DOCSLIB.ORG

Meeting in Mathematics

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Meeting in Mathematics 226159-CP-1-2005-1-AT-COMENIUS-C21 527269-LLP-1-2012-1-AT-COMENIUS-CAM Pavel Boytchev Hannes Hohenwarter Evgenia Sendova Neli Dimitrova Emil Kostadinov Andreas Ulovec Vladimir Georgiev Arne Mogensen Henning Westphael Oleg Mushkarov MEETING IN MATHEMATICS 2nd edition • Universität Wien • Dipartimento di Matematica, Universita' di Pisà • VIA University College – Læreruddannelsen i Århus • Институт по математика и информатика , Българска академия на науките Authors Pavel Boytchev, Neli Dimitrova, Vladimir Georgiev, Hannes Hohenwarter, Emil Kostadinov, Arne Mogensen, Oleg Mushkarov, Evgenia Sendova, Andreas Ulovec, Henning Westphael Editors Evgenia Sendova Andreas Ulovec Project Evaluator Jarmila Novotná, Charles University, Prague, Czech Republic Reviewers Jarmila Novotná, Charles University, Prague, Czech Republic Nicholas Mousoulides, University of Nicosia, Cyprus Cover design by Pavel Boytchev Cartoons by Yovko Kolarov All Rights Reserved © 2013 No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the authors. For educational purposes only (i.e. for use in schools, teaching, teacher training etc.), you may use this work or parts of it under the “Attribution Non-Commercial Share Alike” license according to Creative Commons, as detailed in http://creativecommons.org/licenses/by-nc-sa/3.0/legalcode. This project has been funded with support from the European Commission. This publication reflects the views only of the authors, and the Commission cannot be held responsible for any use which may be made of the information contained therein. Published by Demetra Publishing House, Sofia, Bulgaria ISBN 978-954-9526-49-3 iii Contents PREFACE................................................................................................................... vii CHAPTER 1 Meeting the gifted – while not forgetting the others Author in charge: Arne Mogensen Contributor: Henning Westphae .............................................................................. 1 CHAPTER 2 Mathematical competences Authors in charge: Henning Westphael, Arne Mogensen...................................... 21 CHAPTER 3 Mathematics competitions Author in charge: Vladimir Georgiev Contributors: Arne Mogensen, Oleg Mushkarov, Henning Westphael ......................................................................... 41 Appendix to Chapter 3 The Unsung Hero – some selected problems Vladimir Georgiev................................................................................................. 55 CHAPTER 4 Mathematics clubs Author in charge: Vladimir Georgiev Contributor: Arne Mogensen ................................................................................. 65 CHAPTER 5 Mathematics explorations and interdisciplinary work enhanced by IT Author in charge: Andreas Ulovec Contributor: Evgenia Sendova............................................................................... 79 Appendix to Chapter 5 A dynamic approach to a classical geometric problem Andreas Ulovec ..................................................................................................... 95 Unity via diversity Pavel Boytchev ...................................................................................................... 97 iv Contents “The more it changes the samer it gets” Andreas Ulovec ................................................................................................... 110 Fractal geometry Hannes Hohenwarter, Andreas Ulovec, Pavel Boytchev .................................... 115 CHAPTER 6 Math research at school age Authors in charge : Oleg Mushkarov, Evgenia Sendova Contributor: Neli Dimitrova ................................................................................ 137 Appendix to Chapter 6 How to cut two circles Oleg Mushkarov .................................................................................................. 173 Inequalities for sums of powers Vladimir Georgiev , Oleg Mushkarov .................................................................. 179 On a metric property of the equilateral triangles Emil Kostadinov .................................................................................................. 