Mathematical Gems
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Investigating Centers of Triangles: the Fermat Point
Investigating Centers of Triangles: The Fermat Point A thesis submitted to the Miami University Honors Program in partial fulfillment of the requirements for University Honors with Distinction by Katherine Elizabeth Strauss May 2011 Oxford, Ohio ABSTRACT INVESTIGATING CENTERS OF TRIANGLES: THE FERMAT POINT By Katherine Elizabeth Strauss Somewhere along their journey through their math classes, many students develop a fear of mathematics. They begin to view their math courses as the study of tricks and often seemingly unsolvable puzzles. There is a demand for teachers to make mathematics more useful and believable by providing their students with problems applicable to life outside of the classroom with the intention of building upon the mathematics content taught in the classroom. This paper discusses how to integrate one specific problem, involving the Fermat Point, into a high school geometry curriculum. It also calls educators to integrate interesting and challenging problems into the mathematics classes they teach. In doing so, a teacher may show their students how to apply the mathematics skills taught in the classroom to solve problems that, at first, may not seem directly applicable to mathematics. The purpose of this paper is to inspire other educators to pursue similar problems and investigations in the classroom in order to help students view mathematics through a more useful lens. After a discussion of the Fermat Point, this paper takes the reader on a brief tour of other useful centers of a triangle to provide future researchers and educators a starting point in order to create relevant problems for their students. iii iv Acknowledgements First of all, thank you to my advisor, Dr. -
On the Fermat Point of a Triangle Jakob Krarup, Kees Roos
280 NAW 5/18 nr. 4 december 2017 On the Fermat point of a triangle Jakob Krarup, Kees Roos Jakob Krarup Kees Roos Department of Computer Science Faculty of EEMCS University of Copenhagen, Denmark Technical University Delft [email protected] [email protected] Research On the Fermat point of a triangle For a given triangle 9ABC, Pierre de Fermat posed around 1640 the problem of finding a lem has been solved but it is not known by 2 point P minimizing the sum sP of the Euclidean distances from P to the vertices A, B, C. whom. As will be made clear in the sec- Based on geometrical arguments this problem was first solved by Torricelli shortly after, tion ‘Duality of (F) and (D)’, Torricelli’s ap- by Simpson in 1750, and by several others. Steeped in modern optimization techniques, proach already reveals that the problems notably duality, however, Jakob Krarup and Kees Roos show that the problem admits a (F) and (D) are dual to each other. straightforward solution. Using Simpson’s construction they furthermore derive a formula Our first aim is to show that application expressing sP in terms of the given triangle. This formula appears to reveal a simple re- of duality in conic optimization yields the lationship between the area of 9ABC and the areas of the two equilateral triangles that solution of (F), straightforwardly. Devel- occur in the so-called Napoleon’s Theorem. oped in the 1990s, the field of conic optimi- zation (CO) is a generalization of the more We deal with two problems in planar geo- Problem (D) was posed in 1755 by well-known field of linear optimization (LO). -
Some Problems in Partitio Numerorum
J. Austral. Math. Soc. (Series A) 27 (1979), 319-331 SOME PROBLEMS IN PARTITIO NUMERORUM P. ERDOS and J. H. LOXTON (Received 24 April 1977) Communicated by J. Pitman Abstract We consider some unconventional partition problems in which the parts of the partition are restricted by divisibility conditions, for example, partitions n = ax +... + a* into positive integers «!, ..., ak such that ax | a2 I ••• I ak. Some rather weak estimates for the various partition functions are obtained. Subject classification (Amer. Math. Soc. (MOS) 1970): 10 A 45, 10 J 20. 1. Introduction In this paper, we shall consider various partition problems in which the parts of the partitions are restricted by divisibility conditions. Most of our remarks concern the following two situations: (i) 'Chain partitions', that is partitions n = a1 + ... +ak into positive integers a1,...,ak such that a^a^ ...