DOCSLIB.ORG

Problems and Solutions in Euclidean Geometry

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Problems and Solutions in Euclidean Geometry M. N. AREF AND WILLIAM WERNICK PROBLEMS & INSSOLUTIONS EUCLIDEAN GEOMETRY DOVER BOOKS ON MATHEMATICS HANDBOOK OF MATHEMATICAL FUNCTIONS: WITH FORMULAS, GRAPHS, AND MATHEMATICAL TABLES, Edited by Milton Abramowitz and Irene A. Stegun. (0-486-61272-4) ABSTRACT AND CONCRETE CATEGORIES: THE JOY OF CATS, Jiri Adamek, Horst Herrlich, George E. Strecker. (0-486-46934-4) NONSTANDARD METHODS IN STOCHASTIC ANALYSIS AND MATHEMATICAL PHYSICS, Sergio Albeverio, Jens Erik Fenstad, Raphael Hoegh-Krohn and Tom Lindstrom. (0-486-46899-2) MATHEMATICS: ITS CONTENT, METHODS AND MEANING, A. D. Aleksandrov, A. N. Kolmogo- rov, and M. A. Lavrent'ev. (0-486-40916-3) COLLEGE GEOMETRY: AN INTRODUCTION TO THE MODERN GEOMETRY OF THE TRIANGLE AND THE CIRCLE,NathanAltshiller-Court. (0-486-45805-9) THE WORKS OF ARCHIMEDES, Archimedes. Translated by Sir Thomas Heath. (0-486-42084-1) REAL VARIABLES WITH BASIC METRIC SPACE TOPOLOGY, Robert B. Ash. (0-486-47220-5) INTRODUCTION TO DIFFERENTIABLE MANIFOLDS, Louis Auslander and Robert E. MacKenzie. (0-486-47172-1) PROBLEM SOLVING THROUGH RECREATIONAL MATHEMATICS, Bonnie Averbach and Orin Chein. (0-486-40917-1) THEORY OF LINEAR OPERATIONS, Stefan Banach. Translated by F. Jellett. (0-486-46983-2) VECTOR CALCULUS,Peter Baxandall and HansLiebeck. (0-486-46620-5) INTRODUCTION TO VECTORS AND TENSORS: SECOND EDITION-TWO VOLUMES BOUND AS ONE, Ray M. Bowen and C: C. Wang. (0-486-46914-X) ADVANCED TRIGONOMETRY, C. V. Durell and A. Robson. (0-486-43229-7) FOURIER ANALYSIS IN SEVERAL COMPLEX VARIABLES, Leon Ehrenpreis. (0-486-44975-0) THE THIRTEEN BOOKS OF THE ELEMENTS, VOL. 1, Euclid. Edited by Thomas L. Heath. (0-486-60088-2) THE THIRTEEN BOOKS OF THE ELEMENTS, VOL. 2, Euclid. (0-486-60089-0) THE THIRTEEN BOOKS OF THE ELEMENTS, VOL. 3, Euclid. Edited by Thomas L. Heath. (0-486-60090-4) AN INTRODUCTION TO DIFFERENTIAL EQUATIONS AND THEIR APPLICATIONS, Stanley J. Farlow. (0-486-44595-X) PARTIAL DIFFERENTIAL EQUATIONS FOR SCIENTISTS AND ENGINEERS, Stanley J. Farlow. (0-486-67620-X) STOCHASTIC DIFFERENTIAL EQUATIONS AND APPLICATIONS, Avner Friedman. (0-486-45359-6) ADVANCED CALCULUS, Avner Friedman. (0-486-45795-8) POINT SET TOPOLOGY, Steven A. Gaal. (0-486-47222-1) DISCOVERING MATHEMATICS: THE ART OF INVESTIGATION, A. Gardiner. (0-486-45299-9) LATTICE THEORY: FIRST CONCEPTS AND DISTRIBUTIVE LATTICES, George Gratzer. (0-486-47173-X) (continued on back flap) PROBLEMSANDSOLUTIONS IN EUCLIDEAN GEOMETRY PROBLEMSANDSOLUTIONS IN EUCLIDEAN GEOMETRY M. N. AREF Alexandria University WILLIAM WERNICK City College, NewYork DOVER PUBLICATIONS, INC MINEOLA, NEW YORK Copyright Copyright © 1968 by Dover Publications, Inc. All rights reserved. Bibliographical Note This Dover edition, first published in 2010, is a reissue of a work originally published by Dover Publications, Inc, in 1968. International Standard Book Number ISBN-13: 978-0-486-47720-6 ISBN-10: 0-486-47720-7 Manufactured in the United States by Courier Corporation 47720701 www.doverpublications.com Dedicated to my father who has been always of lifelong inspiration and encouragement to me M.N.A. CONTENTS PREFACE xi SYMBOLS EMPLOYED IN THIS BOOK xiii CHAPTER 1 - Triangles and Polygons Theorems and corollaries I Solved problems 4 Miscellaneous exercises 25 CHAPTER 2 - Areas, Squares, and Rectangles Theorems and corollaries 33 Solved problems 35 Miscellaneous exercises 56 CHAPTER 3 - Circles and Tangency Theorems and corollaries 64 Solved problems 67 Miscellaneous exercises 95 CHAPTER 4 -Ratio and Proportion Theorems and corollaries 105 Solved problems io8 Miscellaneous exercises 133 CHAPTER 5 Loci and Transversals Definitions and theorems 144 Solved problems 145 Miscellaneous exercises 169 CHAPTER 6 -Geometry of Lines and Rays HARMONIC RANGES AND PENCILS