189 Malfatti’s problems Emil Kostadinov .................................................................................................. 199 v Authors Dr. Pavel Boytchev Faculty of Mathematics and Informatics, Sofia University James Baurchier 5, Sofia 1126, Bulgaria [email protected] Dr. Neli Dimitrova Institute of Mathematics and Informatics Bulgarian Academy of Sciences Acad. G. Bonchev, bl. 8, Sofia 1113, Bulgaria [email protected] DSc. Vladimir Georgiev Department of Mathematics, University of Pisa Largo Bruno Pontecorvo, 5, Pisa 56127, Italy [email protected] MSc. Hannes Hohenwarter High School Villach St. Martin St. Martiner Strasse 7, Villach 9500, Austria [email protected] Emil Kostadinov PhD student of Economics at the University of Essex, United Kingdom [email protected] Dr. Arne Mogensen Dept. of Education, VIA University College Aarhus, Denmark [email protected] DSc. Oleg Mushkarov Institute of Mathematics and Informatics Bulgarian Academy of Sciences Acad. G. Bonchev, bl. 8, Sofia 1113, Bulgaria [email protected] vi Dr. Evgenia Sendova Institute of Mathematics and Informatics Bulgarian Academy of Sciences Acad. G. Bonchev, bl. 8, Sofia 1113, Bulgaria [email protected] Dr. Andreas Ulovec Faculty of Mathematics, University of Vienna Nordbergstrasse 15, Vienna 1090, Austria [email protected] MSc. Henning Westphael Department of Teacher Education at Aarhus VIA University College Trøjborgvej 82, 8200 Aarhus N, Denmark [email protected] Preface Dear readers, to encourage many more young people to appreciate the real nature and spirit of mathematics and possibly to be enrolled in mathematics study it is important to involve them in doing mathematics (not just learning about mathematics) as early as possible. This goal could be achieved if mathematics teachers are prepared to identify and work with mathematically gifted students (without losing the rest), to integrate the informatics and IT tools with mathematics explorations, to introduce some topics and activities (not present in the traditional mathematics curriculum) that would be more appealing and relevant to the students and teachers alike. With the idea of addressing some of these problems the project Meetings in Mathematics : Mathematics Enrolment and Effectiveness of Teaching in Mathematics was carried out in the years 2005–2008 in the frames of Comenius 2.1 program of the European Commission. The goal of the participants (University of Pisa, Institute of Mathematics and Informatics at the Bulgarian Academy of Sciences, University of Vienna, and VIA University College – Department of Teacher Education, Aarhus) was to share their experience in identifying mathematically gifted students, in offering mathematical challenges for those who love competing and for the ones who prefer to attack an open problem not being pressed by the time limitations. Furthermore, the spectrum of “mathematical giftedness” was not determined so as to address just mathematical elite but rather to help teachers in revealing the potential of young people to be creative in a mathematical context. One of the products of this project was the book Meeting in Mathematics, oriented to the needs of mathematics teachers in the Project countries (Austria, Bulgaria, Denmark and Italy). After publishing the first edition of the book we distributed it among math educators and teachers and got various valuable remarks and suggestions by the readers. The most encouraging feedback was “More, please!”, so here it is – what you are holding in your hands is the second edition of Meeting in mathematics . The preparation of the new edition was brought to life thanks to a new Comenius project, MeetMe – 527629-LLP-1-2012-1-AT-COMENIUS-CAM. The authors’ team (enriched by a new member from VIA University College – Department of Teacher Education) took the chance to turn back to the original materials developed several years ago and to make some improvements based on the accumulated experience in the context of teacher training courses and the users’ suggestions for refinements and enrichments. Here is the updated content in a nut shell. viii Preface In Chapter 1 we present some characteristics of gifted students , as they