\ak. (ii) 'Umbrella partitions', that is partitions into positive integers such that every part divides the largest one. Our aim is to estimate the partition functions which arise in each case for partitions with distinct parts and for partitions in which repetitions are allowed. This work arose from a question of R. W. Robinson about chain partitions with repetitions which, in turn, came from attempts to count a certain kind of tree. This particular partition problem is closely connected with wj-ary partitions, that is partitions as sums of powers of a fixed integer m, which are obvious instances of the types of partitions described above. In another direction, the problem of representing numbers by umbrella partitions has some connections with the 'practical numbers' of Srinivasan. -
IX Geometrical Olympiad in Honour of I.F.Sharygin Final Round. Ratmino, 2013, August 1 Solutions First Day. 8 Grade 8.1. (N
IX Geometrical Olympiad in honour of I.F.Sharygin Final round. Ratmino, 2013, August 1 Solutions First day. 8 grade 8.1. (N. Moskvitin) Let ABCDE be a pentagon with right angles at vertices B and E and such that AB = AE and BC = CD = DE. The diagonals BD and CE meet at point F. Prove that FA = AB. First solution. The problem condition implies that the right-angled triangles ABC and AED are equal, thus the triangle ACD is isosceles (see fig. 8.1a). Then ∠BCD = ∠BCA + ∠ACD = = ∠EDA + ∠ADC = ∠CDE. Therefore, the isosceles triangles BCD and CDE are equal. Hence ∠CBD = ∠CDB = ∠ECD = ∠DEC. Since the triangle CFD is isosceles and BD = CE, we obtain that BF = FE. Therefore BFE 180◦ − 2 FCD 4ABF = 4AEF. Then AFB = ∠ = ∠ = 90◦ − ECD = 90◦ − DBC = ABF, ∠ 2 2 ∠ ∠ ∠ hence AB = AF, QED. BB BB CC CC PP FF FF AA AA AA DD AA DD EE EE Fig. 8.1а Fig. 8.1b Second solution. Let BC meet DE at point P (see fig. 8.1b). Notice that ∠CBD = ∠CDB = = ∠DBE, i.e., BD is the bisector of ∠CBE. Thus F is the incenter of 4PBE. Since the quadrilateral PBAE is cyclic and symmetrical, we obtain that A is the midpoint of arc BE of the circle (PBE). Therefore, by the trefoil theorem we get AF = AB, QED. Remark. The problem statement holds under the weakened condition of equality of side lengths. It is sufficient to say that AB = AE and BC = CD = DE. 8.2. (D. Shvetsov) Two circles with centers O1 and O2 meet at points A and B. -
Integer Sequences
UHX6PF65ITVK Book > Integer sequences Integer sequences Filesize: 5.04 MB Reviews A very wonderful book with lucid and perfect answers. It is probably the most incredible book i have study. Its been designed in an exceptionally simple way and is particularly just after i finished reading through this publication by which in fact transformed me, alter the way in my opinion. (Macey Schneider) DISCLAIMER | DMCA 4VUBA9SJ1UP6 PDF > Integer sequences INTEGER SEQUENCES Reference Series Books LLC Dez 2011, 2011. Taschenbuch. Book Condition: Neu. 247x192x7 mm. This item is printed on demand - Print on Demand Neuware - Source: Wikipedia. Pages: 141. Chapters: Prime number, Factorial, Binomial coeicient, Perfect number, Carmichael number, Integer sequence, Mersenne prime, Bernoulli number, Euler numbers, Fermat number, Square-free integer, Amicable number, Stirling number, Partition, Lah number, Super-Poulet number, Arithmetic progression, Derangement, Composite number, On-Line Encyclopedia of Integer Sequences, Catalan number, Pell number, Power of two, Sylvester's sequence, Regular number, Polite number, Ménage problem, Greedy algorithm for Egyptian fractions, Practical number, Bell number, Dedekind number, Hofstadter sequence, Beatty sequence, Hyperperfect number, Elliptic divisibility sequence, Powerful number, Znám's problem, Eulerian number, Singly and doubly even, Highly composite number, Strict weak ordering, Calkin Wilf tree, Lucas sequence, Padovan sequence, Triangular number, Squared triangular number, Figurate number, Cube, Square triangular -
On Zumkeller Numbers
