Euclidean Geometry:An Introduction to Mathematical Work

Total Page：16

File Type：pdf, Size：1020Kb

Euclidean Geometry: An Introduction to Mathematical Work Math 3600 Spring 2019 Circles, Coming ’Round Again One of the most useful results about circles is Proposition III.20 which relates an inscribed angle in a circle to a central angle in that circle. Let us try to see what happens when the angle does not sit on the circumference of the circle. 10.1 Conjecture. Let ¡ be a circle with center O. Let X be a point in the interior of the circle, and suppose that two lines ` and m intersect at X so that ` meets ¡ at points A and A0 and m meets ¡ at B and B 0. Then twice angle AXB is congruent to angle AOB and angle A0OB 0 taken together. 10.2 Question. Consider the situation from the last conjecture, but instead assume that X lies outside ¡. What happens here? Formulate a conjecture. 10.3 Conjecture. If two chords of a circle subtend different acute angles at points of a circle, then the smaller angle belongs to the shorter chord. 10.4 Conjecture. If a triangle has two different angles, then the smaller angle has the longer angle bisector (measured from the vertex to the opposite side). 10.5 Conjecture (Steiner-Lehmus). If a triangle has two angle bisectors which are congruent (measured from the vertex to the opposite side), then the triangle is isosceles. 10.6 Conjecture. Let BC be a chord of circle C , let BCÙ be the arc of C which is bounded by B and C and does not contain the center of C . Let M be the midpoint of BCÙ. For a point A on the arc BCÙ, show that as A moves along the arc from B to M, the sums AB AC increase. Å The next theorem is very pretty, and is commonly attributed to Archimedes. 10.7 Conjecture (Archimedes’ Theorem of the Broken Chord). Let AB and BC be two chords of a circle C , where BC is greater than AB. (Such a con- ﬁguration is sometimes called a “broken chord.”) Let M be the midpoint of arc ABC and F the foot of the perpendicular from M to chord BC. Then F is the midpoint of the broken chord, that is, AB and BF taken together are congruent to FC..

Copy Link

Recommended publications

A Genetic Context for Understanding the Trigonometric Functions Danny Otero Xavier University, [email protected]
Ursinus College Digital Commons @ Ursinus College Transforming Instruction in Undergraduate Pre-calculus and Trigonometry Mathematics via Primary Historical Sources (TRIUMPHS) Spring 3-2017 A Genetic Context for Understanding the Trigonometric Functions Danny Otero Xavier University, [email protected] Follow this and additional works at: https://digitalcommons.ursinus.edu/triumphs_precalc Part of the Curriculum and Instruction Commons, Educational Methods Commons, Higher Education Commons, and the Science and Mathematics Education Commons Click here to let us know how access to this document benefits oy u. Recommended Citation Otero, Danny, "A Genetic Context for Understanding the Trigonometric Functions" (2017). Pre-calculus and Trigonometry. 1. https://digitalcommons.ursinus.edu/triumphs_precalc/1 This Course Materials is brought to you for free and open access by the Transforming Instruction in Undergraduate Mathematics via Primary Historical Sources (TRIUMPHS) at Digital Commons @ Ursinus College. It has been accepted for inclusion in Pre-calculus and Trigonometry by an authorized administrator of Digital Commons @ Ursinus College. For more information, please contact [email protected]. A Genetic Context for Understanding the Trigonometric Functions Daniel E. Otero∗ July 22, 2019 Trigonometry is concerned with the measurements of angles about a central point (or of arcs of circles centered at that point) and quantities, geometrical and otherwise, that depend on the sizes of such angles (or the lengths of the corresponding arcs). It is one of those subjects that has become a standard part of the toolbox of every scientist and applied mathematician. It is the goal of this project to impart to students some of the story of where and how its central ideas ﬁrst emerged, in an attempt to provide context for a modern study of this mathematical theory.
