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The article offers a guide for applying the proof without words theorem in a right angle using the GeoGebra software. It suggests the use of the HINT buttons to help students see the pairs of tangents from the vertices of the triangle, the hypotenuse of a right triangle and the lengths of the corresponding segments.

This article deals with the production of meanings around the right triangle during the resolution of a construction problem with GeoGebra, by pre-service mathematics teachers. Assuming a multimodal perspective of learning, we analyze the semiotic activity of the participants, from four categories that emphasize the semiotic and embodied nature of thinking in geometry. We conclude that the explanation of a construction procedure with GeoGebra is linked to the understanding of the geometric object to be built, due to the workspace used and the teacher educator plays an important role for achieving this understanding. [ABSTRACT FROM AUTHOR]

Geometry is one of the important topics in school mathematics. Many people solve daily life problems using geometry, including the concept of triangles. However, many students still have obstacles in learning the concept of triangles. This study aims to analyze students' learning obstacles on the concept of triangle. A qualitative research along with a case study was conducted on 45 students of 9th grade, studying at one of the junior high schools in Majalengka, West Java, Indonesia. The data were collected using triangulation which included document analysis, test, and interview. The results showed that students experienced several types of learning obstacles, including students facing issues in incomplete or inadequate presentation of triangle material (didactical obstacle); their inability to use the triangle concept in the context of non-routine questions (epistemological obstacle); and their inability to follow the learning demands about the definition of triangle, the types of triangles, and the area of triangle (ontogenic obstacle), so that it is indicated that students have problems in understanding the concept of triangle. The results of this study can be used as a reference for developing a more optimal didactical design. [ABSTRACT FROM AUTHOR]

This study explores the epistemological framings of undergraduate students when solving an area problem of a degenerate triangle, without or with a triangle drawing. Through mixed research with a triangulation design, the resolution processes and responses of students were analyzed. The aim was to analyze how students' epistemological framing changes during the problem-solving process depending on whether the task contains the drawing of the triangle or not. Quantitative results show significant differences between students who solve the problem without a triangle drawing and those who do. Qualitative results evidence that students who solved the problem with the drawing established an initial epistemological framing that contained an "obvious fact": the non-zero area of the triangle. They hardly modified this epistemological framing during the solving process, forcing the response to be a positive number. In contrast, students who solved the problem without the drawing easily modified their initial epistemological framing by observing that the area of the triangle was zero. Students' perceptions of the level of difficulty of the problem are discussed, too. [ABSTRACT FROM AUTHOR]

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