DOCSLIB.ORG

The Dual Theorem Concerning Aubert Line

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
The Dual Theorem Concerning Aubert Line The Dual Theorem concerning Aubert Line Professor Ion Patrascu, National College "Buzeşti Brothers" Craiova - Romania Professor Florentin Smarandache, University of New Mexico, Gallup, USA In this article we introduce the concept of Bobillier transversal of a triangle with respect to a point in its plan; we prove the Aubert Theorem about the collinearity of the orthocenters in the triangles determined by the sides and the diagonals of a complete quadrilateral, and we obtain the Dual Theorem of this Theorem. Theorem 1 (E. Bobillier) Let 퐴퐵퐶 be a triangle and 푀 a point in the plane of the triangle so that the perpendiculars taken in 푀, and 푀퐴, 푀퐵, 푀퐶 respectively, intersect the sides 퐵퐶, 퐶퐴 and 퐴퐵 at 퐴푚, 퐵푚 and 퐶푚. Then the points 퐴푚, 퐵푚 and 퐶푚 are collinear. 퐴푚퐵 Proof We note that = 퐴푚퐶 aria (퐵푀퐴푚) (see Fig. 1). aria (퐶푀퐴푚) 1 Area (퐵푀퐴푚) = ∙ 퐵푀 ∙ 푀퐴푚 ∙ 2 sin(퐵푀퐴푚̂ ). 1 Area (퐶푀퐴푚) = ∙ 퐶푀 ∙ 푀퐴푚 ∙ 2 sin(퐶푀퐴푚̂ ). Since 1 3휋 푚(퐶푀퐴푚̂ ) = − 푚(퐴푀퐶̂ ), 2 it explains that sin(퐶푀퐴푚̂ ) = − cos(퐴푀퐶̂ ); 휋 sin(퐵푀퐴푚̂ ) = sin (퐴푀퐵̂ − ) = − cos(퐴푀퐵̂ ). 2 Therefore: 퐴푚퐵 푀퐵 ∙ cos(퐴푀퐵̂ ) = (1). 퐴푚퐶 푀퐶 ∙ cos(퐴푀퐶̂ ) In the same way, we find that: 퐵푚퐶 푀퐶 cos(퐵푀퐶̂ ) = ∙ (2); 퐵푚퐴 푀퐴 cos(퐴푀퐵̂ ) 퐶푚퐴 푀퐴 cos(퐴푀퐶̂ ) = ∙ (3). 퐶푚퐵 푀퐵 cos(퐵푀퐶̂ ) The relations (1), (2), (3), and the reciprocal Theorem of Menelaus lead to the collinearity of points 퐴푚, 퐵푚, 퐶푚. Note Bobillier's Theorem can be obtained – by converting the duality with respect to a circle – from the theorem relative to the concurrency of the heights of a triangle. Definition 1 It is called Bobillier transversal of a triangle 퐴퐵퐶 with respect to the point 푀 the line containing the intersections of the perpendiculars taken in 푀 on 퐴푀, 퐵푀, and 퐶푀 respectively, with sides 퐵퐶, CA and 퐴퐵. 2 Note The Bobillier transversal is not defined for any point 푀 in the plane of the triangle 퐴퐵퐶, for example, where 푀 is one of the vertices or the orthocenter 퐻 of the triangle. Definition 2 If 퐴퐵퐶퐷 is a convex quadrilateral and 퐸, 퐹 are the intersections of the lines 퐴퐵 and 퐶퐷, 퐵퐶 and 퐴퐷 respectively, we say that the figure 퐴퐵퐶퐷퐸퐹 is a complete quadrilateral. The complete quadrilateral sides are 퐴퐵, 퐵퐶, 퐶퐷, 퐷퐴, and 퐴퐶, 퐵퐷 and 퐸퐹 are diagonals. Theorem 2 (Newton-Gauss) The diagonals’ means of a complete quadrilateral are three collinear points. To prove Theorem 2, refer to [1]. Note It is called Newton-Gauss Line of a quadrilateral the line to which the diagonals’ means of a complete quadrilateral belong. Teorema 3 (Aubert) If 퐴퐵퐶퐷퐸퐹 is a complete quadrilateral, then the orthocenters 퐻1, 퐻2, 퐻3, 퐻4 of the triangles 퐴퐵퐹, 퐴퐸퐷, 퐵퐶퐸, and 퐶퐷퐹 respectively, are collinear points. Proof Let 퐴1, 퐵1, 퐹1 be the feet of the heights of the triangle 퐴퐵퐹 and 퐻1 its orthocenter (see Fig. 2). Considering the power of the point 퐻1 relative to the circle circumscribed to the triangle ABF, and given the triangle orthocenter’s property according to which its symmetrics to the triangle sides belong to the circumscribed circle, we find that: 퐻1퐴 ∙ 퐻1퐴1 = 퐻1퐵 ∙ 퐻1퐵1 = 퐻1퐹 ∙ 퐻1퐹1. 