Contextualization

Geometry is a fascinating area of mathematics that deals with the shape, size, and position of things around us. One of the most interesting things about geometry is that it has practical applications in many areas of our daily lives, from building bridges and buildings to designing packaging and graphics. In this project, we will explore an aspect of geometry known as 'Cevians'.

Cevians are line segments that connect a vertex of a triangle to the opposite side (or its extension), whether or not they lie on the opposite side. They have unique and interesting properties that allow us to describe and solve complex geometry problems. There are special types of cevians, such as the median, the bisector, the altitude, and the perpendicular bisector, each with their respective properties.

Studying Cevians and their properties is essential for understanding many advanced mathematical concepts. Furthermore, they have practical applications in many areas of knowledge, such as engineering, graphic design, architecture, and physics. For example, engineers can use the properties of cevians to calculate the distribution of forces in triangular structures, such as bridges and towers. On the other hand, graphic designers and architects can use the properties of cevians to create innovative and functional designs.

To deepen your knowledge on the subject, we suggest the following resources:

1. Cevians: what they are, types, and properties - This article offers a comprehensive introduction to cevians, including their definitions, types, properties, and examples.

2. Flat and Spatial Geometry - Carlos Alberto Campagnole dos Santos - This book covers advanced topics in geometry, including cevians, and includes many practical exercises to help you improve your skills.

3. Khan Academy: Cevians - This is a free online course that provides a detailed explanation of cevians, along with examples and practical exercises.

We hope these resource suggestions help you develop a solid understanding of the topic and prepare students to carry out the project.

Practical Activity

Activity Title: 'Exploring the World of Cevians'

Project Objective:

The objective of this project is to allow students to explore the concept of Cevians, their properties, types, and practical applications. Notably, students will:

1. Investigate the properties of Cevians in relation to different types of triangles.
2. Explain, in their own words, the relevance of Cevians in the real world.
3. Develop, demonstrate, and explain a unique and complex problem involving Cevians.
4. Develop research and writing skills through the preparation of a written report.
5. Improve communication and collaboration skills by working in groups.

Detailed Project Description:

Students will be divided into groups of 3 to 5 and will be tasked with conducting an in-depth investigation of Cevians. The project will be divided into two main parts:

Part 1: Investigation and Exploration of Cevians In this part, students should research the properties and characteristics of Cevians. They should also discover the different types of Cevians (altitude, median, bisector, perpendicular bisector) and their properties. They should explore how different types of Cevians interact with different types of triangles.

Part 2: Practical Application of Cevians Groups are expected to create a complex problem involving Cevians and explain step by step how to solve it. This problem should be original and challenging, demonstrating a deep understanding and practical application of the concepts learned.

Required Materials:

1. Textbooks and/or online reading materials on Cevians.
2. Digital drawing software, such as GeoGebra or AutoCAD, to create diagrams and solve Cevians problems.
3. Internet access for research and group collaboration.
4. Writing materials for the report.

Detailed Step-by-Step for Activity Execution:

1. Divide the class into groups of 3 to 5 students.

2. Each group will research the concepts and properties of Cevians, including different types and practical applications.

3. The groups will then explore the interaction of Cevians with different types of triangles using digital drawing software.

4. Each group will develop an original and complex problem related to Cevians. This problem should include a detailed diagram and a step-by-step explanation of how to solve the problem.

5. Finally, each group will write a report documenting their findings and learnings. The report should include the sections: Introduction, Development, Conclusion, and Bibliography. The Introduction will include an explanation of the concept of Cevians and their practical applications. The Development will contain a detailed description of the problem and the methodology used to solve it. The Conclusion will discuss what the students learned during the project and how the acquired knowledge can be applied in real contexts. Finally, the Bibliography will list all the resources that the students used to carry out the project.

Important: The project has an estimated duration of 15 hours per student.

Project Deliverables:

Each group must deliver:

1. Developed problem with detailed and explained solution.
2. Written document containing:
   • Introduction
   • Development
   • Conclusion
   • Bibliography

The written document should complement the problem that the groups created, highlighting the logical process and practical application of Cevians. It should contain all the necessary information to understand the proposed problem and its solution, as well as the students' personal learnings and possible applications.