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The Euler Line in Non-Euclidean Geometry California State University, San Bernardino CSUSB ScholarWorks Theses Digitization Project John M. Pfau Library 2003 The Euler Line in non-Euclidean geometry Elena Strzheletska Follow this and additional works at: https://scholarworks.lib.csusb.edu/etd-project Part of the Mathematics Commons Recommended Citation Strzheletska, Elena, "The Euler Line in non-Euclidean geometry" (2003). Theses Digitization Project. 2443. https://scholarworks.lib.csusb.edu/etd-project/2443 This Thesis is brought to you for free and open access by the John M. Pfau Library at CSUSB ScholarWorks. It has been accepted for inclusion in Theses Digitization Project by an authorized administrator of CSUSB ScholarWorks. For more information, please contact [email protected]. THE EULER LINE IN NON-EUCLIDEAN GEOMETRY A Thesis Presented to the Faculty of California State University, San Bernardino In Partial Fulfillment of the Requirements for the Degree Master of Arts in Mathematics by Elena Strzheletska December 2003 THE EULER LINE IN NON-EUCLIDEAN GEOMETRY A Thesis Presented to the Faculty of California State University, San Bernardino by Elena Strzheletska December 2003 Approved by: Robert Stein, Committee Member Susan Addington, Committee Member Peter Williams, Chair Terry Hallett, Department of Mathematics Graduate Coordinator Department of Mathematics ABSTRACT In Euclidean geometry, the circumcenter and the centroid of a nonequilateral triangle determine a line called the Euler line. The orthocenter of the triangle, the point of intersection of the altitudes, also belongs to this line. The main purpose of this thesis is to explore the conditions of the existence and the properties of the Euler line of a triangle in the hyperbolic plane. As a model of hyperbolic geometry Poincare’s conformal disk model is going to be used. For the algebraic representation of the geometric objects on the hyperbolic plane, we chose to use Hermitian matrices. In this paper we will find Hermitian representations of the three important points of a hyperbolic triangle (the circumcenter, centroid and orthocenter) as well as the Hermitian matrix that represents the Euler line, and discuss the conditions of their existence. We will complete our discussion by proving, that if the circumcenter and orthocenter of the hyperbolic triangle exist, the Euler line goes through the orthocenter if and only if the triangle is isosceles. iii ACKNOWLEDGEMENTS I would like to express my deepest appreciation to Dr. John Sarli for all of his time, assistance and input in the writing and completion of my thesis. During my graduate studies, I have found him to be challenging and encouraging, insightful and patient; his door has always been open. I would like to thank Dr. Robert Stein and Dr. Susan Addington for their work as members of my thesis committee. In addition, I would like to express my gratitude to all of my professors at California State University, San Bernardino as well as those at M.P. Drahomanov National Teacher’s Training University, who helped me develop a strong foundation in mathematics. Finally, I would like to thahk my husband for all of the assistance and support he has provided in the completion of my thesis. iv TABLE OF CONTENTS ABSTRACT................................ iii ACKNOWLEDGEMENTS.................................................................... iv LIST OF FIGURES........................................................................................................vii CHAPTER ONE: HISTORY AND BACKGROUND 1.1 History of Non-Euclidean Geometry.................... 1 1.2 Poincare's Version of Hyperbolic Geometry - Conformal Disk Model . 3 1.3 Hermitian Matrices..................................................................................... 5 1.4 Pencils of Circles........................................................................................ 8 1.5 Notations......................................................................................................8 CHAPTER TWO: A HYPERBOLIC TRIANGLE AND ITS CEVIANS 2.1 A Hyperbolic Triangle ............................................................................10 2.2 Perpendicular Bisectors ............................................................................12 2.3 Medians ....................................................................................................15 2.4 Altitudes ................................................................................................... 21 CHAPTER THREE: HERMITIAN REPRESENTATIONS 3.1 Circumcenter ............................................................................................25 3.2 Centroid ....................................................................................................28 3.3 Orthocenter................................................................................................32 CHAPTER FOUR: THE ANGLES OF PARALLELISM 4.1 Howto Define the Kind of Pencil ........................................................... 36 4.2 Circumparallelism ............................................................ 40 v 4.3 Orthoparallelism....................................................................................... 42 CHAPTER FIVE: EULER LINE AND ITS PROPERTIES 5.1 Euler Line ............................................................................ 47 5.2 When does the Orthocenter Belong to the Euler Line ........................... 53 CHAPTER SIX: SUMMARY ....................................................................................59 REFERENCES ............................................................................................................63 vi LIST OF FIGURES Figure 2.1........................................................ 14 Figure 2.2............................................................................................ 19 Figure 2.3........................................................................................................................24 Figure 4.1........................................................................................................................36 Figure 4.2......................................................................................................................40 Figure 4.3........................................................................................................................43 Figure 4.4........................................................................................................................46 vii CHAPTER ONE HISTORY AND BACKGROUND 1.1 History of Non-Euclidean Geometry The notion that there exists a geometry different from Euclidean geometry is relatively new for mathematics. It was not until the 19th century that "a new geometry contrary to Euclid’s" was found to be "logically possible." [12] Consider the Euclidean Parallel Postulate: “If a straight line falling on two straight lines makes the two interior angles on the same side of it taken together less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles together less than two right angles. ” [9] This proposition was stated in Euclid’s Elements as the fifth (parallel) postulate. Modern books on geometry more often use the equivalent of the postulate: Through a point not on a given line there is a unique straight line parallel to the given line. The discussion about the parallel postulate lasted over 2000 years. Thinking that the fifth postulate depended on the other four, scientists continually tried to prove it. Non-Euclidean geometry was born as soon as the question about the independence of Euclid’s parallel postulate was resolved. One of the world’s greatest mathematicians, Karl Friedrich Gauss (1777-1855), was probably the first (around 1813) to discover the existence of consistent geometry where Euclid’s postulate was replaced by the contrary statement. To such a geometry, Gauss gave the name "non-Euelidean". Gauss would eventually be considered the founder 1 of non-Euclidean geometry, even though the German mathematician, physicist and astronomer, with international reputation, was fearful of being misunderstood, and never published his discovery. Later on, Ferdinand Schweikart (1780-1859), professor of jurisprudence, basically came to the same conclusion as Gauss, but also failed to publish his views. ? , The great Russian mathematician Nicolay Lobachevskiy (1792-1896) approached the problem in a different way. To show the dependence of the parallel postulate on the other four axioms, he chose to use the proof by contradiction. His thought was the assumption that through a given pointnot on a given line could go at least two straight lines parallel to the given line will lead him to a contradiction. The longer he worked, the more he believed that there was no contradiction in the new, geometry he discovered. Having realized that geometry built on the assumption that Euclid’s parallel axiom was not true does not contain a contradiction, Lobachevsky began to develop a new non-Euclidean geometry, and made a great impact on its foundation. In 1830 Lobachevsky was the first to publish a work paper on non-Euclidean geometry, On the Principles of Geoimetiy . Even though the Hungarian mathematician Janos Bolyai (1802-1860) had actually made the same discovery of the new geometry before Lobachevsky, his was published later, in 1832, and only as.an appendix to his father, Farcas Bolyai’s, mathematical book. That is why, during a 20-year period, non-Euclidean geometry was independently
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