3 This relationship shows that the orthocenter 퐻1 has equal power with respect to the circles of diameters [퐴퐶], [퐵퐷], [퐸퐹]. As well, we establish that the orthocenters 퐻2, 퐻3, 퐻4 have equal powers to these circles. Since the circles of diameters [퐴퐶], [퐵퐷], [퐸퐹] have collinear centers (belonging to the Newton-Gauss line of the 퐴퐵퐶퐷퐸퐹 quadrilateral), it follows that the points 퐻1, 퐻2, 퐻3, 퐻4 belong to the radical axis of the circles, and they are, therefore, collinear points. Notes 1. It is called the Aubert Line or the line of the complete quadrilateral’s orthocenters the line to which the orthocenters 퐻1, 퐻2, 퐻3, 퐻4 belong. 4 2. The Aubert Line is perpendicular on the Newton-Gauss line of the quadrilateral (radical axis of two circles is perpendicular to their centers’ line). Theorem 4 (The Dual Theorem of the Theorem 3) If 퐴퐵퐶퐷 is a convex quadrilateral and 푀 is a point in its plane for which there are the Bobillier transversals of triangles 퐴퐵퐶, 퐵퐶퐷, 퐶퐷퐴 and 퐷퐴퐵; thereupon these transversals are concurrent. Proof Let us transform the configuration in Fig. 2, by duality with respect to a circle of center 푀. By the considered duality, the lines 푎, 푏, 푐, 푑, 푒 and 푓 correspond to the points 퐴, 퐵, 퐶, 퐷, 퐸, 퐹 (their polars). It is known that polars of collinear points are concurrent lines, therefore we have: 푎 ∩ 푏 ∩ 푒 = {퐴′}, 푏 ∩ 푐 ∩ 푓 = {퐵′}, 푐 ∩ 푑 ∩ 푒 = {퐶′}, 푑 ∩ 푓 ∩ 푎 = {퐷′}, 푎 ∩ 푐 = {퐸′}, 푏 ∩ 푑 = {퐹′}. Consequently, by applicable duality, the points 퐴′, 퐵′, 퐶′, 퐷′, 퐸′ and 퐹′ correspond to the straight lines 퐴퐵, 퐵퐶, 퐶퐷, 퐷퐴, 퐴퐶, 퐵퐷. To the orthocenter 퐻1 of the triangle 퐴퐸퐷, it corresponds, by duality, its ′ ′ ′ polar, which we denote 퐴1 – 퐵1 − 퐶1, and which is the Bobillier transversal of the triangle 퐴’퐶’퐷’ in relation to the point 푀. Indeed, the point 퐶’ corresponds to the line 퐸퐷 by duality; to the height from 퐴 of the triangle 퐴퐸퐷, also by ′ duality, it correspond its pole, which is the point 퐶1 located on 퐴’퐷’ such that ̂ ′ 0 푚(퐶′푀퐶1) = 90 . 5 ′ To the height from 퐸 of the triangle 퐴퐸퐷, it corresponds the point 퐵1 ∈ ̂ ′ 0 퐴′퐶′ such that 푚(퐷′푀퐵1) = 90 . ′ Also, to the height from 퐷, it corresponds 퐴1 ∈ 퐶′퐷’ on C such that ′ ′ 0 푚(퐴 푀퐴1) = 90 . To the orthocenter 퐻2 of the triangle 퐴퐵퐹, it will ′ ′ ′ correspond, by applicable duality, the Bobillier transversal 퐴2 − 퐵2 − 퐶2 in ′ ′ ′ the triangle 퐴 퐵 퐷 relative to the point 푀. To the orthocenter 퐻3 of the ′ triangle 퐵퐶퐸, it