Definitions and propositions 178 Solved problems 182 ISOGONAL AND SYMMEDIAN LINES-BROCARD POINTS Definitions and propositions 186 Solved problems 189 Miscellaneous exercises 191 ix X CONTENTS CHAPTER 7 -Geometry of the Circle SIMSON LINE Definitions and propositions Solved problems RADICAL AXIS-COAXAL CIRCLES Definitions and propositions Solved problems POLES AND POLARS Definitions and propositions Solved problems SIMILITUDE AND INVERSION Definitions and propositions Solved problems Miscellaneous exercises CHAPTER 8 -Space Geometry Theorems and corollaries Solved problems Miscellaneous exercises INDEX PREFACE This book is intended as a second course in Euclidean geometry. Its purpose is to give the reader facility in applying the theorems of Euclid to the solution of geometrical problems. Each chapter begins with a brief account of Euclid's theorems and corollaries for simpli- city of reference, then states and proves a number of important propositions. Chapters close with a section of miscellaneous problems of increasing complexity, selected from an immense mass of material for their usefulness and interest. Hints of solutions for a large number of problems are also included. Since this is not intended for the beginner in geometry, such familiar concepts as point, line, ray, segment, angle, and polygon are used freely without explicit definition. For the purpose of clarity rather than rigor the general term line is used to designate sometimes a ray, sometimes a segment, sometimes the length of a segment; the meaning intended will be clear from the context. Definitions of less familiar geometrical concepts, such as those of modern and space geometry, are added for clarity, and since the use of symbols might prove an additional difficulty to some readers, geometrical notation is introduced gradually in each chapter. January, 1968 M. N. AREF WILLIAM WERNICK xi SYMBOLS EMPLOYED IN THIS BOOK L angle Q circle O parallelogram o quadrilateral 0 rectangle square 0 triangle a = ba equals b a > ba is greater than b a < ba is less than b a b a is parallel to b a b a is perpendicular to b a / b a divided by b a : bthe ratio of a to b AB2 the square of the distance from A to B because therefore xiii CHAPTER 1 TRIANGLES AND POLYGONS Theorems and Corollaries LINES AND ANGLES I.I. If a straight line meets another straight line, the sum of the two adjacent angles is two right angles. COROLLARY 1. If any number of straight lines are drawn from a given point, the sum of the consecutive angles so formed is, four right angles. COROLLARY 2. If a straight line meets another straight line, the bisectors of the two adjacent angles are at right angles to one another. 1.2. If the sum of two adjacent angles is two right angles, their non- coincident arms are in the same straight line. 1.3. If two straight lines intersect, the vertically opposite angles are equal. 1.4. If a straight line cuts two other straight lines so as to make the alternate angles equal, the two straight lines are parallel. 1.5. If a straight line cuts two other straight lines so as to make: (i) two corresponding angles equal; or (ii) the interior angles, on the same side of the line, supplementary, the two straight lines are parallel. 1.6. If a straight line intersects two parallel straight lines, it makes: (i) alternate angles equal; (ii) corresponding angles equal; (iii) two interior angles on the same side of the line supplementary. COROLLARY. Two angles whose respective arms are either parallel or perpendicular to one another are either equal or supplementary. 1.7. Straight lines which are parallel to the same straight line are parallel to one another. TRIANGLES AND THEIR CONGRUENCE 1.8. If one side of a triangle is produced, (i) the exterior angle is equal to the sum of the interior non-adjacent angles; (ii) the sum of the three angles of a triangle is two right angles. COROLLARY 1. If two angles of one triangle are respectively equal to two angles of another triangle, the third angles are equal and the triangles are equiangular. COROLLARY 2. If one side of a triangle is produced, the exterior angle is greater than either of the interior non-adjacent angles. 