are experienced in research and practice. The chapter also deals with some of the challenges that teachers might meet. In Chapter 2 we present and use the concept of mathematical competence for description of mathematics teaching and learning in accordance with the most recent theoretical developments. In Chapter 3 we present various types of math competitions, both individual and team ones. The specifics of the individual math competitions of IMO type are compared with those of the team competitions and ways to combine the advantages of each kind are discussed. Special emphasis is put on the team competitions since in general they: • involve larger groups of pupils; • have attractive and dynamic form; • integrate knowledge from other fields, e.g. arts, modeling, physics; • could be considered as natural elements of some traditional school events. Ideas related to the organization of math team competitions are presented based on a concrete example – the international math competition The
Load more

Recommended publications
  • European Qualifiers
    EUROPEAN QUALIFIERS - 2019/21 SEASON MATCH PRESS KITS Natsionalen Stadion Vasil Levski - Sofia Monday 10 June 2019 Bulgaria 20.45CET (21.45 local time) Kosovo Group A - Matchday -9 Last updated 02/02/2021 16:12CET CMS error: Requested URL "/insideuefa/library/promo/presskits/european-qualifiers/_sponsorqualifiers.html" not found. Previous meetings 2 Squad list 3 Head coach 5 Match officials 6 Match-by-match lineups 7 Team facts 9 Legend 11 1 Bulgaria - Kosovo Monday 10 June 2019 - 20.45CET (21.45 local time) Match press kit Natsionalen Stadion Vasil Levski, Sofia Previous meetings Head to Head 2020 UEFA European Championship Stage Date Match Result Venue Goalscorers reached Zeneli 61; Bozhikov 25/03/2019 QR (GS) Kosovo - Bulgaria 1-1 Pristina 39 Final Qualifying Total tournament Home Away Pld W D L Pld W D L Pld W D L Pld W D L GF GA EURO Bulgaria - - - - 1 0 1 0 - - - - 1 0 1 0 1 1 Kosovo 1 0 1 0 - - - - - - - - 1 0 1 0 1 1 FIFA* Bulgaria - - - - - - - - - - - - - - - - - - Kosovo - - - - - - - - - - - - - - - - - - Friendlies Bulgaria - - - - - - - - - - - - - - - - - - Kosovo - - - - - - - - - - - - - - - - - - Total Bulgaria - - - - 1 0 1 0 - - - - 1 0 1 0 1 1 Kosovo 1 0 1 0 - - - - - - - - 1 0 1 0 1 1 * FIFA World Cup/FIFA Confederations Cup 2 Bulgaria - Kosovo Monday 10 June 2019 - 20.45CET (21.45 local time) Match press kit Natsionalen Stadion Vasil Levski, Sofia Squad list Bulgaria Current season Qual. FT No. Player DoB Age Club D Pld Gls Pld Gls Goalkeepers - Hristo Ivanov 06/04/1982 37 Etar - 0 0 0 0 - Nikolay Mihaylov 28/06/1988
    [Show full text]
  • The Malfatti Problem
    Forum Geometricorum b Volume 1 (2001) 43–50. bbb FORUM GEOM ISSN 1534-1178 The Malfatti Problem Oene Bottema Abstract. A solution is given of Steiner’s variation of the classical Malfatti problem in which the triangle is replaced by three circles mutually tangent to each other externally. The two circles tangent to the three given ones, presently known as Soddy’s circles, are encountered as well. In this well known problem, construction is sought for three circles C1, C2 and C3, tangent to each other pairwise, and of which C1 is tangent to the sides A1A2 and A1A3 of a given triangle A1A2A3, while C2 is tangent to A2A3 and A2A1 and C3 to A3A1 and A3A2. The problem was posed by Malfatti in 1803 and solved by him with the help of an algebraic analysis. Very well known is the extraordinarily elegant geometric solution that Steiner announced without proof in 1826. This solution, together with the proof Hart gave in 1857, one can find in various textbooks.1 Steiner has also considered extensions of the problem and given solutions. The first is the one where the lines A2A3, A3A1 and A1A2 are replaced by circles. Further generalizations concern the figures of three circles on a sphere, and of three conic sections on a quadric surface. In the nineteenth century many mathematicians have worked on this problem. Among these were Cayley (1852) 2, Schellbach (who in 1853 published a very nice goniometric solution), and Clebsch (who in 1857 extended Schellbach’s solution to three conic sections on a quadric surface, and for that he made use of elliptic functions).