On Zumkeller Numbers K.P.S. Bhaskara Rao Yuejian Peng Department of Mathematics and Computer Science Indiana State University Terre Haute, IN, 47809, USA Email: [email protected]; [email protected] Abstract Generalizing the concept of a perfect number, Sloane’s sequences of integers A083207 lists the sequence of integers n with the property: the positive factors of n can be partitioned into two disjoint parts so that the sums of the two parts are equal. Following [4] Clark et al., we shall call such integers, Zumkeller numbers. Generalizing this, in [4] a number n is called a half-Zumkeller number if the positive proper factors of n can be partitioned into two disjoint parts so that the sums of the two parts are equal. An extensive study of properties of Zumkeller numbers, half-Zumkeller numbers and their relation to practical numbers is undertaken in this paper. In [4] Clark et al., announced results about Zumkellers numbers and half- Zumkeller numbers and suggested two conjectures. In the present paper we shall settle one of the conjectures, prove the second conjecture in some special cases and arXiv:0912.0052v1 [math.NT] 1 Dec 2009 prove several results related to the second conjecture. We shall also show that if there is an even Zumkeller number that is not half-Zumkeller it should be bigger than 7.2334989 × 109. 1 Introduction A positive integer n is called a perfect number if n equals the sum of its proper positive factors. Generalizing this concept in 2003, Zumkeller published in Sloane’s sequences of inte- gers A083207 a sequence of integers n with the property that the positive factors of n 1 can be partitioned into two disjoint parts so that the sums of the two parts are equal. -
From Euler to Ffiam: Discovery and Dissection of a Geometric Gem
From Euler to ffiam: Discovery and Dissection of a Geometric Gem Douglas R. Hofstadter Center for Research on Concepts & Cognition Indiana University • 510 North Fess Street Bloomington, Indiana 47408 December, 1992 ChapterO Bewitched ... by Circles, Triangles, and a Most Obscure Analogy Although many, perhaps even most, mathematics students and other lovers of mathematics sooner or later come across the famous Euler line, somehow I never did do so during my student days, either as a math major at Stanford in the early sixties or as a math graduate student at Berkeley for two years in the late sixties (after which I dropped out and switched to physics). Geometry was not by any means a high priority in the math curriculum at either institution. It passed me by almost entirely. Many, many years later, however, and quite on my own, I finally did become infatuated - nay, bewitched- by geometry. Just plane old Euclidean geometry, I mean. It all came from an attempt to prove a simple fact about circles that I vaguely remembered from a course on complex variables that I took some 30 years ago. From there on, the fascination just snowballed. I was caught completely off guard. Never would I have predicted that Doug Hofstadter, lover of number theory and logic, would one day go off on a wild Euclidean-geometry jag! But things are unpredictable, and that's what makes life interesting. I especially came to love triangles, circles, and their unexpectedly profound interrelations. I had never appreciated how intimately connected these two concepts are. Associated with any triangle are a plentitude of natural circles, and conversely, so many beautiful properties of circles cannot be defined except in terms of triangles. -
Concurrency of Four Euler Lines
Forum Geometricorum b Volume 1 (2001) 59–68. bbb FORUM GEOM ISSN 1534-1178 Concurrency of Four Euler Lines Antreas P. Hatzipolakis, Floor van Lamoen, Barry Wolk, and Paul Yiu Abstract. Using tripolar coordinates, we prove that if P is a point in the plane of triangle ABC such that the Euler lines of triangles PBC, AP C and ABP are concurrent, then their intersection lies on the Euler line of triangle ABC. The same is true for the Brocard axes and the lines joining the circumcenters to the respective incenters. We also prove that the locus of P for which the four Euler lines concur is the same as that for which the four Brocard axes concur. These results are extended to a family Ln of lines through the circumcenter. The locus of P for which the four Ln lines of ABC, PBC, AP C and ABP concur is always a curve through 15 finite real points, which we identify. 1. Four line concurrency Consider a triangle ABC with incenter I. It is well known [13] that the Euler lines of the triangles IBC, AIC and ABI concur at a point on the Euler line of ABC, the Schiffler point with homogeneous barycentric coordinates 1 a(s − a) b(s − b) c(s − c) : : . b + c c + a a + b There are other notable points which we can substitute for the incenter, so that a similar statement can be proven relatively easily. Specifically, we have the follow- ing interesting theorem. Theorem 1. Let P be a point in the plane of triangle ABC such that the Euler lines of the component triangles PBC, AP C and ABP are concurrent. -
Equilateral Triangles Formed by the Centers of Erected Triangles
