DOCSLIB.ORG

Configurations on Centers of Bankoff Circles 11

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Configurations on Centers of Bankoff Circles 11 CONFIGURATIONS ON CENTERS OF BANKOFF CIRCLES ZVONKO CERINˇ Abstract. We study configurations built from centers of Bankoff circles of arbelos erected on sides of a given triangle or on sides of various related triangles. 1. Introduction For points X and Y in the plane and a positive real number λ, let Z be the point on the segment XY such that XZ : ZY = λ and let ζ = ζ(X; Y; λ) be the figure formed by three mj utuallyj j tangenj t semicircles σ, σ1, and σ2 on the same side of segments XY , XZ, and ZY respectively. Let S, S1, S2 be centers of σ, σ1, σ2. Let W denote the intersection of σ with the perpendicular to XY at the point Z. The figure ζ is called the arbelos or the shoemaker's knife (see Fig. 1). σ σ1 σ2 PSfrag replacements X S1 S Z S2 Y XZ Figure 1. The arbelos ζ = ζ(X; Y; λ), where λ = jZY j . j j It has been the subject of studies since Greek times when Archimedes proved the existence of the circles !1 = !1(ζ) and !2 = !2(ζ) of equal radius such that !1 touches σ, σ1, and ZW while !2 touches σ, σ2, and ZW (see Fig. 2). 1991 Mathematics Subject Classification. Primary 51N20, 51M04, Secondary 14A25, 14Q05. Key words and phrases. arbelos, Bankoff circle, triangle, central point, Brocard triangle, homologic. 1 2 ZVONKO CERINˇ σ W !1 σ1 !2 W1 PSfrag replacements W2 σ2 X S1 S Z S2 Y Figure 2. The Archimedean circles !1 and !2 together. In [1] Bankoff discovered that the circumcircle !3 = !3(ζ) of the triangle MNZ has the same radius as the Archimedean twin circles !1 and ! , where M = σ σ and N = σ σ and σ is the circle that 2 3 \ 1 3 \ 2 3 touches σ from inside and σ1 and σ2 from outside (see Fig. 3). σ σ3 σ2 PSfrag replacements S3 σ1 M N W3 !3 X S1 Z S S2 Y Figure 3. The Bankoff circle !3 and the circle σ3. The purpose of this paper is to study triangles on centers of the Bankoff circles of arbelos either on sides of an arbitrary triangle ABC or on sides of some of its associated triangles. More precisely, our first goal is to prove the following theorem (see Fig. 4). All other results in this paper are similar. Let τ denote the base triangle ABC. Then τb is a short notation for its first Brocard triangle AbBbCb. Its vertices are the orthogonal projections of the symmedian point K onto the perpendicular bisectors of sides (see [6]). Let ζa = ζa(τ) = ζ(B; C; λ), ζb = ζb(τ) = ζ(C; A; λ) CONFIGURATIONS ON CENTERS OF BANKOFF CIRCLES 3 a b c and ζc = ζc(τ) = ζ(A; B; λ). Let W , W , W denote centers of !3(ζa), !3(ζb), !3(ζc), respectively. Recall that triangles τ and θ = XY Z are homologic and we write τ θ provided lines AX, BY , and CZ are concurrent. The point Q in whichi h they concur is their homology center and the line ` containing intersections of pairs of lines (BC; Y Z), (CA; ZX), and (AB; XY ) is their homology axis. In this situation we also use the notation τ Q; ` θ where ` and/or Q can be omitted. For Q and ` sometimes we useh τ; iθ and τ; θ . In stead of homologic, homology center, and homologyh i axis hhmanyiiauthors use perspective, perspector, and perspectrix. Theorem 1. For every λ 0 the triangle W aW bW c on the centers of Bankoff circles of arbelos ≥on sides of a triangle τ is homologic to its Brocard triangle τb. The centre of this homology lies on the Brocard circle of τ (i.e., on the circumcircle of τb). The triangles τ, τb, and W aW bW c have the same centroid. C W a Bb b W Ab Cb A B W c PSfrag replacements Figure 4. The triangle W aW bW c and the first Brocard triangle AbBbCb are homologic. 2. Bankoff circle !3 x+y a+b Let X(x; a) and Y (y; b). Then S( 2 ; 2 ) is the midpoint of the XZ x+λ y a+λ b j j segment XY . Since ZY = λ, the point Z is λ+1 ; λ+1 . More- j j (λ+2)x+λ y (λ+2) a+λ b over, semicircles σ1 and σ2 have centres at 2(λ+1) ; 2(λ+1) and x+(2 λ+1) y a+(2 λ+1) b 2(λ+1) ; 2(λ+1) (the midpoints of segments XZ and ZY ). 