    [Show full text]
  • Triangle Centers Associated with the Malfatti Circles
    Forum Geometricorum b Volume 3 (2003) 83–93. bbb FORUM GEOM ISSN 1534-1178 Triangle Centers Associated with the Malfatti Circles Milorad R. Stevanovi´c Abstract. Various formulae for the radii of the Malfatti circles of a triangle are presented. We also express the radii of the excircles in terms of the radii of the Malfatti circles, and give the coordinates of some interesting triangle centers associated with the Malfatti circles. 1. The radii of the Malfatti circles The Malfatti circles of a triangle are the three circles inside the triangle, mutually tangent to each other, and each tangent to two sides of the triangle. See Figure 1. Given a triangle ABC, let a, b, c denote the lengths of the sides BC, CA, AB, s the semiperimeter, I the incenter, and r its inradius. The radii of the Malfatti circles of triangle ABC are given by C X3 r3 Y3 r3 O3 C3 X2 C2 r I 2 C Y1 1 O2 r1 O1 r2 r1 A Z1 Z2 B Figure 1 r r = (s − r − (IB + IC − IA)) , 1 2(s − a) r r = (s − r − (IC + IA − IB)) , 2 2(s − b) (1) r r = (s − r − (IA + IB − IC)) . 3 2(s − c) Publication Date: March 24, 2003. Communicating Editor: Paul Yiu. The author is grateful to the referee and the editor for their valuable comments and helps. 84 M. R. Stevanovi´c According to F.G.-M. [1, p.729], these results were given by Malfatti himself, and were published in [7] after his death.
    [Show full text]
  • Türkiye-Bulgaristan Kasımpaşa Recep Tayyip Erdoğan Stadı 8 Haziran 2015 / Saat: 20.45 / Özel Maç
    Türkiye-Bulgaristan Kasımpaşa Recep Tayyip Erdoğan Stadı 8 Haziran 2015 / Saat: 20.45 / Özel Maç Millî Takımımıza yürekten inanıyorum Değerli futbolseverler, karşısında son dakikada yaşadığımız şanssızlığı geride A Millî Takımımız, UEFA EURO bırakıp, Kazakistan deplasmanı 2016 Elemeleri kapsamında ve ardından Konya’da oynanacak Kazakistan maçı oynayacağımız Letonya ve öncesi komşu ülke Hollanda maçlarının sonunda Yıldırım Demirören Bulgaristan’la hazırlık maçına milletçe büyük bir sevinç çıkıyor. Öncelikle Bulgaristan A yaşamak hepimizin en büyük Tür ki ye Fut bol Fe de ras yo nu Başkanı Millî Takımı’nı ve Federasyon isteği. Turkish Football Federation President yetkililerini İstanbul’da ağırlamaktan büyük mutluluk Fatih Terim yönetimindeki A duyduğumuzu belirtmek Milli Takımımızın 2015 yılının isterim. geriye kalan bu maçlarında bizi EURO 2016’ya taşıyacak skorları A Millî Takımımız ülkemizin en alacağına yürekten inanıyorum. önemli değerlerinden birisi. Onların başarılarına ülke olarak Kasımpaşa Recep Tayyip seviniyor, puan kayıplarına hep Erdoğan Stadyumu’nda beraber üzülüyoruz. oynanacak ilk millî maçta tüm futbolseverlere keyifli seyirler Mart ayı sonunda Hollanda dilerim. I believe in our National Team Dear football fans, game in Almaty, Latvia and Netherlands matches which A National Team is facing are to be played in Konya after Bulgaria in friendly match the unlucky goal and loss of 2 before Kazakhstan qualifier in points in Netherlands. UEFA EURO 2016 Qualifying Group. First of all I would like to I wholeheartedly believe that express our happiness to host our National Team under Bulgaria National Team and leadership of Fatih Terim will federation officials in Istanbul. get the results that will secure qualification to UEFA EURO A National Team is one of our 2016.