[Show full text]
Geometric Exercises in Paper Folding
MATH/STAT. T. SUNDARA ROW S Geometric Exercises in Paper Folding Edited and Revised by WOOSTER WOODRUFF BEMAN PROFESSOR OF MATHEMATICS IN THE UNIVERSITY OF MICHIGAK and DAVID EUGENE SMITH PROFESSOR OF MATHEMATICS IN TEACHERS 1 COLLEGE OF COLUMBIA UNIVERSITY WITH 87 ILLUSTRATIONS THIRD EDITION CHICAGO ::: LONDON THE OPEN COURT PUBLISHING COMPANY 1917 & XC? 4255 COPYRIGHT BY THE OPEN COURT PUBLISHING Co, 1901 PRINTED IN THE UNITED STATES OF AMERICA hn HATH EDITORS PREFACE. OUR attention was first attracted to Sundara Row s Geomet rical Exercises in Paper Folding by a reference in Klein s Vor- lesungen iiber ausgezucihlte Fragen der Elementargeometrie. An examination of the book, obtained after many vexatious delays, convinced us of its undoubted merits and of its probable value to American teachers and students of geometry. Accordingly we sought permission of the author to bring out an edition in this country, wnich permission was most generously granted. The purpose of the book is so fully set forth in the author s introduction that we need only to say that it is sure to prove of interest to every wide-awake teacher of geometry from the graded school to the college. The methods are so novel and the results so easily reached that they cannot fail to awaken enthusiasm. Our work as editors in this revision has been confined to some slight modifications of the proofs, some additions in the way of references, and the insertion of a considerable number of half-tone reproductions of actual photographs instead of the line-drawings of the original. W. W.
[Show full text]
Chapter 12, Lesson 1
i8o Chapter 12, Lesson 1 Chapter 12, Lesson 1 •14. They are perpendicular. 15. The longest chord is always a diameter; so Set I (pages 486-487) its length doesn't depend on the location of According to Matthys Levy and Mario Salvador! the point. The length of the shortest chord in their book titled \Nhy Buildings Fall Down depends on how far away from the center (Norton, 1992), the Romans built semicircular the point is; the greater the distance, the arches of the type illustrated in exercises 1 through shorter the chord. 5 that spanned as much as 100 feet. These arches Theorem 56. were used in the bridges of 50,000 miles of roads that the Romans built all the way from Baghdad 16. Perpendicular lines form right angles. to London! • 17. Two points determine a line. The conclusions of exercises 13 through 15 are supported by the fact that a diameter of a circle is •18. All radii of a circle are equal. its longest chord and by symmetry considerations. 19. Reflexive. As the second chord rotates away from the diameter in either direction about the point of 20. HL. intersection, it becomes progressively shorter. 21. Corresponding parts of congruent triangles are equal. Roman Arch. •22. If a line divides a line segment into two 1. Radii. equal parts, it bisects the segment. •2. Chords. Theorem 57. •3. A diameter. 23. If a line bisects a line segment, it divides the 4. Isosceles. Each has two equal sides because segment into two equal parts. the radii of a circle are equal.
[Show full text]
Constructibility
Appendix A Constructibility This book is dedicated to the synthetic (or axiomatic) approach to geometry, di- rectly inspired and motivated by the work of Greek geometers of antiquity. The Greek geometers achieved great work in geometry; but as we have seen, a few prob- lems resisted all their attempts (and all later attempts) at solution: circle squaring, trisecting the angle, duplicating the cube, constructing all regular polygons, to cite only the most popular ones. We conclude this book by proving why these various problems are insolvable via ruler and compass constructions. However, these proofs concerning ruler and compass constructions completely escape the scope of syn- thetic geometry which motivated them: the methods involved are highly algebraic and use theories such as that of fields and polynomials. This is why we these results are presented as appendices. Moreover, we shall freely use (with precise references) various algebraic results developed in [5], Trilogy II. We first review the theory of the minimal polynomial of an algebraic element in a field extension. Furthermore, since it will be essential for the applications, we prove the Eisenstein criterion for the irreducibility of (such) a (minimal) polynomial over the field of rational numbers. We then show how to associate a field with a given geometric configuration and we prove a formal criterion telling us—in terms of the associated fields—when one can pass from a given geometrical configuration to a more involved one, using only ruler and compass constructions. A.1 The Minimal Polynomial This section requires some familiarity with the theory of polynomials in one variable over a field K.