2 PROBLEMS AND SOLUTIONS IN EUCLIDEAN GEOMETRY COROLLARY 3. The sum of any two angles of a triangle is less than two right angles. 1.9. If all the sides of a polygon of n sides are produced in order, the sum of the exterior angles is four right angles. COROLLARY. The sum of all the interior angles of a polygon of n sides is (2n - 4) right angles. 1.10. Two triangles are congruent if two sides and the included angle of one triangle are respectively equal to two sides and the included angle of the other. 1.11. Two triangles are congruent if two angles and a side of one triangle are respectively equal to two angles and the corresponding side of the other. 1.12. If two sides of a triangle are equal, the angles opposite to these sides are equal. COROLLARY 1. The bisector of the vertex angle of an isosceles triangle, (i) bisects the base; (ii) is perpendicular to the base. COROLLARY 2. An equilaterial triangle is also equiangular. 1.13. If two angles of a triangle are equal, the sides which subtend these angles are equal. COROLLARY. An equiangular triangle is also equilateral. 1.14. Two triangles are congruent if the three sides of one triangle are respectively equal to the three sides of the other. 1.15. Two right-angled triangles are congruent if the hypotenuse and a side of one are respectively equal to the hypotenuse and a side of the other. INEQUALITIES 1.16. If two sides of a triangle are unequal, the greater side has the greater angle opposite to it. 1.17. If two angles of a triangle are unequal, the greater angle has the greater side opposite to it. 1.18. Any two sides of a triangle are together greater than the third. 1.19. If two triangles have two sides of the one respectively equal to two sides of the other and the included angles unequal, then the third side of that with the greater angle is greater than the third side of the other.
Load more

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]
  • Algebra/Geometry/Trigonometry App Samples
    Algebra/Geometry/Trigonometry App Samples Holt McDougal Algebra 1 HMH Fuse: Algebra 1- HMH Fuse is the first core K-12 education solution developed exclusively for the iPad. The portability of a complete classroom course on an iPad enables students to learn in the classroom, on the bus, or at home—anytime, anywhere—with engaging content that provides an individually-tailored learning experience. Students and educators using HMH Fuse: will benefit from: •Instructional videos that teach or re-teach all key concepts •Math Motion is a step-by-step interactive demonstration that displays the process to solve complex equations •Homework Help provides at-home support for intricate problems by providing hints for each step in the solution •Vocabulary support throughout with links to a complete glossary that includes audio definitions •Tips, hints, and links that enable students to acquire the help they need to understand the lessons every step of the way •Quizzes that assess student’s skills before they begin a concept and at strategic points throughout the chapters. Instant, automatic grading of quizzes lets students know exactly how they have performed •Immediate assessment results sent to teachers so they can better differentiate instruction. Sample- Cost is Free, Complete App price- $59.99 Holt McDougal HMH Fuse: Geometry- Following our popular HMH Fuse: Algebra 1 app, HMH Fuse: Geometry is the newest offering in the HMH Fuse series. HMH Fuse: Geometry will allow you a sneak peek at the future of mobile geometry curriculum and includes a FREE sample chapter. HMH Fuse is the first core K-12 education solution developed exclusively for the iPad.