4 ZVONKO CERINˇ Our goal now is to find the center and the radius of the unique circle !3 in the arbelos ζ which touches the semicircles σ1 and σ2 from outside and the semicircle σ from inside. Let its centre be the point S3(p; q) and the radius a positive real number %. Let A = x y and B = a b. − − Since σ3 touches σ from inside, the distance S3S is equal to the 1 p 2 2 j j difference 2 A + B % of their radii. This condition leads to the relation − 2 2 x + y 2 a + b pA2 + B2 (1) p + q = % : − 2 − 2 2 − Since σ3 touches σ1 and σ2 from outside, the distances S3S1 and S3S2 λ pA2+B2 pA2+B2 j j j j are equal to the sums % + 2 (λ+1) and % + 2 (λ+1) of their radii. It follows that 2 2 2 (λ + 2) x + λ y (λ + 2) a + λ b λ pA2 + B2 (2) p + q = % + ; − 2 (λ + 1) − 2 (λ + 1) 2 (λ + 1) 2 2 2 x + (2 λ + 1) y a + (2 λ + 1) b pA2 + B2 (3) p + q = % + : − 2 (λ + 1) − 2 (λ + 1) 2 (λ + 1) These equations have two solutions in p, q, and %. We shall use the λ pA2+B2 solution % = 2 (λ2+λ+1) and (λ + 2) x + λ (2 λ + 1) y 2 λ (a b) (λ + 2) a + λ (2 λ + 1) b + 2 λ (x y) S3 − − ; − : 2 (λ2 + λ + 1) 2 (λ2 + λ + 1) It is now easy to find the coordinates of the points M and N because we know that they divide the segments S1S3 and S2S3 in the ratio of the radii of the circles σ1, σ3 and σ2, σ3. Hence, (λ + 2) x + λ (λ + 1) y λ (a b) (λ + 2) a + λ (λ + 1) b + λ (x y) M − − ; − ; λ2 + 2 λ + 2 λ2 + 2 λ + 2 (λ + 1) x + λ (2 λ + 1) y λ (a b) (λ + 1) a + λ (2 λ + 1) b + λ (x y) N − − ; − : 2 λ2 + 2 λ + 1 2 λ2 + 2 λ + 1 λ pA2+B2 The circumcircle !3 of the triangle MNZ has the radius 2 (λ+1)2 (of the Archimedean twin circles) and its center is at the point x + λ y λ (a b) a + λ b λ (x y) W − ; + − : 3 λ + 1 − 2 (λ + 1)2 λ + 1 2 (λ + 1)2 Remark 1. It is possible to prove the above by inversion (see [9, p. 224]). But, since in this paper we use analytic approach, we need functions that describe coordinates of the center of the Bankoff's cir- cle. Behind the scene we work in rectangular coordinates with compli- cated expressions and then transfer the results into trilinear coordinates where the expressions are often much simpler. CONFIGURATIONS ON CENTERS OF BANKOFF CIRCLES 5 3. Proof of Theorem 1 In our proofs we shall use trilinear coordinates. Recall that the actual trilinear coordinates of a point P with respect to the triangle ABC are signed distances f, g, and h of P from the lines BC, CA, and AB. We shall regard P as lying on the positive side of BC if P lies on the same side of BC as A. Similarly, we shall regard P as lying on the positive side of CA if it lies on the same side of CA as B, and similarly with regard to the side AB. Ordered triples x : y : z of real numbers proportional to (f; g; h) (that is such that x = µf, y = µg, and z = µh, for some real number µ different from zero) are called trilinear coordinates of P . The advantage of their use is that a high degree of symmetry is present so that it usually suffices to describe part of the information and the rest is self evident. For example, when we write X1(1) or I(1) or simply say I is 1 this indicates that the incenter has trilinear coordinates 1 : 1 : 1. We gave only the first coordinate while the other two are cyclic permutations of the first. Similarly, 1 1 1 1 1 X2( a ) or G( a ) say that the centroid has has trilinears a : b : c , where a, b, c are lengths of sides of ABC. We use Xn to denote the n-th central point of the triangle ABC (see [8]). The expressions in terms of sides a, b, and c can be shortened using the following notation. da = b c; db = c a; dc = a b; za = b + c; zb = c + a; zc = a + b; − − − t = a + b + c; ta = b + c a; tb = c + a b; tc = a + b c; − − − 1 m = abc; ma = bc; mb = ca; mc = ab; T = 4 pttatbtc; n n n n n For an integer n, let tn = a + b + c and dna = b c and similarly for other cases.
