# Exploring Advanced Euclidean Geometry with GeoGebra

Gerard A. Venema
Edition: 1
Pages: 146
https://www.jstor.org/stable/10.4169/j.ctt5hh9ht

1. Front Matter
(pp. i-vi)
2. Preface
(pp. vii-xii)
(pp. xiii-xvi)
4. 0 A Quick Review of Elementary Euclidean Geometry
(pp. 1-12)

This preliminary chapter reviews basic results from elementary Euclidean geometry that will be needed in the remainder of the book. It assumes that readers have already studied elementary Euclidean geometry; the purpose of the chapter is to clarify which results will be used later and to introduce consistent notation. Readers who are using this book as a supplement to a course in the foundations of geometry can probably omit most of the chapter and simply refer to it as needed for a summary of the notation and terminology that are used in the remainder of the book.

The theorems in...

5. 1 The Elements of GeoGebra
(pp. 13-22)

The tool we will use to facilitate our exploration of advanced Euclidean geometry is the computer program GeoGebra. No prior knowledge of the software is assumed; this chapter takes a quick tour of GeoGebra and provides all the information you need to get started. The basic commands that are explained in the chapter are enough for you to begin creating sketches of your own. More advanced GeoGebra techniques, such as the creation of custom tools and the use of check boxes, will be discussed in Chapter 3.

GeoGebra is open source software that you can obtain free of charge. If...

6. 2 The Classical Triangle Centers
(pp. 23-30)

We begin our study of advanced Euclidean geometry by looking at several points associated with a triangle. These points are all calledtriangle centersbecause each of them can claim to be at the center of the triangle in a certain sense. They areclassicalin that they were known to the ancient Greeks. The classical triangle centers form a bridge between elementary and advanced Euclidean geometry. They also provide an excellent setting in which to develop proficiency with GeoGebra.

While the three classical triangle centers were known to the ancient Greeks, the ancients missed a simple relationship between them....

7. 3 Advanced Techniques in GeoGebra
(pp. 31-38)

This chapter introduces two features of GeoGebra that enhance its functionality. By far the most useful property of GeoGebra for our purposes is the ability to create user-defined tools, which allow routine constructions to be carried out quickly and efficiently. Check boxes make it possible to produce GeoGebra documents that illustrate a process, not just a static finished sketch.

One of the most effective ways to tap the power of GeoGebra is to create your own custommade tools to perform constructions that you will want to repeat several times. Once you learn how to make tools of your own, you...

8. 4 Circumscribed, Inscribed, and Escribed Circles
(pp. 39-46)

Next we explore properties of several circles associated with a triangle. This study leads naturally to definitions of still more triangle centers. The emphasis in this chapter is again mainly GeoGebra exploration, with formal proofs of most results postponed until later. The chapter ends with a proof of Heron’s formula for the area of a triangle.

The first circle we study is the circumscribed circle. It can be thought of as the smallest circle that contains a given triangle.

Definition. A circle that contains all three vertices of the triangle ΔABCis said tocircumscribethe triangle. The circle is...

9. 5 The Medial and Orthic Triangles
(pp. 47-52)

We now investigate constructions of new triangles from old. We begin by studying two specific examples of such triangles, the medial triangle and the orthic triangle, and then generalize the constructions in two different ways.

Note on terminology. A median of a triangle was defined to be the segment from a vertex of the triangle to the midpoint of the opposite side. While this is usually the appropriate definition, there are occasions in this chapter when it is more convenient to define the median to be the line determined by the vertex and midpoint rather than the segment joining them....

(pp. 53-56)

The geometric objects we have studied until now have been relatively simple: just lines, triangles, and circles. In the remainder of the book we will need to use polygons with more sides; specifically, we will study four-sided and six-sided figures. This chapter contains the necessary information about four-sided polygons.