    [Show full text]
  • Canada Archives Canada Published Heritage Direction Du Branch Patrimoine De I'edition
    Rhetoric more geometrico in Proclus' Elements of Theology and Boethius' De Hebdomadibus A Thesis submitted in Candidacy for the Degree of Master of Arts in Philosophy Institute for Christian Studies Toronto, Ontario By Carlos R. Bovell November 2007 Library and Bibliotheque et 1*1 Archives Canada Archives Canada Published Heritage Direction du Branch Patrimoine de I'edition 395 Wellington Street 395, rue Wellington Ottawa ON K1A0N4 Ottawa ON K1A0N4 Canada Canada Your file Votre reference ISBN: 978-0-494-43117-7 Our file Notre reference ISBN: 978-0-494-43117-7 NOTICE: AVIS: The author has granted a non­ L'auteur a accorde une licence non exclusive exclusive license allowing Library permettant a la Bibliotheque et Archives and Archives Canada to reproduce, Canada de reproduire, publier, archiver, publish, archive, preserve, conserve, sauvegarder, conserver, transmettre au public communicate to the public by par telecommunication ou par Plntemet, prefer, telecommunication or on the Internet, distribuer et vendre des theses partout dans loan, distribute and sell theses le monde, a des fins commerciales ou autres, worldwide, for commercial or non­ sur support microforme, papier, electronique commercial purposes, in microform, et/ou autres formats. paper, electronic and/or any other formats. The author retains copyright L'auteur conserve la propriete du droit d'auteur ownership and moral rights in et des droits moraux qui protege cette these. this thesis. Neither the thesis Ni la these ni des extraits substantiels de nor substantial extracts from it celle-ci ne doivent etre imprimes ou autrement may be printed or otherwise reproduits sans son autorisation. reproduced without the author's permission.
    [Show full text]
  • Lesson 3: Rectangles Inscribed in Circles
    NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 3 M5 GEOMETRY Lesson 3: Rectangles Inscribed in Circles Student Outcomes . Inscribe a rectangle in a circle. Understand the symmetries of inscribed rectangles across a diameter. Lesson Notes Have students use a compass and straightedge to locate the center of the circle provided. If necessary, remind students of their work in Module 1 on constructing a perpendicular to a segment and of their work in Lesson 1 in this module on Thales’ theorem. Standards addressed with this lesson are G-C.A.2 and G-C.A.3. Students should be made aware that figures are not drawn to scale. Classwork Scaffolding: Opening Exercise (9 minutes) Display steps to construct a perpendicular line at a point. Students follow the steps provided and use a compass and straightedge to find the center of a circle. This exercise reminds students about constructions previously . Draw a segment through the studied that are needed in this lesson and later in this module. point, and, using a compass, mark a point equidistant on Opening Exercise each side of the point. Using only a compass and straightedge, find the location of the center of the circle below. Label the endpoints of the Follow the steps provided. segment 퐴 and 퐵. Draw chord 푨푩̅̅̅̅. Draw circle 퐴 with center 퐴 . Construct a chord perpendicular to 푨푩̅̅̅̅ at and radius ̅퐴퐵̅̅̅. endpoint 푩. Draw circle 퐵 with center 퐵 . Mark the point of intersection of the perpendicular chord and the circle as point and radius ̅퐵퐴̅̅̅. 푪. Label the points of intersection .
