In this article, we state an interesting geometric conservation property between the three angle bisectors of three similar right triangles and provide a proof without words for its justification. A GeoGebra applet is also presented to help with the understanding of the progression of the proof from inductive to deductive stage.

Hands-on experiments with overturning some prisms (partially filled with water) lead students to a conjecture which can be confirmed by using a 3D geometry programme and reinterpreting the process of "overturning of a prism" in an appropriate way. But such confirmations are not a proof and particularly cannot answer the question…

Problem solving and proofs have always played a major role in mathematics. They are, in fact, the heart and soul of the discipline. The using of a number of different proof techniques for one specific problem can display the beauty, and elegance of mathematics. In this paper, we present one specific, interesting geometry problem, and present four…

GeoGebra is a dynamic software that is frequently used and of increasing importance in mathematics teaching processes in our digital age. Accordingly, in this study a new perspective has been brought to the proofs of the "two square difference identity" expressed for the square, which is a flat polygon, made with different approaches.…

PWW style are presented. The difference of evidence is based on various additional auxiliary constructions, which in itself is a "mathematical art" that the student can master as a result of much practice. The PWW are accompanied by GeoGebra applets containing HINT buttons that allow the student to get step-by-step help in completing the…

The complexity of the art of teaching and learning, especially in the field of mathematics, calls for a broader approach than the traditional status quote. Proof without words in geometry coupled with the dynamic and interactive GeoGebra applet have the capacity to be used in all levels of mathematics education from grade schools to higher…

Mathematical proof, often regarded as the heart of mathematics, is essential for interconnected mathematical knowledge. However, proof-writing skills do not develop inherently. Effective learning environments are vital for university students to enhance these skills. This study investigates the impact of the ISMAT model on pre-service teachers'…

This article focuses on the possible relationship between predictions, developed during the resolution of geometrical tasks within a Paper-and-Pencil Environment (PPE), and subsequent explorations within a Dynamic Geometry Environment (DGE). Building on Fischbein's Theory of Figural Concepts, and to gain insight into the transition of predictions…

In the article, we prove 18 inequalities involving inradius, a length of one side and one additional element of a given triangle. 14 of these inequalities are the necessary and sufficient conditions for the existence and uniqueness of such a triangle. All proofs are based on standard methods of calculus and can serve as a good demonstration of the…

We present an investigation of the infinite sequences of numbers formed by calculating the pairwise averages of three given numbers. The problem has an interesting geometric interpretation related to the sequence of triangles with equal perimeters which tend to an equilateral triangle. Investigative activities of the problem are carried out in…

The problem in this study is to evaluate the improvement of students' geometry skills using the GeoGebra application by focusing on two key indicators: logical thinking and drawing skills. This study employs both quantitative and qualitative methods. In the first stage, quantitative data were collected through pretests and posttests to measure…

In this study, we explore automated reasoning tools (ART) in geometry education and we argue that these tools are part of a wider, nascent ecosystem for computer-supported geometric reasoning. To provide some context, we set out to summarize the capabilities of ART in GeoGebra (GGb), and we discuss the first research proposals of its use in the…

The present paper describes a dynamic investigation of polygons obtained by reflecting an arbitrary point located inside or outside a given polygon through the midpoints of its sides. The activity was based on hypothesizing on the shape of the reflection polygon that would be obtained, testing the hypotheses using dynamic software, and finding a…

Windmill images and shapes have a long history in geometry and can be found in problems in different mathematical contexts. In this paper, we share and discuss various problems involving windmill shapes and solutions from geometry, algebra, to elementary number theory. These problems can be used, separately or together, for students to explore…

With the ever-evolving implementation of technology in the teaching and learning of geometry, the task of designing and/or finding suitable questions or problems that can facilitate and promote mathematical reasoning and accurately assess students' understanding of the subject matter can be a challenge. It is the aim of this article to share the…