will correspond the Bobillier transversal 퐴3′ − 퐵 3 − 퐶3′ in the ′ ′ ′ triangle 퐴 퐵 퐶 relative to the point 푀, and to the orthocenter 퐻4 of the ′ ′ ′ triangle 퐶퐷퐹, it will correspond the transversal 퐴4 − 퐵4 − 퐶4 in the triangle 퐶′퐷′퐵′ relative to the point 푀. ′ ′ ′ ̅̅̅̅ The Bobillier transversals 퐴푖 − 퐵푖 − 퐶푖 , 푖 = 1,4 correspond to the collinear points 퐻푖, 푖 = 1̅̅,̅4̅. These transversals are concurrent in the pole of the line of the orthocenters towards the considered duality. It results that, given the quadrilateral 퐴’퐵’퐶’퐷’, the Bobillier transversals of the triangles 퐴’퐶’퐷’, 퐴’퐵’퐷’, 퐴’퐵’퐶’ and 퐶’퐷’퐵’ relative to the point 푀 are concurrent. References [1] Florentin Smarandache, Ion Patrascu: „The Geometry of Homological Triangles”. The Education Publisher Inc., Columbus, Ohio, USA – 2012. [2] Ion Patrascu, Florentin Smarandache: „Variance on Topics of Plane Geometry”. The Education Publisher Inc., Columbus, Ohio, USA – 2013. 6 .
Load more

Recommended publications
  • Projective Geometry: a Short Introduction
    Projective Geometry: A Short Introduction Lecture Notes Edmond Boyer Master MOSIG Introduction to Projective Geometry Contents 1 Introduction 2 1.1 Objective . .2 1.2 Historical Background . .3 1.3 Bibliography . .4 2 Projective Spaces 5 2.1 Definitions . .5 2.2 Properties . .8 2.3 The hyperplane at infinity . 12 3 The projective line 13 3.1 Introduction . 13 3.2 Projective transformation of P1 ................... 14 3.3 The cross-ratio . 14 4 The projective plane 17 4.1 Points and lines . 17 4.2 Line at infinity . 18 4.3 Homographies . 19 4.4 Conics . 20 4.5 Affine transformations . 22 4.6 Euclidean transformations . 22 4.7 Particular transformations . 24 4.8 Transformation hierarchy . 25 Grenoble Universities 1 Master MOSIG Introduction to Projective Geometry Chapter 1 Introduction 1.1 Objective The objective of this course is to give basic notions and intuitions on projective geometry. The interest of projective geometry arises in several visual comput- ing domains, in particular computer vision modelling and computer graphics. It provides a mathematical formalism to describe the geometry of cameras and the associated transformations, hence enabling the design of computational ap- proaches that manipulates 2D projections of 3D objects. In that respect, a fundamental aspect is the fact that objects at infinity can be represented and manipulated with projective geometry and this in contrast to the Euclidean geometry. This allows perspective deformations to be represented as projective transformations. Figure 1.1: Example of perspective deformation or 2D projective transforma- tion. Another argument is that Euclidean geometry is sometimes difficult to use in algorithms, with particular cases arising from non-generic situations (e.g.