Load more

Recommended publications
  • Kaleidoscopic Symmetries and Self-Similarity of Integral Apollonian Gaskets
    Kaleidoscopic Symmetries and Self-Similarity of Integral Apollonian Gaskets Indubala I Satija Department of Physics, George Mason University , Fairfax, VA 22030, USA (Dated: April 28, 2021) Abstract We describe various kaleidoscopic and self-similar aspects of the integral Apollonian gaskets - fractals consisting of close packing of circles with integer curvatures. Self-similar recursive structure of the whole gasket is shown to be encoded in transformations that forms the modular group SL(2;Z). The asymptotic scalings of curvatures of the circles are given by a special set of quadratic irrationals with continued fraction [n + 1 : 1; n] - that is a set of irrationals with period-2 continued fraction consisting of 1 and another integer n. Belonging to the class n = 2, there exists a nested set of self-similar kaleidoscopic patterns that exhibit three-fold symmetry. Furthermore, the even n hierarchy is found to mimic the recursive structure of the tree that generates all Pythagorean triplets arXiv:2104.13198v1 [math.GM] 21 Apr 2021 1 Integral Apollonian gaskets(IAG)[1] such as those shown in figure (1) consist of close packing of circles of integer curvatures (reciprocal of the radii), where every circle is tangent to three others. These are fractals where the whole gasket is like a kaleidoscope reflected again and again through an infinite collection of curved mirrors that encodes fascinating geometrical and number theoretical concepts[2]. The central themes of this paper are the kaleidoscopic and self-similar recursive properties described within the framework of Mobius¨ transformations that maps circles to circles[3]. FIG. 1: Integral Apollonian gaskets.
    [Show full text]
  • Strophoids, a Family of Cubic Curves with Remarkable Properties
    Hellmuth STACHEL STROPHOIDS, A FAMILY OF CUBIC CURVES WITH REMARKABLE PROPERTIES Abstract: Strophoids are circular cubic curves which have a node with orthogonal tangents. These rational curves are characterized by a series or properties, and they show up as locus of points at various geometric problems in the Euclidean plane: Strophoids are pedal curves of parabolas if the corresponding pole lies on the parabola’s directrix, and they are inverse to equilateral hyperbolas. Strophoids are focal curves of particular pencils of conics. Moreover, the locus of points where tangents through a given point contact the conics of a confocal family is a strophoid. In descriptive geometry, strophoids appear as perspective views of particular curves of intersection, e.g., of Viviani’s curve. Bricard’s flexible octahedra of type 3 admit two flat poses; and here, after fixing two opposite vertices, strophoids are the locus for the four remaining vertices. In plane kinematics they are the circle-point curves, i.e., the locus of points whose trajectories have instantaneously a stationary curvature. Moreover, they are projections of the spherical and hyperbolic analogues. For any given triangle ABC, the equicevian cubics are strophoids, i.e., the locus of points for which two of the three cevians have the same lengths. On each strophoid there is a symmetric relation of points, so-called ‘associated’ points, with a series of properties: The lines connecting associated points P and P’ are tangent of the negative pedal curve. Tangents at associated points intersect at a point which again lies on the cubic. For all pairs (P, P’) of associated points, the midpoints lie on a line through the node N.