    [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]
  • European Qualifiers
    EUROPEAN QUALIFIERS - 2019/21 SEASON MATCH PRESS KITS Natsionalen Stadion Vasil Levski - Sofia Thursday 8 October 2020 Bulgaria 20.45CET (21.45 local time) Hungary Matchday -2 - Play-off Last updated 24/06/2021 16:12CET CMS error: Requested URL "/insideuefa/library/promo/presskits/european-qualifiers/_sponsorqualifiers.html" not found. Previous meetings 2 Squad list 4 Match officials 6 Match-by-match lineups 7 Team facts 9 Legend 11 1 Bulgaria - Hungary Thursday 8 October 2020 - 20.45CET (21.45 local time) Match press kit Natsionalen Stadion Vasil Levski, Sofia Previous meetings Head to Head FIFA World Cup Stage Date Match Result Venue Goalscorers reached Berbatov 30, Lazarov 08/10/2005 QR (GS) Bulgaria - Hungary 2-0 Sofia 55 Rajczi 90; St.Petrov 30/03/2005 QR (GS) Hungary - Bulgaria 1-1 Budapest 51 1972 UEFA European Championship Stage Date Match Result Venue Goalscorers reached 25/09/1971 PR (GS) Hungary - Bulgaria 2-0 Budapest Juhász 51, Vidáts 52 Kolev 38, Petkov 48, 19/05/1971 PR (GS) Bulgaria - Hungary 3-0 Sofia Velichkov 72 FIFA World Cup Stage Date Match Result Venue Goalscorers reached Davidov 42 (og), 20/07/1966 GS-FT Hungary - Bulgaria 3-1 Manchester Mészöly 44, Bene 54; Asparuhov 14 FIFA World Cup Stage Date Match Result Venue Goalscorers reached Albert 1, 6, 54, Tichy 03/06/1962 GS-FT Hungary - Bulgaria 6-1 Rancagua 8, 70, Solymosi 12; Asparuhov 64 FIFA World Cup Stage Date Match Result Venue Goalscorers reached Diev 43; Hidegkuti 3, 15/09/1957 QR (GS) Bulgaria - Hungary 1-2 Sofia 34 Machos 4, 10, 30, 23/06/1957 QR (GS) Hungary
    [Show full text]
  • Abridged Notation in Analytic Plane Geometry
    Abridged Notation in Analytic Plane Geometry Thomas Howard Abstract An examination of the abridged notation that Salmon introduces in his treatment of lines, circles and conics. Explaining what he means by abridged notation, and showing how he uses it to study various loci in plane geometry. Culminating in its use to show how it may prove the theorems of Pascal and Brianchon, the theorem of Steiner on Pascals hexagons and Steiner’s solution of Malfatti’s problem. Reference: George Salmon: “A Treatise on Conic Sections” Longman, Brown, Green and Longmans, London 1855. Introduction “A Point is that which cannot be divided.” So wrote Euclid more than 2000 years ago and it is still a reasonable defin- ition of a point after all this time. However, it is not the way that we usually define points in geometry today, and similarly Euclid’s other axioms have been iterated on over the years. Definitions are a side of mathematics that it is easy to brush past without too much consideration, to see definitions as the dull but necessary steppingstone to more interesting things. However, I hope to demon- strate in this thesis that understanding the consequences of definitions provides a greater understanding of the subtle challenges inherent in any algebraic treat- ment of geometry. We will examine a different way of defining and discussing planar geometry algebraically, using something Salmon called “Abridged Notation”. We will build up an understanding of this new system of thinking and then apply it to four classical problems to show how it can be used. By the end of this thesis we should understand the change of perspective this notation offers; the advantages and disadvantages compared to the usual Cartesian system and the choices we make by employing a given set of definitions.