    [Show full text]
  • Robot Vision: Projective Geometry
    Robot Vision: Projective Geometry Ass.Prof. Friedrich Fraundorfer SS 2018 1 Learning goals . Understand homogeneous coordinates . Understand points, line, plane parameters and interpret them geometrically . Understand point, line, plane interactions geometrically . Analytical calculations with lines, points and planes . Understand the difference between Euclidean and projective space . Understand the properties of parallel lines and planes in projective space . Understand the concept of the line and plane at infinity 2 Outline . 1D projective geometry . 2D projective geometry ▫ Homogeneous coordinates ▫ Points, Lines ▫ Duality . 3D projective geometry ▫ Points, Lines, Planes ▫ Duality ▫ Plane at infinity 3 Literature . Multiple View Geometry in Computer Vision. Richard Hartley and Andrew Zisserman. Cambridge University Press, March 2004. Mundy, J.L. and Zisserman, A., Geometric Invariance in Computer Vision, Appendix: Projective Geometry for Machine Vision, MIT Press, Cambridge, MA, 1992 . Available online: www.cs.cmu.edu/~ph/869/papers/zisser-mundy.pdf 4 Motivation – Image formation [Source: Charles Gunn] 5 Motivation – Parallel lines [Source: Flickr] 6 Motivation – Epipolar constraint X world point epipolar plane x x’ x‘TEx=0 C T C’ R 7 Euclidean geometry vs. projective geometry Definitions: . Geometry is the teaching of points, lines, planes and their relationships and properties (angles) . Geometries are defined based on invariances (what is changing if you transform a configuration of points, lines etc.) . Geometric transformations
    [Show full text]
  • Definition Concurrent Lines Are Lines That Intersect in a Single Point. 1. Theorem 128: the Perpendicular Bisectors of the Sides
    14.3 Notes Thursday, April 23, 2009 12:49 PM Definition 1. Concurrent lines are lines that intersect in a single point. j k m Theorem 128: The perpendicular bisectors of the sides of a triangle are concurrent at a point that is equidistant from the vertices of the triangle. This point is called the circumcenter of the triangle. D E F Theorem 129: The bisectors of the angles of a triangle are concurrent at a point that is equidistant from the sides of the triangle. This point is called the incenter of the triangle. A B Notes Page 1 C A B C Theorem 130: The lines containing the altitudes of a triangle are concurrent. This point is called the orthocenter of the triangle. A B C Theorem 131: The medians of a triangle are concurrent at a point that is 2/3 of the way from any vertex of the triangle to the midpoint of the opposite side. This point is called the centroid of the of the triangle. Example 1: Construct the incenter of ABC A B C Notes Page 2 14.4 Notes Friday, April 24, 2009 1:10 PM Examples 1-3 on page 670 1. Construct an angle whose measure is equal to 2A - B. A B 2. Construct the tangent to circle P at point A. P A 3. Construct a tangent to circle O from point P. Notes Page 3 3. Construct a tangent to circle O from point P. O P Notes Page 4 14.5 notes Tuesday, April 28, 2009 8:26 AM Constructions 9, 10, 11 Geometric mean Notes Page 5 14.6 Notes Tuesday, April 28, 2009 9:54 AM Construct: ABC, given {a, ha, B} a Ha B A b c B C a Notes Page 6 14.1 Notes Tuesday, April 28, 2009 10:01 AM Definition: A locus is a set consisting of all points, and only the points, that satisfy specific conditions.
    [Show full text]
  • Geometry: Neutral MATH 3120, Spring 2016 Many Theorems of Geometry Are True Regardless of Which Parallel Postulate Is Used
    Geometry: Neutral MATH 3120, Spring 2016 Many theorems of geometry are true regardless of which parallel postulate is used. A neutral geom- etry is one in which no parallel postulate exists, and the theorems of a netural geometry are true for Euclidean and (most) non-Euclidean geomteries. Spherical geometry is a special case of Non-Euclidean geometries where the great circles on the sphere are lines. This leads to spherical trigonometry where triangles have angle measure sums greater than 180◦. While this is a non-Euclidean geometry, spherical geometry develops along a separate path where the axioms and theorems of neutral geometry do not typically apply. The axioms and theorems of netural geometry apply to Euclidean and hyperbolic geometries. The theorems below can be proven using the SMSG axioms 1 through 15. In the SMSG axiom list, Axiom 16 is the Euclidean parallel postulate. A neutral geometry assumes only the first 15 axioms of the SMSG set. Notes on notation: The SMSG axioms refer to the length or measure of line segments and the measure of angles. Thus, we will use the notation AB to describe a line segment and AB to denote its length −−! −! or measure. We refer to the angle formed by AB and AC as \BAC (with vertex A) and denote its measure as m\BAC. 1 Lines and Angles Definitions: Congruence • Segments and Angles. Two segments (or angles) are congruent if and only if their measures are equal. • Polygons. Two polygons are congruent if and only if there exists a one-to-one correspondence between their vertices such that all their corresponding sides (line sgements) and all their corre- sponding angles are congruent.