    [Show full text]
  • 9 · the Growth of an Empirical Cartography in Hellenistic Greece
    9 · The Growth of an Empirical Cartography in Hellenistic Greece PREPARED BY THE EDITORS FROM MATERIALS SUPPLIED BY GERMAINE AUJAe There is no complete break between the development of That such a change should occur is due both to po­ cartography in classical and in Hellenistic Greece. In litical and military factors and to cultural developments contrast to many periods in the ancient and medieval within Greek society as a whole. With respect to the world, we are able to reconstruct throughout the Greek latter, we can see how Greek cartography started to be period-and indeed into the Roman-a continuum in influenced by a new infrastructure for learning that had cartographic thought and practice. Certainly the a profound effect on the growth of formalized know­ achievements of the third century B.C. in Alexandria had ledge in general. Of particular importance for the history been prepared for and made possible by the scientific of the map was the growth of Alexandria as a major progress of the fourth century. Eudoxus, as we have seen, center of learning, far surpassing in this respect the had already formulated the geocentric hypothesis in Macedonian court at Pella. It was at Alexandria that mathematical models; and he had also translated his Euclid's famous school of geometry flourished in the concepts into celestial globes that may be regarded as reign of Ptolemy II Philadelphus (285-246 B.C.). And it anticipating the sphairopoiia. 1 By the beginning of the was at Alexandria that this Ptolemy, son of Ptolemy I Hellenistic period there had been developed not only the Soter, a companion of Alexander, had founded the li­ various celestial globes, but also systems of concentric brary, soon to become famous throughout the Mediter­ spheres, together with maps of the inhabited world that ranean world.
    [Show full text]
  • Volume 6 (2006) 1–16
    FORUM GEOMETRICORUM A Journal on Classical Euclidean Geometry and Related Areas published by Department of Mathematical Sciences Florida Atlantic University b bbb FORUM GEOM Volume 6 2006 http://forumgeom.fau.edu ISSN 1534-1178 Editorial Board Advisors: John H. Conway Princeton, New Jersey, USA Julio Gonzalez Cabillon Montevideo, Uruguay Richard Guy Calgary, Alberta, Canada Clark Kimberling Evansville, Indiana, USA Kee Yuen Lam Vancouver, British Columbia, Canada Tsit Yuen Lam Berkeley, California, USA Fred Richman Boca Raton, Florida, USA Editor-in-chief: Paul Yiu Boca Raton, Florida, USA Editors: Clayton Dodge Orono, Maine, USA Roland Eddy St. John’s, Newfoundland, Canada Jean-Pierre Ehrmann Paris, France Chris Fisher Regina, Saskatchewan, Canada Rudolf Fritsch Munich, Germany Bernard Gibert St Etiene, France Antreas P. Hatzipolakis Athens, Greece Michael Lambrou Crete, Greece Floor van Lamoen Goes, Netherlands Fred Pui Fai Leung Singapore, Singapore Daniel B. Shapiro Columbus, Ohio, USA Steve Sigur Atlanta, Georgia, USA Man Keung Siu Hong Kong, China Peter Woo La Mirada, California, USA Technical Editors: Yuandan Lin Boca Raton, Florida, USA Aaron Meyerowitz Boca Raton, Florida, USA Xiao-Dong Zhang Boca Raton, Florida, USA Consultants: Frederick Hoffman Boca Raton, Floirda, USA Stephen Locke Boca Raton, Florida, USA Heinrich Niederhausen Boca Raton, Florida, USA Table of Contents Khoa Lu Nguyen and Juan Carlos Salazar, On the mixtilinear incircles and excircles,1 Juan Rodr´ıguez, Paula Manuel and Paulo Semi˜ao, A conic associated with the Euler line,17 Charles Thas, A note on the Droz-Farny theorem,25 Paris Pamfilos, The cyclic complex of a cyclic quadrilateral,29 Bernard Gibert, Isocubics with concurrent normals,47 Mowaffaq Hajja and Margarita Spirova, A characterization of the centroid using June Lester’s shape function,53 Christopher J.