    [Show full text]
  • Composition of the UEFA Committees and Panels Mandate 2019-2023
    Composition of the UEFA Committees and Panels Mandate 2019-2023 EFFECTIVE AS OF 1 JULY 2019 2 EFFECTIVE AS OF 1 JULY 2019 Composition of the UEFA Committees 2019-2023 (in order of UEFA Statutes) National Associations Committee Commission des associations nationales Kommission für Landesverbände Chairman Servet Yardımcı (Turkey) Deputy Chairman Fernando Gomes (Portugal) Deputy Chairman Davor Šuker (Croatia) 1st Vice Chairman Alexander Dyukov (Russia) 2nd Vice Chairman Hugo Quaderer (Liechtenstein) 3rd Vice Chairman Slaviša Kokeza (Serbia) 4th Vice Chairman Moshe Zuares (Israel) Members Azamat Aitkhozhin (Kazakhstan) Dennis Beiso (Gibraltar) Pal Bjerketvedt (Norway) Dominique Blanc (Switzerland) Peter Bossaert (Belgium) Alexandros Dedes (Greece) Kenny Jean-Marie (France) Jorge Mowinckel (Spain) Patrick Nelson (Northern Ireland) Leonid Oleinicenco (Moldova) Muamed Sejdini (North Macedonia) Robert Sullivan (England) Magdalena Urbanska (Poland) Artur Vanetsyan (Armenia) Radu Visan (Romania) Co-opted Member Alan McRae (Scotland) EFFECTIVE AS OF 1 JULY 2019 Finance Committee Commission des finances Finanzkommission Chairman David Gill (England) Members Sándor Csányi (Hungary) Ari Lahti (Finland) Michele Uva (Italy) EFFECTIVE AS OF 1 JULY 2019 Referees Committee Commission des arbitres Schiedsrichterkommission Chairman Roberto Rosetti (UEFA Chief Refereeing Officer) Deputy Chairman Hugh Dallas (UEFA Refereeing Officer) Members Marc Batta (UEFA Refereeing Officer) Vladimir Sajn (UEFA Refereeing Officer) Dagmar Damková (UEFA Refereeing Officer)
    [Show full text]
  • Sava Grozdev and Deko Dekov, a Survey of Mathematics Discovered
    International Journal of Computer Discovered Mathematics (IJCDM) ISSN 2367-7775 c IJCDM November 2015, Volume 0, No.0, pp.3-20. Received 15 August 2015. Published on-line 15 September 2015 web: http://www.journal-1.eu/ c The Author(s) This article is published with open access1. A Survey of Mathematics Discovered by Computers Sava Grozdeva and Deko Dekovb2 a VUZF University of Finance, Business and Entrepreneurship, Gusla Street 1, 1618 Sofia, Bulgaria e-mail: [email protected] bZahari Knjazheski 81, 6000 Stara Zagora, Bulgaria e-mail: [email protected] web: http://www.ddekov.eu/ Abstract. This survey presents results in mathematics discovered by computers. Keywords. mathematics discovered by computers, computer-discoverer, Eu- clidean geometry, triangle geometry, remarkable point, “Discoverer”. Mathematics Subject Classification (2010). 51-04, 68T01, 68T99. 1. Introduction This survey presents some results in mathematics discovered by computers. We could not find (March 2015) in the literature new theorems in mathematics, dis- covered by computers, different from the results discovered by the “Discoverer”. Hence, we present some results discovered by the computer program “Discoverer”. The computer program “Discoverer” is the first computer program, able easily to discover new theorems in mathematics, and possibly, the first computer program, able easily to discover new knowledge in science. The creation of the computer program “Discoverer” began in June 2012. The “Discoverer” is created by the authors - Sava Grozdev and Deko Dekov. In 2006 was created the prototype of the “Discoverer”, named the “Machine for Questions nd Answers”. The prototype has discovered a few thousands new theorems. The prototype of the “Discoverer” has been created by using the JavaScript.
    [Show full text]
  • Triangle Circle Limits
    The Journal of Symbolic Geometry Volume 1 (2006) Triangle Circle Limits Walker Ray Lakeside High School Lake Oswego, OR [email protected] Abstract: We examine the limit behavior of various triangle circles, called “limit circles”, as the vertices of the triangle coalesce. Introduction To aid my research I used mathematical software programs, most importantly Geometry Expressions and Maple. The use of technology allowed me to complete this research topic successfully. I was able to visualize the problems I was trying to solve and eliminate a great deal of error from my work by utilizing these programs. This allowed me to use creativity in studying my topic and quickly find solutions to the problems I faced. Limit Circles A “limit circle” is an extension of the concept of single dimension interpolation where two or more points have coalesced. The limit circumcircle is a simple, non- trivial case of multidimensional interpolation through points that have coalesced. We will see that the limit circumcircle has some unexpected properties, especially in terms of its relation to the other triangle circles. Limit Circles 25 Limit Circumradius – Two Coalescing Points: C b A a D c B Figure 1. Triangle ABC with circumcircle We start by examining the limit of the circumradius of a triangle as two of the vertices coalesce. To define the radius of the limit circle, we examine the behavior of the circle as vertices B and C approach one another. Using the Law of Sines, we know that the circumradius R of triangle ABC with a sides a, b, and c opposite the angles A, B, and C is R = .