    [Show full text]
  • Squaring the Circle a Case Study in the History of Mathematics the Problem
    Squaring the Circle A Case Study in the History of Mathematics The Problem Using only a compass and straightedge, construct for any given circle, a square with the same area as the circle. The general problem of constructing a square with the same area as a given figure is known as the Quadrature of that figure. So, we seek a quadrature of the circle. The Answer It has been known since 1822 that the quadrature of a circle with straightedge and compass is impossible. Notes: First of all we are not saying that a square of equal area does not exist. If the circle has area A, then a square with side √A clearly has the same area. Secondly, we are not saying that a quadrature of a circle is impossible, since it is possible, but not under the restriction of using only a straightedge and compass. Precursors It has been written, in many places, that the quadrature problem appears in one of the earliest extant mathematical sources, the Rhind Papyrus (~ 1650 B.C.). This is not really an accurate statement. If one means by the “quadrature of the circle” simply a quadrature by any means, then one is just asking for the determination of the area of a circle. This problem does appear in the Rhind Papyrus, but I consider it as just a precursor to the construction problem we are examining. The Rhind Papyrus The papyrus was found in Thebes (Luxor) in the ruins of a small building near the Ramesseum.1 It was purchased in 1858 in Egypt by the Scottish Egyptologist A.
    [Show full text]
  • Circumference and Area of Circles 353
    Circumference and 7-1 Area of Circles MAIN IDEA Find the circumference Measure and record the distance d across the circular and area of circles. part of an object, such as a battery or a can, through its center. New Vocabulary circle Place the object on a piece of paper. Mark the point center where the object touches the paper on both the object radius and on the paper. chord diameter Carefully roll the object so that it makes one complete circumference rotation. Then mark the paper again. pi Finally, measure the distance C between the marks. Math Online glencoe.com • Extra Examples • Personal Tutor • Self-Check Quiz in. 1234 56 1. What distance does C represent? C 2. Find the ratio _ for this object. d 3. Repeat the steps above for at least two other circular objects and compare the ratios of C to d. What do you observe? 4. Graph the data you collected as ordered pairs, (d, C). Then describe the graph. A circle is a set of points in a plane center radius circumference that are the same distance from a (r) (C) given point in the plane, called the center. The segment from the center diameter to any point on the circle is called (d) the radius. A chord is any segment with both endpoints on the circle. The diameter of a circle is twice its radius or d 2r. A diameter is a chord that passes = through the center. It is the longest chord. The distance around the circle is called the circumference.
    [Show full text]
  • And Are Congruent Chords, So the Corresponding Arcs RS and ST Are Congruent
    9-3 Arcs and Chords ALGEBRA Find the value of x. 3. SOLUTION: 1. In the same circle or in congruent circles, two minor SOLUTION: arcs are congruent if and only if their corresponding Arc ST is a minor arc, so m(arc ST) is equal to the chords are congruent. Since m(arc AB) = m(arc CD) measure of its related central angle or 93. = 127, arc AB arc CD and . and are congruent chords, so the corresponding arcs RS and ST are congruent. m(arc RS) = m(arc ST) and by substitution, x = 93. ANSWER: 93 ANSWER: 3 In , JK = 10 and . Find each measure. Round to the nearest hundredth. 2. SOLUTION: Since HG = 4 and FG = 4, and are 4. congruent chords and the corresponding arcs HG and FG are congruent. SOLUTION: m(arc HG) = m(arc FG) = x Radius is perpendicular to chord . So, by Arc HG, arc GF, and arc FH are adjacent arcs that Theorem 10.3, bisects arc JKL. Therefore, m(arc form the circle, so the sum of their measures is 360. JL) = m(arc LK). By substitution, m(arc JL) = or 67. ANSWER: 67 ANSWER: 70 eSolutions Manual - Powered by Cognero Page 1 9-3 Arcs and Chords 5. PQ ALGEBRA Find the value of x. SOLUTION: Draw radius and create right triangle PJQ. PM = 6 and since all radii of a circle are congruent, PJ = 6. Since the radius is perpendicular to , bisects by Theorem 10.3. So, JQ = (10) or 5. 7. Use the Pythagorean Theorem to find PQ.