[Show full text]
Geometry, the Common Core, and Proof
Geometry, the Common Core, and Proof John T. Baldwin, Andreas Mueller Geometry, the Common Core, and Proof Overview From Geometry to Numbers John T. Baldwin, Andreas Mueller Proving the field axioms Interlude on Circles December 14, 2012 An Area function Side-splitter Pythagorean Theorem Irrational Numbers Outline Geometry, the Common Core, and Proof 1 Overview John T. Baldwin, 2 From Geometry to Numbers Andreas Mueller 3 Proving the field axioms Overview From Geometry to 4 Interlude on Circles Numbers Proving the 5 An Area function field axioms Interlude on Circles 6 Side-splitter An Area function 7 Pythagorean Theorem Side-splitter Pythagorean 8 Irrational Numbers Theorem Irrational Numbers Agenda Geometry, the Common Core, and Proof John T. Baldwin, Andreas 1 G-SRT4 { Context. Proving theorems about similarity Mueller 2 Proving that there is a field Overview 3 Areas of parallelograms and triangles From Geometry to Numbers 4 lunch/Discussion: Is it rational to fixate on the irrational? Proving the 5 Pythagoras, similarity and area field axioms Interlude on 6 reprise irrational numbers and Golden ratio Circles 7 resolving the worries about irrationals An Area function Side-splitter Pythagorean Theorem Irrational Numbers Logistics Geometry, the Common Core, and Proof John T. Baldwin, Andreas Mueller Overview From Geometry to Numbers Proving the field axioms Interlude on Circles An Area function Side-splitter Pythagorean Theorem Irrational Numbers Common Core Geometry, the Common Core, and Proof John T. Baldwin, G-SRT: Prove theorems involving similarity Andreas Mueller 4. Prove theorems about triangles. Theorems include: a line Overview parallel to one side of a triangle divides the other two From proportionally, and conversely; the Pythagorean Theorem Geometry to Numbers proved using triangle similarity.
[Show full text]
Theorems of Euclidean Geometry Through Calculus
Theorems of Euclidean Geometry through Calculus Martin Buysse∗ Facultéd'architecture, d'ingénieriearchitecturale, d'urbanisme { LOCI, UCLouvain We re-derive Thales, Pythagoras, Apollonius, Stewart, Heron, al Kashi, de Gua, Terquem, Ptolemy, Brahmagupta and Euler's theorems as well as the inscribed angle theorem, the law of sines, the circumradius, inradius and some angle bisector formulae, by assuming the existence of an unknown relation between the geometric quantities at stake, observing how the relation behaves under small deviations of those quantities, and naturally establishing differential equations that we integrate out. Applying the general solution to some specific situation gives a particular solution corresponding to the expected theorem. We also establish an equivalence between a polynomial equation and a set of partial differential equations. We finally comment on a differential equation which arises after a small scale transformation and should concern all relations between metric quantities. I. THALES OF MILETUS II. PYTHAGORAS OF SAMOS Imagine that Newton was born before Thales. When If he was born before Thales, Newton was born before considering a triangle with two sides of lengths x and Pythagoras too, so that we do not have to make any y, he could have fantasized about moving the third further unlikely hypothesis. Imagine that driven by his side parallel to itself and thought: "Well, I am not success in suspecting the existence of a Greek theorem, an ancient Greek geometer but I am rather good in he moved to consider a right triangle of legs of lengths x calculus and I feel there might be some connection and y and of hypotenuse of length z.
[Show full text]
Chapter 4 Euclidean Geometry
Chapter 4 Euclidean Geometry Based on previous 15 axioms, The parallel postulate for Euclidean geometry is added in this chapter. 4.1 Euclidean Parallelism, Existence of Rectangles De¯nition 4.1 Two distinct lines ` and m are said to be parallel ( and we write `km) i® they lie in the same plane and do not meet. Terminologies: 1. Transversal: a line intersecting two other lines. 2. Alternate interior angles 3. Corresponding angles 4. Interior angles on the same side of transversal 56 Yi Wang Chapter 4. Euclidean Geometry 57 Theorem 4.2 (Parallelism in absolute geometry) If two lines in the same plane are cut by a transversal to that a pair of alternate interior angles are congruent, the lines are parallel. Remark: Although this theorem involves parallel lines, it does not use the parallel postulate and is valid in absolute geometry. Proof: Assume to the contrary that the two lines meet, then use Exterior Angle Inequality to draw a contradiction. 2 The converse of above theorem is the Euclidean Parallel Postulate. Euclid's Fifth Postulate of Parallels If two lines in the same plane are cut by a transversal so that the sum of the measures of a pair of interior angles on the same side of the transversal is less than 180, the lines will meet on that side of the transversal. In e®ect, this says If m\1 + m\2 6= 180; then ` is not parallel to m Yi Wang Chapter 4. Euclidean Geometry 58 It's contrapositive is If `km; then m\1 + m\2 = 180( or m\2 = m\3): Three possible notions of parallelism Consider in a single ¯xed plane a line ` and a point P not on it.
