Elegant Property of the Equilateral Triangle

Summary Figures are used to show that for an equilateral triangle ABC and some point D on the ray CB, the distances from the point D to the vertices of the triangle satisfy different polynomial equations depending on whether the point D is between B and C or not.

Investigating derivatives by means of combinatorial analysis of the components of the function

Given a composite function of the form h(x) = f(g(x)), difficulties are often encountered in calculating the value of the nth derivative at some point x = x0 when one attempts to determine whether its nth derivative becomes zero at this point, or attempts to find the sign of the nth derivative by differentiating it n times and substituting x0. This present paper offers an alternative method that allows the investigation of the nth derivative of function h(x) based on the investigation of functions f (x) and g(x) only. Several examples are given, which implement the conclusions on the properties of the relation.