    [Show full text]
  • The Generalization of Miquels Theorem
    THE GENERALIZATION OF MIQUEL’S THEOREM ANDERSON R. VARGAS Abstract. This papper aims to present and demonstrate Clifford’s version for a generalization of Miquel’s theorem with the use of Euclidean geometry arguments only. 1. Introduction At the end of his article, Clifford [1] gives some developments that generalize the three circles version of Miquel’s theorem and he does give a synthetic proof to this generalization using arguments of projective geometry. The series of propositions given by Clifford are in the following theorem: Theorem 1.1. (i) Given three straight lines, a circle may be drawn through their intersections. (ii) Given four straight lines, the four circles so determined meet in a point. (iii) Given five straight lines, the five points so found lie on a circle. (iv) Given six straight lines, the six circles so determined meet in a point. That can keep going on indefinitely, that is, if n ≥ 2, 2n straight lines determine 2n circles all meeting in a point, and for 2n +1 straight lines the 2n +1 points so found lie on the same circle. Remark 1.2. Note that in the set of given straight lines, there is neither a pair of parallel straight lines nor a subset with three straight lines that intersect in one point. That is being considered all along the work, without further ado. arXiv:1812.04175v1 [math.HO] 11 Dec 2018 In order to prove this generalization, we are going to use some theorems proposed by Miquel [3] and some basic lemmas about a bunch of circles and their intersections, and we will follow the idea proposed by Lebesgue[2] in a proof by induction.
    [Show full text]
  • Linear Features in Photogrammetry
    Linear Features in Photogrammetry by Ayman Habib Andinet Asmamaw Devin Kelley Manja May Report No. 450 Geodetic Science and Surveying Department of Civil and Environmental Engineering and Geodetic Science The Ohio State University Columbus, Ohio 43210-1275 January 2000 Linear Features in Photogrammetry By: Ayman Habib Andinet Asmamaw Devin Kelley Manja May Report No. 450 Geodetic Science and Surveying Department of Civil and Environmental Engineering and Geodetic Science The Ohio State University Columbus, Ohio 43210-1275 January, 2000 ABSTRACT This research addresses the task of including points as well as linear features in photogrammetric applications. Straight lines in object space can be utilized to perform aerial triangulation. Irregular linear features (natural lines) in object space can be utilized to perform single photo resection and automatic relative orientation. When working with primitives, it is important to develop appropriate representations in image and object space. These representations must accommodate for the perspective projection relating the two spaces. There are various options for representing linear features in the above applications. These options have been explored, and an optimal representation has been chosen. An aerial triangulation technique that utilizes points and straight lines for frame and linear array scanners has been implemented. For this task, the MSAT (Multi Sensor Aerial Triangulation) software, developed at the Ohio State University, has been extended to handle straight lines. The MSAT software accommodates for frame and linear array scanners. In this research, natural lines were utilized to perform single photo resection and automatic relative orientation. In single photo resection, the problem is approached with no knowledge of the correspondence of natural lines between image space and object space.
    [Show full text]
  • 1.3 Collinear, Betweenness, and Assumptions Date: Goals: How Are Diagrams Interpreted in the Study of Geometry
    1.3 Collinear, Betweenness, and Assumptions Date: Goals: How are diagrams interpreted in the study of geometry Collinear/Noncollinear: Three or more points that lie on the same line are collinear. Points that do not lie on the same line are noncollinear. Two points are always collinear. For a point to be between two other points, all three points must be collinear. Example 1: Use the diagram to the right to answer the following questions. a. Are points A, B, and C collinear? A Yes, B is between A and C. b. Is point B between A and D? B No, the three points are noncollinear. c. Name 3 noncollinear points. AEC, , and E D C d. Are points A and D collinear? Yes, two points are always collinear. The Triangle Inequality: For any three points, there are only two possibilies: 1. They are collinear. One point must be between the other two. Two of the distances must add to the third. 2. They are noncollinear. The three points will form a triangle. Any two of the lengths will add to be larger than the third. Example 2: If AB=12, BC=13, and AC=25, can we determine if the points are collinear? Yes, the lengths 12 and 13 add to 25, therefore the points are collinear. How to Interpret a Diagram: You Should Assume: You Should Not Assume: Straight lines and angles Right Angles Collinearity of points Congruent Segments Betweenness of points Congruent Angles Relative position of points Relative sizes of segments and angles Example 3: Use the diagram to the right to answer the following questions.