International Journal of Computer Discovered Mathematics (IJCDM) ISSN 2367-7775 ©IJCDM Volume 6, 2021, pp. 43{67 Received 15 February 2021. Published on-line 15 April 2021 web: http://www.journal-1.eu/ ©The Author(s) This article is published with open access1. Equilateral Triangles formed by the Centers of Erected Triangles Stanley Rabinowitza and Ercole Suppab a 545 Elm St Unit 1, Milford, New Hampshire 03055, USA e-mail: [email protected] web: http://www.StanleyRabinowitz.com/ b Via B. Croce 54, 64100 Teramo, Italia e-mail: [email protected] Abstract. Related triangles are erected outward on the sides of a triangle. A triangle center is constructed in each of these triangles. We use a computer to find instances where these three centers form an equilateral triangle. Keywords. triangle geometry, erected triangles, equilateral triangles, computer- discovered mathematics, GeometricExplorer. Mathematics Subject Classification (2020). 51M04, 51-08. 1. Introduction The well-known result, known as Napoleon's Theorem, states that if equilateral triangles are erected outward on the sides of an arbitrary triangle ABC, then their centers (G, H, and I) form an equilateral triangle (Figure 1). Figure 1. Outer Napoleon Triangle 1This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited. 43 44 Equilateral Triangles formed by the Centers of Erected Triangles Triangle GHI is called the outer Napoleon triangle of 4ABC. As is also well known, the equilateral triangles can be erected inward (Figure 2). -
Handbook of Number Theory Ii
HANDBOOK OF NUMBER THEORY II by J. Sandor´ Babes¸-Bolyai University of Cluj Department of Mathematics and Computer Science Cluj-Napoca, Romania and B. Crstici formerly the Technical University of Timis¸oara Timis¸oara Romania KLUWER ACADEMIC PUBLISHERS DORDRECHT / BOSTON / LONDON A C.I.P. Catalogue record for this book is available from the Library of Congress. ISBN 1-4020-2546-7 (HB) ISBN 1-4020-2547-5 (e-book) Published by Kluwer Academic Publishers, P.O. Box 17, 3300 AA Dordrecht, The Netherlands. Sold and distributed in North, Central and South America by Kluwer Academic Publishers, 101 Philip Drive, Norwell, MA 02061, U.S.A. In all other countries, sold and distributed by Kluwer Academic Publishers, P.O. Box 322, 3300 AH Dordrecht, The Netherlands. Printed on acid-free paper All Rights Reserved C 2004 Kluwer Academic Publishers No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Printed in the Netherlands. Contents PREFACE 7 BASIC SYMBOLS 9 BASIC NOTATIONS 10 1 PERFECT NUMBERS: OLD AND NEW ISSUES; PERSPECTIVES 15 1.1 Introduction .............................. 15 1.2 Some historical facts ......................... 16 1.3 Even perfect numbers ......................... 20 1.4 Odd perfect numbers ......................... 23 1.5 Perfect, multiperfect and multiply perfect numbers ......... 32 1.6 Quasiperfect, almost perfect, and pseudoperfect numbers ............................... -
A Note on the Anticomplements of the Fermat Points
Forum Geometricorum b Volume 9 (2009) 119–123. b b FORUM GEOM ISSN 1534-1178 A Note on the Anticomplements of the Fermat Points Cosmin Pohoata Abstract. We show that each of the anticomplements of the Fermat points is common to a triad of circles involving the triangle of reflection. We also gen- erate two new triangle centers as common points to two other triads of circles. Finally, we present several circles passing through these new centers and the anticomplements of the Fermat points. 1. Introduction The Fermat points F± are the common points of the lines joining the vertices of a triangle T to the apices of the equilateral triangles erected on the corresponding sides. They are also known as the isogonic centers (see [2, pp.107, 151]) and are among the basic triangle centers. In [4], they appear as the triangle centers X13 and X14. Not much, however, is known about their anticomplements, which are the points P± which divide F±G in the ratio F±G : GP± =1:2. Given triangle T with vertices A, B, C, (i) let A′, B′, C′ be the reflections of the vertices A, B, C in the respective opposite sides, and (ii) for ε = ±1, let Aε, Bε, Cε be the apices of the equilateral triangles erected on the sides BC, CA, AB of triangle ABC respectively, on opposite or the same sides of the vertices according as ε = 1 or −1 (see Figures 1A and 1B). ′ ′ ′ Theorem 1. For ε = ±1, the circumcircles of triangles A BεCε, B CεAε, C AεBε are concurrent at the anticomplement P−ε of the Fermat point F−ε. -
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.