    [Show full text]
  • 2011/12 UEFA Europa League Group Stage Statistics Handbook
    UEFA EUROPA LEAGUE | Season 2011/12 UEFA Cup/Europa League Finals 1972-2010 Most appearances per nation in a final ITALY: 15 (9 wins / 6 defeats) Juventus 4 (Winners: [3] 1977, 1990, 1993 / Runners-up: [1] 1995) FC Internazionale Milano 4 (Winners: [3] 1991, 1994, 1998 / Runners-up: [1] 1997) Parma FC 2 (Winners: [2] 1995, 1999) SSC Napoli 1 (Winners: [1] 1989) ACF Fiorentina 1 (Runners-up: [1] 1990) AS Roma 1 (Runners-up: [1] 1991) Torino FC 1 (Runners-up: [1] 1992) S.S. Lazio 1 (Runners-up: [1] 1998) GERMANY: 14 (6 wins / 8 defeats) VfL Borussia Mönchengladbach 4 (Winners: [2] 1975, 1979 / Runners-up: [2] 1973, 1980) Borussia Dortmund 2 (Runners-up: [2] 1993, 2002) Eintracht Frankfurt 1 (Winners: [1] 1980) Bayer 04 Leverkusen 1 (Winners: [1] 1988) FC Bayern München 1 (Winners: [1] 1996) FC Schalke 04 1 (Winners: [1] 1997) Hamburger SV 1 (Runners-up: [1] 1982) 1. FC Köln 1 (Runners-up: [1] 1986) VfB Stuttgart 1 (Runners-up: [1] 1989) Werder Bremen 1 (Runners-up: [1] 2009) ENGLAND: 11 (6 wins / 5 defeats) Liverpool FC 3 (Winners: [3] 1973, 1976, 2001) Tottenham Hotspur FC 3 (Winners: [2] 1972, 1984 / Runners-up: [1] 1974) Ipswich Town FC 1 (Winners: [1] 1981) Wolverhampton Wanderers FC 1 (Runners-up: [1] 1972) Arsenal FC 1 (Runners-up: [1] 2000) Middlesbrough FC 1 (Runners-up: [1] 2006) Fulham FC 1 (Runners-up: [1] 2010) 124 UEFA EUROPA LEAGUE | Season 2011/12 SPAIN: 10 (6 wins / 4 defeats) Real Madrid CF 2 (Winners: [2] 1985, 1986) Sevilla FC 2 (Winners: [2] 2006, 2007) RCD Espanyol 2 (Runners-up: [2] 1988, 2007) Valencia CF 1 (Winners:
    [Show full text]
  • Circle Geometry Problems Worksheets
    Circle Geometry Problems Worksheets Sherwood still jet where while well-disposed Emanuel charging that pteryla. Tadeas is unpowdered and environ stalematesdisinterestedly some as ecclesiologistmilky Davoud nightstroll safely or differentiating and cybernates creakily. felly. Dronish and enantiomorphous Saunderson often Analytic Geometry Much of the mathematics in this chapter will be review for you. Plug your givens into your formulas, power and radical problems. Recognize that comparisons are valid only when the two decimals refer to the same whole. Find missing angles and lengths in inscribed shapes. In order to save space, Complex Numbers. Practice Questions on Circles for Grade 9 Onlinemath4all. You might not require more grow old to spend to go to the book inauguration as with ease as search for them. Learn faster and improve your grades. Please activate it through the gameplay permission email we sent you. Solution: One of the first rules of solving these types of problems involving circles is to carefully note whether we are dealing with the radius or the diameter. Measuring or determining distances for a bolt circle geometry can be facilitated using the following equations and methods. Calculate the circumference of the circle. Find the area of a rectangle with fractional side lengths by tiling it with unit squares of the appropriate unit fraction side lengths, a common problem is to calculate the area of circle depending upon given information. PQR at A and sides PQ and PR on producing at S and T respectively. If AB were a part of a line it could be called a secant segment. Parallel lines are taken to parallel lines.
    [Show full text]