    [Show full text]
  • Apollonian Circles Patterns in Musical Scales Posing Problems Triangles
    Summer/Autumn 2017 A Problem Fit for a PrincessApollonian Apollonian Circles Gaskets Polygons and PatternsPrejudice in Exploring Musical Social Scales Issues Daydreams in MusicPosing Patterns Problems in Scales ProblemTriangles, Posing Squares, Empowering & Segregation Participants A NOTE FROM AIM #playwithmath Dear Math Teachers’ Circle Network, In this issue of the MTCircular, we hope you find some fun interdisciplinary math problems to try with your Summer is exciting for us, because MTC immersion MTCs. In “A Problem Fit for a Princess,” Chris Goff workshops are happening all over the country. We like traces the 2000-year history of a fractal that inspired his seeing the updates in real time, on Twitter. Your enthu- MTC’s logo. In “Polygons and Prejudice,” Anne Ho and siasm for all things math and problem solving is conta- Tara Craig use a mathematical frame to guide a con- gious! versation about social issues. In “Daydreams in Music,” Jeremy Aikin and Cory Johnson share a math session Here are some recent tweets we enjoyed from MTC im- motivated by patterns in musical scales. And for those mersion workshops in Cleveland, OH; Greeley, CO; and of you looking for ways to further engage your MTC San Jose, CA, respectively: participants’ mathematical thinking, Chris Bolognese and Mike Steward’s “Using Problem Posing to Empow- What happens when you cooperate in Blokus? er MTC Participants” will provide plenty of food for Try and create designs with rotational symmetry. thought. #toocool #jointhemath – @CrookedRiverMTC — Have MnMs, have combinatorial games Helping regions and states build networks of MTCs @NoCOMTC – @PaulAZeitz continues to be our biggest priority nationally.
    [Show full text]
  • Efficiently Constructing Tangent Circles
    Mathematics Magazine ISSN: 0025-570X (Print) 1930-0980 (Online) Journal homepage: https://www.tandfonline.com/loi/umma20 Efficiently Constructing Tangent Circles Arthur Baragar & Alex Kontorovich To cite this article: Arthur Baragar & Alex Kontorovich (2020) Efficiently Constructing Tangent Circles, Mathematics Magazine, 93:1, 27-32, DOI: 10.1080/0025570X.2020.1682447 To link to this article: https://doi.org/10.1080/0025570X.2020.1682447 Published online: 22 Jan 2020. Submit your article to this journal View related articles View Crossmark data Full Terms & Conditions of access and use can be found at https://www.tandfonline.com/action/journalInformation?journalCode=umma20 VOL. 93, NO. 1, FEBRUARY 2020 27 Efficiently Constructing Tangent Circles ARTHUR BARAGAR University of Nevada Las Vegas Las Vegas, NV 89154 [email protected] ALEX KONTOROVICH Rutgers University New Brunswick, NJ 08854 [email protected] The Greek geometers of antiquity devised a game—we might call it geometrical solitaire—which ...mustsurelystand at the very top of any list of games to be played alone. Over the ages it has attracted hosts of players, and though now well over 2000 years old, it seems not to have lost any of its singular charm or appeal. –HowardEves The Problem of Apollonius is to construct a circle tangent to three given ones in a plane. The three circles may also be limits of circles, that is, points or lines; and “con- struct” means using a straightedge and compass. Apollonius’s own solution did not survive antiquity [8], and we only know of its existence through a “mathscinet review” by Pappus half a millennium later.