    [Show full text]
  • Advanced Euclidean Geometry
    Advanced Euclidean Geometry Paul Yiu Summer 2016 Department of Mathematics Florida Atlantic University July 18, 2016 Summer 2016 Contents 1 Some Basic Theorems 101 1.1 The Pythagorean Theorem . ............................ 101 1.2 Constructions of geometric mean . ........................ 104 1.3 The golden ratio . .......................... 106 1.3.1 The regular pentagon . ............................ 106 1.4 Basic construction principles ............................ 108 1.4.1 Perpendicular bisector locus . ....................... 108 1.4.2 Angle bisector locus . ............................ 109 1.4.3 Tangency of circles . ......................... 110 1.4.4 Construction of tangents of a circle . ............... 110 1.5 The intersecting chords theorem ........................... 112 1.6 Ptolemy’s theorem . ................................. 114 2 The laws of sines and cosines 115 2.1 The law of sines . ................................ 115 2.2 The orthocenter ................................... 116 2.3 The law of cosines .................................. 117 2.4 The centroid ..................................... 120 2.5 The angle bisector theorem . ............................ 121 2.5.1 The lengths of the bisectors . ........................ 121 2.6 The circle of Apollonius . ............................ 123 3 The tritangent circles 125 3.1 The incircle ..................................... 125 3.2 Euler’s formula . ................................ 128 3.3 Steiner’s porism ................................... 129 3.4 The excircles ....................................
    [Show full text]
  • Arxiv:2101.02592V1 [Math.HO] 6 Jan 2021 in His Seminal Paper [10]
    International Journal of Computer Discovered Mathematics (IJCDM) ISSN 2367-7775 ©IJCDM Volume 5, 2020, pp. 13{41 Received 6 August 2020. Published on-line 30 September 2020 web: http://www.journal-1.eu/ ©The Author(s) This article is published with open access1. Arrangement of Central Points on the Faces of a Tetrahedron Stanley Rabinowitz 545 Elm St Unit 1, Milford, New Hampshire 03055, USA e-mail: [email protected] web: http://www.StanleyRabinowitz.com/ Abstract. We systematically investigate properties of various triangle centers (such as orthocenter or incenter) located on the four faces of a tetrahedron. For each of six types of tetrahedra, we examine over 100 centers located on the four faces of the tetrahedron. Using a computer, we determine when any of 16 con- ditions occur (such as the four centers being coplanar). A typical result is: The lines from each vertex of a circumscriptible tetrahedron to the Gergonne points of the opposite face are concurrent. Keywords. triangle centers, tetrahedra, computer-discovered mathematics, Eu- clidean geometry. Mathematics Subject Classification (2020). 51M04, 51-08. 1. Introduction Over the centuries, many notable points have been found that are associated with an arbitrary triangle. Familiar examples include: the centroid, the circumcenter, the incenter, and the orthocenter. Of particular interest are those points that Clark Kimberling classifies as \triangle centers". He notes over 100 such points arXiv:2101.02592v1 [math.HO] 6 Jan 2021 in his seminal paper [10]. Given an arbitrary tetrahedron and a choice of triangle center (for example, the circumcenter), we may locate this triangle center in each face of the tetrahedron.
    [Show full text]
  • A Simple Way to Test for Collinearity in Spin Symmetry Broken Wave Functions: General Theory and Application to Generalized Hartree Fock
    A simple way to test for collinearity in spin symmetry broken wave functions: general theory and application to Generalized Hartree Fock David W. Small and Eric J. Sundstrom and Martin Head-Gordon Department of Chemistry, University of California, Berkeley, California 94720 and Chemical Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720 (Dated: February 17, 2015) Abstract We introduce a necessary and sufficient condition for an arbitrary wavefunction to be collinear, i.e. its spin is quantized along some axis. It may be used to obtain a cheap and simple computa- tional procedure to test for collinearity in electronic structure theory calculations. We adapt the procedure for Generalized Hartree Fock (GHF), and use it to study two dissociation pathways in CO2. For these dissociation processes, the GHF wave functions transform from low-spin Unre- stricted Hartree Fock (UHF) type states to noncollinear GHF states and on to high-spin UHF type states, phenomena that are succinctly illustrated by the constituents of the collinearity test. This complements earlier GHF work on this molecule. 1 I. INTRODUCTION Electronic structure practitioners have long relied on spin-unrestricted Hartree Fock (UHF) and Density Functional Theory (DFT) for open-shell and strongly correlated sys- tems. A key reason for this is that the latter's multiradical nature is partially accommo- dated by unrestricted methods: the paired spin-up and spin-down electrons are allowed to separate, while maintaining their spin alignment along a three-dimensional axis, i.e. their spin collinearity. But, considering the great potential for diversity in multitradical systems, it is not a stretch to concede that this approach is not always reasonable.