[Show full text]
Complements to Classic Topics of Circles Geometry
Ion Patrascu | Florentin Smarandache Complements to Classic Topics of Circles Geometry Pons Editions Brussels | 2016 Complements to Classic Topics of Circles Geometry Ion Patrascu | Florentin Smarandache Complements to Classic Topics of Circles Geometry 1 Ion Patrascu, Florentin Smarandache In the memory of the first author's father Mihail Patrascu and the second author's mother Maria (Marioara) Smarandache, recently passed to eternity... 2 Complements to Classic Topics of Circles Geometry Ion Patrascu | Florentin Smarandache Complements to Classic Topics of Circles Geometry Pons Editions Brussels | 2016 3 Ion Patrascu, Florentin Smarandache © 2016 Ion Patrascu & Florentin Smarandache All rights reserved. This book is protected by copyright. No part of this book may be reproduced in any form or by any means, including photocopying or using any information storage and retrieval system without written permission from the copyright owners. ISBN 978-1-59973-465-1 4 Complements to Classic Topics of Circles Geometry Contents Introduction ....................................................... 15 Lemoine’s Circles ............................................... 17 1st Theorem. ........................................................... 17 Proof. ................................................................. 17 2nd Theorem. ......................................................... 19 Proof. ................................................................ 19 Remark. ............................................................ 21 References.
[Show full text]
Complements to Classic Topics of Circles Geometry
University of New Mexico UNM Digital Repository Mathematics and Statistics Faculty and Staff Publications Academic Department Resources 2016 Complements to Classic Topics of Circles Geometry Florentin Smarandache University of New Mexico, [email protected] Ion Patrascu Follow this and additional works at: https://digitalrepository.unm.edu/math_fsp Part of the Algebra Commons, Algebraic Geometry Commons, Applied Mathematics Commons, Geometry and Topology Commons, and the Other Mathematics Commons Recommended Citation Smarandache, Florentin and Ion Patrascu. "Complements to Classic Topics of Circles Geometry." (2016). https://digitalrepository.unm.edu/math_fsp/264 This Book is brought to you for free and open access by the Academic Department Resources at UNM Digital Repository. It has been accepted for inclusion in Mathematics and Statistics Faculty and Staff Publications by an authorized administrator of UNM Digital Repository. For more information, please contact [email protected], [email protected], [email protected]. Ion Patrascu | Florentin Smarandache Complements to Classic Topics of Circles Geometry Pons Editions Brussels | 2016 Complements to Classic Topics of Circles Geometry Ion Patrascu | Florentin Smarandache Complements to Classic Topics of Circles Geometry 1 Ion Patrascu, Florentin Smarandache In the memory of the first author's father Mihail Patrascu and the second author's mother Maria (Marioara) Smarandache, recently passed to eternity... 2 Complements to Classic Topics of Circles Geometry Ion Patrascu | Florentin Smarandache Complements to Classic Topics of Circles Geometry Pons Editions Brussels | 2016 3 Ion Patrascu, Florentin Smarandache © 2016 Ion Patrascu & Florentin Smarandache All rights reserved. This book is protected by copyright. No part of this book may be reproduced in any form or by any means, including photocopying or using any information storage and retrieval system without written permission from the copyright owners.
