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Four-Fold Symmetry in Universal Triangle Geometry

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SCIENTIA MANU E T MENTE Four-Fold Symmetry in Universal Triangle Geometry 10th August, 2015 A thesis presented to The School of Mathematics and Statistics The University of New South Wales in fulfilment of the thesis requirement for the degree of Doctor of Philosophy by NGUYEN HONG LE Supervisor: A/PROF. NORMAN WILDBERGER ORIGINALITY STATEMENT 'I hereby declare that this submission is my own work and to the best of my knowledge it contains no materials previously published or written by another person, or substantial proportions of material which have been accepted for the award of any other degree or diploma at UNSW or any other educational institution, except where due acknowledgement is made in the thesis. Any contribution made to the research by others, with whom I have worked at UNSW or elsewhere, is explicitly acknowledged in the thesis. I also declare that the intellectual content of this thesis is the product of my own work, except to the extent that assistance from others in the project's design and conception or in style, presentation and linguistic expression is acknowledged.' ~ /-./1 n/J Signed ...... y.u. .r.w.~ .. ........................ Date .. .. 1.01. .~ /2..0 !.5. ............................... COPYRIGHT STATEMENT 'I hereby grant the University of New South Wales or its agents the right to archive and to make available my thesis or dissertation in whole or part in the University libraries in all forms of media, now or here after known, subject to the provisions of the Copyright Act 1968. I retain all proprietary rights, such as patent rights. I also retain the right to use in future works (such as articles or books) all or part of this thesis or dissertation. I also authorise University Microfilms to use the 350 word abstract of my thesis in Dissertation Abstract International (this is applicable to doctoral theses only). I have either used no substantial portions of copyright material in my thesis or I have obtained permission to use copyright material; where permission has not been granted I have applied/will apply for a partial restriction of the digital copy of my thesis or dissertation.' Signed ... J{m.t.XL~.... ...................................... Date ...... ... (.D../..~./.2.0.1).... ...................................... AUTHENTICITY STATEMENT 'I certify that the Library deposit digital copy is a direct equivalent of the final officially approved version of my thesis. No emendation of content has occurred and if there are any minor variations in formatting, they are the result of the conversion to digital format.' Signed ·····-~·- · ···· · ··················· · ·········· Date ........... I. _f?!.. g_ Ilc. 9.!!2. ................................... PLEASE TYPE THE UNIVERSITY OF NEW SOUTH WALES Thesis/Dissertation Sheet Surname or Family name: Le First name: Nguyen Other name/s:Hong Abbreviation for degree as given in the University calendar: PhD School: Mathematics and Statistics Faculty: Science Title: Four-Fold Symmetry in Universal Triangle Geometry Abstract 350 words maximum: (PLEASE TYPE) We develop a generalized triangle geometry, using an arbitrary bilinear form in an affine plane over a general field. By introducing standardized coordinates we find canonical forms for some basic centers and lines. Strong concurrencies formed by quadruples of lines from the lncenter hierarchy are investigated, including joins of corresponding lncenters, Gergonne, Nagel, Spieker points, Mittenpunkts , Bevan points, midpoints of lncenters and the Centroid, and so on. We identify the resulting centers in Kimberling's list. We also use a Kimberling 6-9-13 triangle to connect the triangle centers we have found with Kimberling's list. The diagrams are taken from Euclidean (blue) geometry and relativistic (green) geometry. Chromogeometry brings together planar Euclidean geometry, here called blue geometry, and two relativistic geometries, called red and green. We show that if a triangle has four blue lncenters and four red lncenters, then these eight points lie on a green circle, whose center is the green Orthocenter of the triangle, and similarly for the other colours. Tangents to the incenter circles yield interesting additional standard quadrangles and concurrencies. The proofs use the framework of rational trigonometry together with standard coordinates for triangle geometry, while a dilation argument allows us to extend the results also to Nagel and Spieker points. We investigate the concurrencies of triples of Euler lines (at four points) and the concurrencies of Euler lines and bilines (at 6 points), and prove the pleasant result that these concurrencies (10 points) lie on the Circumcircle. We generalize both the classical definition of the Schiffler point S=X21 and