    [Show full text]
  • Dissertation Hyperovals, Laguerre Planes And
    DISSERTATION HYPEROVALS, LAGUERRE PLANES AND HEMISYSTEMS { AN APPROACH VIA SYMMETRY Submitted by Luke Bayens Department of Mathematics In partial fulfillment of the requirements For the Degree of Doctor of Philosophy Colorado State University Fort Collins, Colorado Spring 2013 Doctoral Committee: Advisor: Tim Penttila Jeff Achter Willem Bohm Chris Peterson ABSTRACT HYPEROVALS, LAGUERRE PLANES AND HEMISYSTEMS { AN APPROACH VIA SYMMETRY In 1872, Felix Klein proposed the idea that geometry was best thought of as the study of invariants of a group of transformations. This had a profound effect on the study of geometry, eventually elevating symmetry to a central role. This thesis embodies the spirit of Klein's Erlangen program in the modern context of finite geometries { we employ knowledge about finite classical groups to solve long-standing problems in the area. We first look at hyperovals in finite Desarguesian projective planes. In the last 25 years a number of infinite families have been constructed. The area has seen a lot of activity, motivated by links with flocks, generalized quadrangles, and Laguerre planes, amongst others. An important element in the study of hyperovals and their related objects has been the determination of their groups { indeed often the only way of distinguishing them has been via such a calculation. We compute the automorphism group of the family of ovals constructed by Cherowitzo in 1998, and also obtain general results about groups acting on hyperovals, including a classification of hyperovals with large automorphism groups. We then turn our attention to finite Laguerre planes. We characterize the Miquelian Laguerre planes as those admitting a group containing a non-trivial elation and acting tran- sitively on flags, with an additional hypothesis { a quasiprimitive action on circles for planes of odd order, and insolubility of the group for planes of even order.
    [Show full text]
  • Lemmas in Euclidean Geometry1 Yufei Zhao [email protected]
    IMO Training 2007 Lemmas in Euclidean Geometry Yufei Zhao Lemmas in Euclidean Geometry1 Yufei Zhao [email protected] 1. Construction of the symmedian. Let ABC be a triangle and Γ its circumcircle. Let the tangent to Γ at B and C meet at D. Then AD coincides with a symmedian of △ABC. (The symmedian is the reflection of the median across the angle bisector, all through the same vertex.) A A A O B C M B B F C E M' C P D D Q D We give three proofs. The first proof is a straightforward computation using Sine Law. The second proof uses similar triangles. The third proof uses projective geometry. First proof. Let the reflection of AD across the angle bisector of ∠BAC meet BC at M ′. Then ′ ′ sin ∠BAM ′ ′ BM AM sin ∠ABC sin ∠BAM sin ∠ABD sin ∠CAD sin ∠ABD CD AD ′ = ′ sin ∠CAM ′ = ∠ ∠ ′ = ∠ ∠ = = 1 M C AM sin ∠ACB sin ACD sin CAM sin ACD sin BAD AD BD Therefore, AM ′ is the median, and thus AD is the symmedian. Second proof. Let O be the circumcenter of ABC and let ω be the circle centered at D with radius DB. Let lines AB and AC meet ω at P and Q, respectively. Since ∠PBQ = ∠BQC + ∠BAC = 1 ∠ ∠ ◦ 2 ( BDC + DOC) = 90 , we see that PQ is a diameter of ω and hence passes through D. Since ∠ABC = ∠AQP and ∠ACB = ∠AP Q, we see that triangles ABC and AQP are similar. If M is the midpoint of BC, noting that D is the midpoint of QP , the similarity implies that ∠BAM = ∠QAD, from which the result follows.
    [Show full text]