Context

Mathematics, an ancient exact science, is present in all areas of modern life, invisibly guiding many of our daily activities. Geometry, one of the disciplines of this science, is responsible for studying the shape and size of the things that surround us. Within this universe lies the figure of the triangle, one of the simplest geometric shapes and, at the same time, full of interesting properties and characteristics. One way to explore these characteristics is through the study of cevians.

Cevians are line segments that connect one of the vertices of a triangle to any point on the opposite side. They have fascinating mathematical properties and are extremely important for the understanding and study of various areas of Mathematics. There are special cevians that, due to their properties, have specific names, such as the median, the altitude, the angle bisector, and the perpendicular bisector.

In particular, there are some very famous cevians, such as the Simson lines, which are cevians related to the nine-point circle of a triangle. We also have Nagel's cevians, which connect the vertices of the triangle to the Nagel point, a notable point whose properties are interesting and surprising. Nagel's cevians are used in the construction of solids of revolution, which are three-dimensional geometric figures generated by the rotation of a flat region around an axis.

The study of cevians, besides being an important part of the complete understanding of the geometric universe, has various practical applications. For example, they can be used to calculate the area of a triangle more efficiently, to determine the position of a point within a triangle, or to solve optimization problems. In the real world, cevians are used, for example, in areas such as civil engineering for structural calculations or in computer science for algorithm optimization.

For further exploration on the topic, we suggest the following resources:

• Book: "5000 Problems in Combinatorial Analysis" by Paulo Cezar Pinto Carvalho.
• Website: Brazil School, the Geometry section provides explanations and examples of cevians.
• Video: Khan Academy, a channel with educational resources, including videos on Geometry.

Practical Activity

Activity Title: "Investigating Cevians - The Art and Science of Triangles"

Project Objective

The objective of this project is to understand what cevians are, what are the main cevians, their properties, and notable points through theoretical study and the creation of a 3D model of a triangle with its cevians.

Detailed Project Description

In this project, students will be divided into groups of 3 to 5 members. Each group will delve into the study of cevians, investigating their properties and relating them to the characteristics of triangles.

The activity will consist of two main parts: the theoretical part, where students will study and discuss the properties of cevians, and the practical part, where they will create a 3D model of a triangle with its cevians.

Required Materials

• Books and/or online resources on cevians.
• Wooden board or similar to serve as a base for the 3D model.
• Toothpicks or straws.
• Hot glue or white glue.
• Acrylic paint in different colors.
• Brushes.
• Pencil, pen, eraser, ruler, and compass.
• Cardboard or cardstock.
• Nylon thread.

Step-by-Step for Activity Execution

Theoretical Part:

1. Study the properties of cevians, such as the median, altitude, angle bisector, and perpendicular bisector.
2. Discuss in groups the studied properties and prepare a summary for each type of cevian.
3. Investigate Ceva's theorem and Menelaus's theorem, which are important theorems related to cevians.

Practical Part:

4. Draw a life-size triangle on cardboard or cardstock, marking the vertices and the points where the cevians meet.
5. Use toothpicks or straws to create line segments representing the sides of the triangle and its cevians.
6. Glue the line segments on the cardboard or cardstock.
7. Use different types of paint to identify each type of cevian.
8. Attach the 3D model to the wooden board using hot glue or white glue.
9. Add explanatory captions and titles using colored pens.

Writing the Written Document:

10. Write the report containing the four topics: Introduction (context, relevance, and project objective), Development (theory, activity description, methodology, and results), Conclusions (learnings, key points, and conclusions about the project), and Bibliography (sources used).
11. The report should be formatted clearly, with organization into sections and subsections. It should present the theory in a didactic way, a detailed description of the activity with images of the 3D model, and a discussion of the results obtained.

Project Deliverables

The project deliverables should consist of the 3D model of the triangle with its cevians and the written report containing the four indicated sections. Students must ensure that the report is complete, reviewed, and with all the necessary information for understanding the project. It is important for students to demonstrate, in the report and in the 3D model, their understanding of cevians and their properties, as well as the geometry concepts involved in the construction of the model.

The project deliverables are directly connected to the proposed activities and the theoretical concepts studied. The construction of the 3D model allows students to apply practically the theoretical concepts they have learned, while writing the report allows them to articulate in words their understanding of the topic and the activity carried out.