the approach of L. Emelyanov and T. Emelyanov to obtain in fact four Schiffler points. The four-fold Schiffler points give interesting results, for example the four ln-Schiffler lines form a standard quadrilateral s0 s1 s2s3 , and the four ln-Schiffler lines s0 , s1, s2, and s3 are respectively the tangents at 10 , l1, 12, 13 to the In-Circum conic which passes through the Circumcenter and four lncenters. Declaration relating to disposition of project thesis/dissertation I hereby grant to the University of New South Wales or its agents the right to archive and to make available my thesis or dissertation in whole or in part in the University libraries in all forms of media, now or here after known, subject to the provisions of the Copyright Act 1968. I retain all property rights, such as patent rights. I also retain the right to use in future works (such as articles or books) all or part of this thesis or dissertation. I also authorise University Microfilms to use the 350 word abstract of my thesis in Dissertation Abstracts International (this is applicable to doctoral theses only). ~!!!&. Witness ................... 10/8/2015 ... ..... ·· · ~Signature ·- · Date The University recognises that there may be exceptional circumstances requiring restrictions on copying or conditions on use. Requests for restriction for a period of up to 2 years must be made in writing. Requests for a longer period of restriction may be considered in exceptional circumstances and r uire the a roval of the Dean of Graduate Research. FOR OFFICE USE ONLY Date of completion of requirements for Award: THIS SHEET IS TO BE GLUED TO THE INSIDE FRONT COVER OF THE THESIS Contents Acknowledgements v 1 Introduction 1 1.1 Overview . 1 1.2 Background . 3 1.2.1 Rational Trigonometry and Universal Geometry . 3 1.2.2 Chromogeometry and Euler lines . 4 1.2.3 Triangle Geometry . 5 1.2.4 The classical Incenter hierarchy . 6 1.2.5 Quadrilaterals and quadrangles . 8 1.2.6 Construction of a quadrangle containing a given point with a given diagonal triangle . 9 1.2.7 Construction of a quadrilateral containing a given line with a given diagonal triangle . 10 1.3 Outline of contents . 11 2 Basic Universal Triangle Geometry 18 2.1 Affi ne structure and vectors . 18 2.2 Metrical structure: quadrance and spread . 19 2.3 Triple spread formula . 21 2.4 Altitudes and orthocenters . 22 2.5 Change of coordinates and an explicit example . 23 2.6 Bilines . 25 2.7 Standard coordinates and triangle geometry . 26 2.7.1 Basic affi ne objects in triangle geometry . 28 2.7.2 The Orthocenter hierarchy . 29 2.8 Transformations . 31 2.8.1 Dilations about the Centroid . 32 2.8.2 Reflections and Isogonal conjugates . 33 2.8.3 Isotomic conjugates . 34 2.9 A Kimberling 6-9-13 triangle . 34 3 Universal affi ne triangle geometry and four-fold incenter symmetry 37 3.1 Introduction . 37 3.2 Bilines and Incenters . 40 i CONTENTS ii 3.3 Strong concurrences . 43 3.3.1 Bevan points . 43 3.3.2 Sight Lines, Gergonne and Nagel points . 45 3.3.3 InMid lines and Mittenpunkts . 49 3.3.4 Spieker points . 51 3.3.5 Contact triangle and Weill point . 53 3.3.6 Summary Table . 56 4 Midpoints, four-fold Incenters symmetry and the Euler line 57 4.1 Midpoints of Incenters I and the Centroid G .............. 57 4.2 Midpoints of Incenters I and Circumcenter . 60 4.3 Midpoints of Incenters I and Orthocenter H . 63 4.4 Midpoints of Incenter I and nine-point center X5 . 66 4.5 Midpoints of Incenters I and De-Longchamp point . 70 4.5.1 Summary Table . 74 5 Incenter circles, chromogeometry, and the Omega triangle 75 5.1 Introduction . 75 5.2 The Incenter Circle theorem . 76 5.2.1 Equations of Incenter Circles . 80 5.2.2 Tangent lines of Incenter Circles . 81 5.3 Explicit examples . 87 5.3.1 An example over Q p30, p217, p741, p2470, p82 297 . 87 5.3.2 An example over ........................ 91 F13 5.4 Spieker circles and Nagel circles . 93 6 Euler lines and Schi­ er points 96 6.1 Incenter Euler lines . 96 6.1.1 Euler points P0,P1,P2 and P3 ................... 97 6.1.2 Bi-Euler points . 101 6.2 The classical Schi­ er point X21, and the Emelyanov’spoint of view . 106 6.3 Four-fold views of Schi­ er points . 107 6.3.1 Generalization of the classical Schi­ er point . 107 6.3.2 Generalization of L. Emelyanov and T. Emelyanova theorem . 110 6.3.3 In-Schi­ er lines and points, and standard quadrilaterals and quadrangles . 113 6.4 The In-Circum conic and its tangent lines . 119 6.5 Conclusion and future directions . 122 List of Figures 1.1 Euclidean Bilines and Incenters I0,I1,I2 and I3 of A1A2A3 . 3 1.2 Blue, red and green Euler lines of A1A2A3 ................ 5 1.3 Euclidean Incenter I0, excenters I1,I2,I3, Gergonne point G = X7 and Nagel point N = X8 ............................ 7 1.4 Euclidean Spieker center S and De Longchamps point X20 . 8 1.5 A quadrilateral and its opposite quadrangle .
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