[Show full text]
The Final Eucleidian Solution for the Trisection of Random Acute Angle First Ever Presentation in the History of Geometry
Volume 6, Issue 5, May – 2021 International Journal of Innovative Science and Research Technology ISSN No:-2456-2165 The Final Eucleidian Solution for the Trisection of Random Acute Angle First Ever Presentation in the History of Geometry Author: Giorgios (Gio) Vassiliou Visual artist - Researcher - Founder and Inventor of Transcendental Surrealism in visual arts - Architect Salamina Island - Greece (Dedicated to the memory of my beloved parents Spiros & Stavroula.) Abstract:- A brief introduction about the Eucleidian As we have already mentioned, the limitations of the solution’s “impossibility” of the trisection problem… past becomes the offspring for the future! The historic problem of Eucleidian trisection for a This future has just arrived as present time and random acute angle, was involved humanity, from 6th after so many centuries since antiquity, the Eucleidian century BC without any interruption untill the late 19th solution for angle's trisection is a fact. century, by not finding a satisfied solution according to Eucleidian Geometry. 2500 years of "impossibity" have just ended and by doing so, this fact arrises new hopes to scientific The trisection is an equal achievement of making researcher, amateurs or professionals, for even greater the "impossible" into possible, because there is a huge list accomplishment in the future of humanity. of names, that includes the greatest genius mathematicians of all times, such as: Hippocrates of I. ARCHIMEDES’ NON-EUCLEIDIAN METHOD Chios, Archimedes, Nicomedes, Descartes, Pascal and OF TRISECTION Lagrance that all failed to give a satisfied solution according to Eucleidian Geometry! (ARTICLE TAKEN FROM BRITANNICA ENCYCLOPEDIA) Never the less non-Eucleidian solutions have been Written by: J.L.
[Show full text]
Ptolemy Finds High Noon in Chords of Circles
A Genetic Context for Understanding the Trigonometric Functions Episode 3: Ptolemy Finds High Noon in Chords of Circles Daniel E. Otero∗ December 23, 2020 Trigonometry is concerned with the measurements of angles about a central point (or of arcs of circles centered at that point) and quantities, geometrical and otherwise, which depend on the sizes of such angles (or the lengths of the corresponding arcs). It is one of those subjects that has become a standard part of the toolbox of every scientist and applied mathematician. Today an introduction to trigonometry is normally part of the mathematical preparation for the study of calculus and other forms of mathematical analysis, as the trigonometric functions make common appearances in applications of mathematics to the sciences, wherever the mathematical description of cyclical phenomena is needed. This project is one of a series of curricular units that tell some of the story of where and how the central ideas of this subject first emerged, in an attempt to provide context for the study of this important mathematical theory. Readers who work through the entire collection of units will encounter six milestones in the history of the development of trigonometry. In this unit, we encounter a brief selection from Claudius Ptolemy’s Almagest (second century, CE), in which the author shows how a table of chords can be used to monitor the motion of the Sun in the daytime sky for the purpose of telling the time of day. 1 An (Extremely) Brief History of Time We in the 21st century generally take for granted the omnipresence of synchronized clocks in our phones that monitor the rhythms of our lives.
[Show full text]
The Problem of Apollonius As an Opportunity for Teaching Students to Use Reflections and Rotations to Solve Geometry Problems Via Ga
THE PROBLEM OF APOLLONIUS AS AN OPPORTUNITY FOR TEACHING STUDENTS TO USE REFLECTIONS AND ROTATIONS TO SOLVE GEOMETRY PROBLEMS VIA GA James A. Smith [email protected] ABSTRACT. The beautiful Problem of Apollonius from classical geometry (“Construct all of the circles that are tangent, simultaneously, to three given coplanar circles”) does not appear to have been solved previously by vector methods. It is solved here via GA to show students how they can make use of GA’s capabilities for expressing and manipulating rotations and reflections. As Viete` did when deriving his ruler-and-compass solution, we first transform the problem by shrinking one of the given circles to a point. In the course of solving the transformed problem, guidance is provided to help students “see” geometric content in GA terms. Examples of the guidance that is given include (1) recognizing and formulating useful reflections and rotations that are present in diagrams; (2) using postulates on the equality of multivectors to obtain solvable equations; and (3) recognizing complex algebraic expressions that reduce to simple rotations of multivectors. 1. INTRODUCTION “Stop wasting your time on GA! Nobody at Arizona State University [David Hestenes’s institution] uses it—students consider it too hard”. This opinion, received in an email from one of Professor Hestenes’s colleagues, might well disconcert both the uninitiated whom we hope to introduce to GA, and those of us who recognize the need to begin teaching GA at the high-school level ([1]–[3]). For some time to come, the high-school teachers whom we’ll ask to ”sell” GA to parents, administrators, and students will themselves be largely self-taught, and will at the same time need to learn how to teach GA to their students.
[Show full text]