    [Show full text]
  • Cevians, Symmedians, and Excircles Cevian Cevian Triangle & Circle
    10/5/2011 Cevians, Symmedians, and Excircles MA 341 – Topics in Geometry Lecture 16 Cevian A cevian is a line segment which joins a vertex of a triangle with a point on the opposite side (or its extension). B cevian C A D 05-Oct-2011 MA 341 001 2 Cevian Triangle & Circle • Pick P in the interior of ∆ABC • Draw cevians from each vertex through P to the opposite side • Gives set of three intersecting cevians AA’, BB’, and CC’ with respect to that point. • The triangle ∆A’B’C’ is known as the cevian triangle of ∆ABC with respect to P • Circumcircle of ∆A’B’C’ is known as the evian circle with respect to P. 05-Oct-2011 MA 341 001 3 1 10/5/2011 Cevian circle Cevian triangle 05-Oct-2011 MA 341 001 4 Cevians In ∆ABC examples of cevians are: medians – cevian point = G perpendicular bisectors – cevian point = O angle bisectors – cevian point = I (incenter) altitudes – cevian point = H Ceva’s Theorem deals with concurrence of any set of cevians. 05-Oct-2011 MA 341 001 5 Gergonne Point In ∆ABC find the incircle and points of tangency of incircle with sides of ∆ABC. Known as contact triangle 05-Oct-2011 MA 341 001 6 2 10/5/2011 Gergonne Point These cevians are concurrent! Why? Recall that AE=AF, BD=BF, and CD=CE Ge 05-Oct-2011 MA 341 001 7 Gergonne Point The point is called the Gergonne point, Ge. Ge 05-Oct-2011 MA 341 001 8 Gergonne Point Draw lines parallel to sides of contact triangle through Ge.
    [Show full text]
  • Developing Creative Thinking in Mathematics: Trigonometry
    Secondary Mathematics Developing creative thinking in mathematics: trigonometry Teacher Education through School-based Support in India www.TESS-India.edu.in http://creativecommons.org/licenses/ TESS-India (Teacher Education through School-based Support) aims to improve the classroom practices of elementary and secondary teachers in India through the provision of Open Educational Resources (OERs) to support teachers in developing student-centred, participatory approaches. The TESS-India OERs provide teachers with a companion to the school textbook. They offer activities for teachers to try out in their classrooms with their students, together with case studies showing how other teachers have taught the topic and linked resources to support teachers in developing their lesson plans and subject knowledge. TESS-India OERs have been collaboratively written by Indian and international authors to address Indian curriculum and contexts and are available for online and print use (http://www.tess-india.edu.in/). The OERs are available in several versions, appropriate for each participating Indian state and users are invited to adapt and localise the OERs further to meet local needs and contexts. TESS-India is led by The Open University UK and funded by UK aid from the UK government. Video resources Some of the activities in this unit are accompanied by the following icon: . This indicates that you will find it helpful to view the TESS-India video resources for the specified pedagogic theme. The TESS-India video resources illustrate key pedagogic techniques in a range of classroom contexts in India. We hope they will inspire you to experiment with similar practices. They are intended to complement and enhance your experience of working through the text-based units, but are not integral to them should you be unable to access them.