Let us begin by repeating the definitions from Chapter 0. Four pointsA , B , C ,andDsuch that no three are collinear determine aquadrilateraldenoted by$\square ABCD$. It is defined to be the union of four segments:

$ABCD = \overline {AB} \cup \overline {BC} \cup \overline The four segments are thesidesof the... 11. 7 The Nine-Point Circle (pp. 57-62) One of the most remarkable discoveries in nineteenth century Euclidean geometry is that there is one circle that contains nine significant points associated with a triangle. In 1765 Euler proved that the midpoints of the sides and the feet of the altitudes of a triangle lie on a single circle. In other words, the medial and orthic triangles share the same circumcircle. Furthermore, the center of the common circumcircle lies on the Euler line of the original triangle. More than fifty years later, in 1820, Charles-Julien Brianchon (1783–1864) and Jean-Victor Poncelet (1788–1867) proved that the midpoints of the... 12. 8 Ceva’s Theorem (pp. 63-76) In this chapter we finally complete the proofs of the concurrency theorems that were explored in earlier chapters. We will establish a theorem of Giovanni Ceva (1647–1734) that gives a necessary and sufficient condition for three lines through the vertices of a triangle to be concurrent and then we will derive all of the concurrency theorems as corollaries. We could have givenad hocproofs of each of the concurrency results separately, but it seems better to expose the unifying principle behind them. One satisfying aspect of doing things this way is that the proof of Ceva’s general theorem... 13. 9 The Theorem of Menelaus (pp. 77-80) The theorem we study next is ancient, dating from about the year AD 100. It was originally discovered by Menelaus of Alexandria (70–130), but it did not become well known until it was rediscovered by Ceva in the seventeenth century. The theorem is powerful and has many interesting applications, some of which will be explored in later chapters. In the last chapter we studied the problem of determining when three lines through the vertices of a triangle are concurrent. In this chapter we study the problem of determining when three points on the sidelines of a triangle are collinear.... 14. 10 Circles and Lines (pp. 81-84) Before proceeding to the applications of Menelaus’s theorem we develop some results about circles and lines that we will need. We begin by defining the power of a point with respect to a circle. Definition. Let β be a circle and letObe a point. Choose a line ℓ such thatOlies on ℓ and ℓ intersects β. Define thepower of O with respect to βto be$$p(O,\beta ) - \left\{ \begin{array}{l} {(OP)^2} \\ (OQ)(OR) \\ \end{array} \right.$\$

if ℓ is tangent to β atP, or

if ℓ is intersects β at two pointsQ...

15. 11 Applications of the Theorem of Menelaus
(pp. 85-98)

The theorem of Menelaus is powerful and has interesting consequences in a variety of situations. This chapter contains a sampling of corollaries.

The first applications are simple results about how tangent lines and angle bisectors intersect the sidelines of the triangle. All the proofs in this section rely on the trigonometric form of Menelaus’s theorem.

*11.1.1. Construct a triangle and its circumscribed circle. For each vertex of the triangle, construct the line that is tangent to the circumcircle at that point. Mark the point at which the line that is tangent at a vertex intersects the opposite sideline of the...

16. 12 Additional Topics in Triangle Geometry
(pp. 99-104)

This chapter examines a number of loosely connected topics regarding the geometry of the triangle. We will explore the statements of the theorems using GeoGebra, but will not prove them.

The theorem in this section is commonly attributed to the French emperor Napoleon Bonaparte (1769-1821). Napoleon was an amateur mathematician, with a particular interest in geometry, who took pride in his mathematical talents. Thus this attribution may be based at least partially on historical fact. On the other hand, Coxeter and Greitzer [3, page 63] make the following comment regarding the possibility that the theorem might in fact be due...

17. 13 Inversions in Circles
(pp. 105-110)

We now investigate a class of transformations of the Euclidean plane called inversions in circles. The study of inversions is a standard part of college geometry courses, so we will leave most of the details of the proofs to those courses and will not attempt to give a complete treatment. Instead we will concentrate on the construction of the GeoGebra tools that will be needed in the study of the Poincaré disk model of hyperbolic geometry in the next chapter. We will state the basic definitions and theorems, but will not prove the theorems. Proofs may be found in many...

18. 14 The Poincaré Disk
(pp. 111-120)

In this final chapter we study the Poincaré disk model for hyperbolic geometry. Even though hyperbolic geometry is a non-Euclidean geometry, the topic is nonetheless appropriate for inclusion in a treatment of Euclidean geometry because the Poincaré disk is built within Euclidean geometry. The main tool used in the construction of the Poincaré disk model is inversion in Euclidean circles, so the tools you made in Chapter 13 will be used in this chapter to perform hyperbolic constructions. Many of the constructions in this chapter were inspired by those in the beautiful paper [7] by Chaim Goodman-Strauss.

It was Eugenio...

19. References
(pp. 121-122)
20. Index
(pp. 123-128)