A fascinating and catchy method for proving that a number of special lines concur is using the concept of locus. This is now the classical method for proving the concurrency of the internal angle bisectors and perpendicular side bisectors of a triangle. In this paper, we prove the concurrency of the altitudes and the medians by showing that they…

The perpendicular bisectors of the sides of a hexagon, whose opposite sides are parallel, produce a point symmetric hexagon. Michael de Villiers already gave two proofs of this theorem, firstly an elaborate one with the aid of dynamic geometry and secondly a merely verifying one with the help of coordinates and computer algebra. In this note, we…

Interesting conservation properties that exist in parallelograms are well known, such as: diagonal intersection, pairs of interior triangles that overlap each other, and other properties. This article presents a dynamic study of choosing any point inside and outside the parallelogram and connecting it with two of the vertices of the parallelogram.…

This thesis focuses on the challenge of characterizing multipartite entanglement. While the study of bipartite entanglement is well-documented in scientific literature, recognizing that entanglement can involve more than two parties--i.e. three or more parties---is crucial, as multipartite entanglement enables the completion of more complicated…

Throughout the years of the history of the development of mathematics, starting with the ancient mathematicians, new things were discovered from time to time, which made it a central component of the computerized world, which serves us in many fields, with the young "swimming" in it, and the adults asking: how could we live without it!…

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…

In this work, we delve into geometric analysis, particularly examining the interplay between lower bounds on Ricci curvature and specific functionals. Our exploration begins with an investigation into the implications of Yamabe invariants for asymptotically Poincare-Einstein manifolds and their conformal boundaries under conditions of…

Transitioning from material to theoretical geometry is often considered a significant challenge in compulsory education. Over the last twenty years, research rooted in the Theory of Didactical Situations (TDS) has explored how material geometry can support the teaching and learning of theoretical geometry, promoting continuity in teaching. These…

An interesting geometric conservation problem is presented. Here proof is presented in a 'proof without words' style, with the aim of developing the reader's visual proof ability. The study of the task and its expansion is accompanied by a dynamic sketch to highlight the conservation property.

This study provides a framework for explaining the way individuals might construct the final result of infinite iterative processes in a geometrical context. To this end, 30 undergraduate students were interviewed. The analysis of the data collected yielded two different views: first, there were those who believed that the presented infinite…

The Van Hiele model of geometric reasoning establishes five levels of development, from level 1 (visual) to level 5 (rigor). Despite the fact that this model has been deeply studied, there are few research works concerning the fifth level. However, there are some works that point out the interest of working with activities at this level to promote…

This contribution focuses on reasoning about quadrilaterals provided by lower secondary school students when working with pre-prepared dynamic constructions. On this topic, we present an exploratory empirical qualitative study carried out within a GeoGebra Classroom environment, and our diagnostic instrument consists of a set of dynamic…

Although augmented reality has become one of the most commonly used materials that support learning, especially in learning geometric concepts, it is avoided to be used in the lessons due to its complex structure. At that point, artificial intelligence working as a personal assistant in many fields can help us learn to produce our own model with…

We use Action-Process-Object-Schema (APOS) theory to study students' geometric understanding of partial derivatives of functions of two variables. This study contributes to research on the teaching and learning of differential multivariable calculus and its didactics. This is an important area due to its multiple applications in science,…

The aim of the present study is to explore strategies in enumerating units of three dimensional (3D) arrays. We analyse enumeration strategies of students in grade 3 (ages 8 to 9) in situations of cubical and spherical representations of units of 3D arrays. By exploring students' strategies in these two situations, we find that difficulties in…