    [Show full text]
  • MYSTERIES of the EQUILATERAL TRIANGLE, First Published 2010
    MYSTERIES OF THE EQUILATERAL TRIANGLE Brian J. McCartin Applied Mathematics Kettering University HIKARI LT D HIKARI LTD Hikari Ltd is a publisher of international scientific journals and books. www.m-hikari.com Brian J. McCartin, MYSTERIES OF THE EQUILATERAL TRIANGLE, First published 2010. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission of the publisher Hikari Ltd. ISBN 978-954-91999-5-6 Copyright c 2010 by Brian J. McCartin Typeset using LATEX. Mathematics Subject Classification: 00A08, 00A09, 00A69, 01A05, 01A70, 51M04, 97U40 Keywords: equilateral triangle, history of mathematics, mathematical bi- ography, recreational mathematics, mathematics competitions, applied math- ematics Published by Hikari Ltd Dedicated to our beloved Beta Katzenteufel for completing our equilateral triangle. Euclid and the Equilateral Triangle (Elements: Book I, Proposition 1) Preface v PREFACE Welcome to Mysteries of the Equilateral Triangle (MOTET), my collection of equilateral triangular arcana. While at first sight this might seem an id- iosyncratic choice of subject matter for such a detailed and elaborate study, a moment’s reflection reveals the worthiness of its selection. Human beings, “being as they be”, tend to take for granted some of their greatest discoveries (witness the wheel, fire, language, music,...). In Mathe- matics, the once flourishing topic of Triangle Geometry has turned fallow and fallen out of vogue (although Phil Davis offers us hope that it may be resusci- tated by The Computer [70]). A regrettable casualty of this general decline in prominence has been the Equilateral Triangle. Yet, the facts remain that Mathematics resides at the very core of human civilization, Geometry lies at the structural heart of Mathematics and the Equilateral Triangle provides one of the marble pillars of Geometry.
    [Show full text]
  • Efficiently Constructing Tangent Circles 3
    EFFICIENTLY CONSTRUCTING TANGENT CIRCLES ARTHUR BARAGAR AND ALEX KONTOROVICH 1. Introduction The famous Problem of Apollonius is to construct a circle tangent to three given ones in a plane. The three circles may also be limits of circles, that is, points or lines, and “construct” of course refers to straightedge and compass. In this note, we consider the problem of constructing tangent circles from the point of view of efficiency. By this we mean using as few moves as possible, where a move is the act of drawing a line or circle. (Points are free as they do not harm the straightedge or compass, and all lines are considered endless, so there is no cost to “extending” a line segment.) Our goal is to present, in what we believe is the most efficient way possible, a construction of four mutually tangent circles. (Five circles of course cannot be mutually tangent in the plane, for their tangency graph, the complete graph K5, is non-planar.) We first present our construction before giving some remarks comparing it to others we found in the literature. 2. Baby Cases: One and Two Circles Constructing one circle obviously costs one move: let A and Z be any distinct points in the plane and draw the circle OA with center A and passing through Z. Given OA, constructing a second circle tangent to it costs two more moves: draw a line through AZ, and put an arbitrary point B on this line (say, outside OA). Now draw the circle OB with center B and passing through Z; then OA and OB are obviously tangent at Z, see Figure 1.
    [Show full text]
  • David Benko, Western Kentucky University
    Abstracts of Talks for the 2008 KYMAA Annual Meeting Western Kentucky University, Bowling Green March 28 - 29, 2008 Note: Undergraduate student speakers are indicated by (u), graduate student speakers are indicated by (g), and faculty speakers are indicated by (f). Contributed Talks: Robert Amundson, Murray State University (u) The Lie Group GL( n, Q ) and Lie Algebra gl( n, Q) The Lie algebra gl( n, Q) is something that has not been investigated before. This presentation will give a general overview of the Lie group GL( n, Q ) and the Lie algebra gl( n, Q) and the motivation for finding the Lie group and Lie Algebra. David Benko, Western Kentucky University (f) The Pros and Cons of Ebay We will present the best buying and selling strategies on Ebay - from a mathematical point of view. We will also propose a better way to rate sellers on Ebay. Disclaimer: I do not own shares of Ebay... Timothy M. Brauch, University of Louisville (g) Counting Perfect Matchings This presentation will show an application of combinatorial nullstellensatz for detecting perfect matchings in bipartite graphs. An elegant circular lock idea will be explored to show some beautiful relationships of graph theory with other branches of mathematics. Some known results as well as open problems will be presented. This is joint work with André E. Kézdy (University of Louisville) and Hunter S. Snevily (University of Idaho). Knowledge of one semester of undergraduate linear algebra will be assumed. Woody Burchett, Georgetown College (u) Thinking Inside the Box: Geometric Interpretations of Quadratic Problems in BM 13901 In my talk I will examine some quadratic problems from the Babylonian Mathematical tablet BM 13901.