    [Show full text]
  • Can One Design a Geometry Engine? on the (Un) Decidability of Affine
    Noname manuscript No. (will be inserted by the editor) Can one design a geometry engine? On the (un)decidability of certain affine Euclidean geometries Johann A. Makowsky Received: June 4, 2018/ Accepted: date Abstract We survey the status of decidabilty of the consequence relation in various ax- iomatizations of Euclidean geometry. We draw attention to a widely overlooked result by Martin Ziegler from 1980, which proves Tarski’s conjecture on the undecidability of finitely axiomatizable theories of fields. We elaborate on how to use Ziegler’s theorem to show that the consequence relations for the first order theory of the Hilbert plane and the Euclidean plane are undecidable. As new results we add: (A) The first order consequence relations for Wu’s orthogonal and metric geometries (Wen- Ts¨un Wu, 1984), and for the axiomatization of Origami geometry (J. Justin 1986, H. Huzita 1991) are undecidable. It was already known that the universal theory of Hilbert planes and Wu’s orthogonal geom- etry is decidable. We show here using elementary model theoretic tools that (B) the universal first order consequences of any geometric theory T of Pappian planes which is consistent with the analytic geometry of the reals is decidable. The techniques used were all known to experts in mathematical logic and geometry in the past but no detailed proofs are easily accessible for practitioners of symbolic computation or automated theorem proving. Keywords Euclidean Geometry · Automated Theorem Proving · Undecidability arXiv:1712.07474v3 [cs.SC] 1 Jun 2018 J.A. Makowsky Faculty of Computer Science, Technion–Israel Institute of Technology, Haifa, Israel E-mail: [email protected] 2 J.A.
    [Show full text]
  • The Isogonal Tripolar Conic
    Forum Geometricorum b Volume 1 (2001) 33–42. bbb FORUM GEOM The Isogonal Tripolar Conic Cyril F. Parry Abstract. In trilinear coordinates with respect to a given triangle ABC,we define the isogonal tripolar of a point P (p, q, r) to be the line p: pα+qβ+rγ = 0. We construct a unique conic Φ, called the isogonal tripolar conic, with respect to which p is the polar of P for all P . Although the conic is imaginary, it has a real center and real axes coinciding with the center and axes of the real orthic inconic. Since ABC is self-conjugate with respect to Φ, the imaginary conic is harmonically related to every circumconic and inconic of ABC. In particular, Φ is the reciprocal conic of the circumcircle and Steiner’s inscribed ellipse. We also construct an analogous isotomic tripolar conic Ψ by working with barycentric coordinates. 1. Trilinear coordinates For any point P in the plane ABC, we can locate the right projections of P on the sides of triangle ABC at P1, P2, P3 and measure the distances PP1, PP2 and PP3. If the distances are directed, i.e., measured positively in the direction of −→ −→ each vertex to the opposite side, we can identify the distances α =PP1, β =PP2, −→ γ =PP3 (Figure 1) such that aα + bβ + cγ =2 where a, b, c, are the side lengths and area of triangle ABC. This areal equation for all positions of P means that the ratio of the distances is sufficient to define the trilinear coordinates of P (α, β, γ) where α : β : γ = α : β : γ.
    [Show full text]
  • Understand the Principles and Properties of Axiomatic (Synthetic
    Michael Bonomi Understand the principles and properties of axiomatic (synthetic) geometries (0016) Euclidean Geometry: To understand this part of the CST I decided to start off with the geometry we know the most and that is Euclidean: − Euclidean geometry is a geometry that is based on axioms and postulates − Axioms are accepted assumptions without proofs − In Euclidean geometry there are 5 axioms which the rest of geometry is based on − Everybody had no problems with them except for the 5 axiom the parallel postulate − This axiom was that there is only one unique line through a point that is parallel to another line − Most of the geometry can be proven without the parallel postulate − If you do not assume this postulate, then you can only prove that the angle measurements of right triangle are ≤ 180° Hyperbolic Geometry: − We will look at the Poincare model − This model consists of points on the interior of a circle with a radius of one − The lines consist of arcs and intersect our circle at 90° − Angles are defined by angles between the tangent lines drawn between the curves at the point of intersection − If two lines do not intersect within the circle, then they are parallel − Two points on a line in hyperbolic geometry is a line segment − The angle measure of a triangle in hyperbolic geometry < 180° Projective Geometry: − This is the geometry that deals with projecting images from one plane to another this can be like projecting a shadow − This picture shows the basics of Projective geometry − The geometry does not preserve length
    [Show full text]