    [Show full text]
  • Generalized Apollonian Circles
    Generalized Apollonian Circles Paul Yiu Abstract. Given a triangle, we construct a one-parameter family of triads of coaxal circles orthogonal to circumcircle and mutually tangent to each other at a point on the circumcircle. This is a generalization of the classical Apollonian circles which intersect at the two isodynamic points. The classical Apollonian circles of a triangle are the three circles each with di- ameter the segment between the feet of the two bisectors of an angle on its opposite sides. It is well known that the centers of the three circles are the intercepts of the Lemoine axis on the sidelines, 1 and that they have two points in common, the iso- dynamic points ([1, p.218]). The feet of the internal angle bisectors are the traces of the incenter. Those of the external angle bisectors are the intercepts of the tri- linear polar of the incenter. More generally, given a point P , with traces X, Y , Z on the sidelines BC, CA, AB of triangle ABC, the harmonic conjugates of X in BC, Y in CA, Z in AB are the intercepts X, Y , Z of the trilinear polar of P on these sidelines. The circles with diameters XX, YY, ZZ are the Apollonian cir- cles associated with P . The isodynamic points are the common points of the triad of Apollonian circles associated with the incenter. In this note we characterize the points P for which the three Apollonian circles are mutually tangent to each other, and provide an elegant euclidean construction for such circles. We work with barycentric coordinates with respect to triangle ABC, and their homogenization.
    [Show full text]
  • Stefanie Ursula Eminger Phd Thesis
    CARL FRIEDRICH GEISER AND FERDINAND RUDIO: THE MEN BEHIND THE FIRST INTERNATIONAL CONGRESS OF MATHEMATICIANS Stefanie Ursula Eminger A Thesis Submitted for the Degree of PhD at the University of St Andrews 2015 Full metadata for this item is available in Research@StAndrews:FullText at: http://research-repository.st-andrews.ac.uk/ Please use this identifier to cite or link to this item: http://hdl.handle.net/10023/6536 This item is protected by original copyright Carl Friedrich Geiser and Ferdinand Rudio: The Men Behind the First International Congress of Mathematicians Stefanie Ursula Eminger This thesis is submitted in partial fulfilment for the degree of PhD at the University of St Andrews 2014 Thesis Declaration 1. Candidate’s declarations: I, Stefanie Eminger, hereby certify that this thesis, which is approximately 78,500 words in length, has been written by me, and that it is the record of work carried out by me, or principally by myself in collaboration with others as acknowledged, and that it has not been submitted in any previous application for a higher degree. I was admitted as a research student in September 2010 and as a candidate for the degree of PhD in September 2010; the higher study for which this is a record was carried out in the University of St Andrews between 2010 and 2014. Date ……………… Signature of candidate …………………………………… 2. Supervisor’s declaration: I hereby certify that the candidate has fulfilled the conditions of the Resolution and Regulations appropriate for the degree of PhD in the University of St Andrews and that the candidate is qualified to submit this thesis in application for that degree.
    [Show full text]
  • A Young Person's Guide to the Hopf Fibration
    A YOUNG PERSON'S GUIDE TO THE HOPF FIBRATION ZACHARY TREISMAN The purpose of these notes is to introduce a mathematical structure which goes by the name the Hopf fibration, and demonstrates a number of surprising and beautiful things. The Hopf fibration is a map showing a connection between two spheres, a two dimensional sphere, and a three dimensional sphere. In order to understand this map, we will need to develop a few tools. There will be a few detours along the way, in order to develop enough familiarity with the tools being used so that the student is sufficiently impressed by the structures that are eventually uncovered. The Hopf fibration and the mathematics that are developed along the way makes for some very interesting visual images. The paintings that have been included here are all the work of Lun-Yi Tsai, an artist, a mathematician, and a good friend. The computer generated images I arXiv:0908.1205v1 [math.HO] 9 Aug 2009 Figure 1. Hopf Fibration Lun-Yi Tsai 1 2 ZACHARY TREISMAN have produced using the programs surf and jenn3d, both of which are freely available online, as well as Mathematica. 1. Complex Numbers Our first task is to introduce you to the complex numbers. The in- troduction is geometric, and appeals to the intuitive basis for the real numbers as measurements that scale. One thing that this development is not is historical. This is because the first ways of thinking of some- thing are not always the easiest to understand. In this section you are often asked to show algebraic facts by drawing a picture.
    [Show full text]