American Mathematical Monthly Geometry Problems 1894 –
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13-6 Success for English Language Learners
LESSON Success for English Language Learners 13-6 The Law of Cosines Steps for Success Step I In order to create interest in the lesson opener, point out the following to students. • Trapeze, trapezium (a geometric figure as well as a bone in the wrist), trapezoid (also a geometric figure and a bone in the wrist), and trapezius (a muscle in the back) all come from the same root. They come from the Greek word trapeza, or “four-legged table.” Step II Teach the lesson. • Have students derive the remaining formulas in the Law of Cosines that were not derived in the text. • Technically speaking, a knot is not a nautical mile. A knot is a speed of one nautical mile per hour. • Heron’s Formula is also known as Hero’s Formula. A Heronian triangle is a triangle having rational side lengths and a rational area. Step III Ask English Language Learners to complete the worksheet for this lesson. • Point out that Example 1A in the student textbook is supported by Problem 1 on the worksheet. Remind students that, for example, side a is opposite angle A, not adjacent to it. • Point out that Example 3 in the student textbook is supported by Problem 2 on the worksheet. • Think and Discuss supports the problems on the worksheet. Making Connections • Students comfortableᎏᎏ with matrices may wish to verify the following equation: If ᭝ ϭ ͙s ͑ s Ϫ a ͒ ͑ s Ϫ b ͒ ͑ s Ϫ c ͒ is the area of the triangle under consideration, then 2 Ϫ1 11 a 2 2 2 2 Ϫ 2 ͑ 4 ͒ ϭ a b c 1 1 1 b . -
Downloaded from Bookstore.Ams.Org 30-60-90 Triangle, 190, 233 36-72
Index 30-60-90 triangle, 190, 233 intersects interior of a side, 144 36-72-72 triangle, 226 to the base of an isosceles triangle, 145 360 theorem, 96, 97 to the hypotenuse, 144 45-45-90 triangle, 190, 233 to the longest side, 144 60-60-60 triangle, 189 Amtrak model, 29 and (logical conjunction), 385 AA congruence theorem for asymptotic angle, 83 triangles, 353 acute, 88 AA similarity theorem, 216 included between two sides, 104 AAA congruence theorem in hyperbolic inscribed in a semicircle, 257 geometry, 338 inscribed in an arc, 257 AAA construction theorem, 191 obtuse, 88 AAASA congruence, 197, 354 of a polygon, 156 AAS congruence theorem, 119 of a triangle, 103 AASAS congruence, 179 of an asymptotic triangle, 351 ABCD property of rigid motions, 441 on a side of a line, 149 absolute value, 434 opposite a side, 104 acute angle, 88 proper, 84 acute triangle, 105 right, 88 adapted coordinate function, 72 straight, 84 adjacency lemma, 98 zero, 84 adjacent angles, 90, 91 angle addition theorem, 90 adjacent edges of a polygon, 156 angle bisector, 100, 147 adjacent interior angle, 113 angle bisector concurrence theorem, 268 admissible decomposition, 201 angle bisector proportion theorem, 219 algebraic number, 317 angle bisector theorem, 147 all-or-nothing theorem, 333 converse, 149 alternate interior angles, 150 angle construction theorem, 88 alternate interior angles postulate, 323 angle criterion for convexity, 160 alternate interior angles theorem, 150 angle measure, 54, 85 converse, 185, 323 between two lines, 357 altitude concurrence theorem, -
An Innovative Analysis to Develop New Theorems on Irregular Polygon
International Journal of Physics and Mathematical Sciences ISSN: 2277-2111 (Online) An Online International Journal Available at http://www.cibtech.org/jpms.htm 2013 Vol. 3 (1) January-March, pp.73-81/Kalaimaran Research Article AN INNOVATIVE ANALYSIS TO DEVELOP NEW THEOREMS ON IRREGULAR POLYGON *Kalaimaran Ara Construction & Civil Maintenance Unit, Central Food Technological Research Institute, Mysore-20, Karnataka, India *Author for Correspondence ABSTRACT The irregular Polygon is a four sided polygon of two dimensional geometrical figures. The triangle, square, rectangle, tetragon, pentagon, hexagon, heptagon, octagon, nonagon, dodecagon, parallelogram, rhombus, rhomboid, trapezium or trapezoidal, kite and dart are the members of the irregular polygon family. A polygon is a two dimensional example of the more general prototype in any number of dimensions. However the properties are varied from one to another. The author has attempted to develop two new theorems for the property of irregular polygon for a point anywhere inside of the polygon with necessary illustrations, appropriate examples and derivation of equations for better understanding. Key Words: Irregular Polygon, Triangle, Right-angled triangle, Perpendicular and Vertex INTRODUCTION Polygon (Weisstein, 2003) is a closed two dimensional figure formed by connecting three or more straight line segments, where each line segment end connects to only one end of two other line segments. Polygon is one of the most all-encompassing shapes in two- dimensional geometry. The sum of the interior angles is equal to 180 degree multiplied by number of sides minus two. The sum of the exterior angles is equal to 360 degree. From the simple triangle up through square, rectangle, tetragon, pentagon, hexagon, heptagon, octagon, nonagon, dodecagon (Weisstein, 2003,) and beyond is called n-gon. -
The Brahmagupta Triangles Raymond A
The Brahmagupta Triangles Raymond A. Beauregard and E. R. Suryanarayan Ray Beauregard ([email protected]) received his Ph.D. at the University of New Hampshire in 1968, then joined the University of Rhode Island, where he is a professor of mathematics. He has published many articles in ring theory and two textbooks. Linear Algebra (written with John Fraleigh) is currently in its third edition. Besides babysitting for his grandchild Elyse, he enjoys sailing the New England coast on his sloop, Aleph One, and playing the piano. E. R. Suryanarayan ([email protected]) taught at universities in India before receiving his Ph.D. (1961) at the University of Michigan, under Nathaniel Coburn. He has been at the University of Rhode Island since 1960, where is a professor of mathematics. An author of more than 20 research articles in applied mathematics, crystallography, and the history of mathematics, he lists as his main hobbies music, languages, and aerobic walking. The study of geometric objects has been a catalyst in the development of number theory. For example, the figurate numbers (triangular, square, pentagonal, . ) were a source of many early results in this field [41.Measuring the length of a diagonal of a rectangle led to the problem of approxin~atingfi for a natural number N. The study of triangles has been of particular significance. Heron of Alexandria (c. A.D. 75)-gave the well-known formula for the area A of a triangle in terms of its sides: A = Js(s - a)(s- b)(s- c),where s = (a + b + c)/2 is the semiperimeter of the triangle having sides a,b, c [41.He illustrated this with a triangle whose sides are 13,14,15 and whose area is 84. -
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.................................................................................................................. -
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). -
Some Relations and Properties Concerning Tangential Polygons
View metadata, citation and similar papers at core.ac.uk brought to you by CORE Mathematical Communications 4(1999), 197-206 197 Some relations and properties concerning tangential polygons Mirko Radic´∗ Abstract. The k-tangential polygon is defined, and some of its properties are proved. Key words: k-tangential polygon AMS subject classifications: 51E12 Received October 10, 1998 Accepted June 7, 1999 1. Preliminaries A polygon with the vertices A1, ..., An (in this order) will be denoted by A1...An. The lengths of the sides of the polygon A1...An will be denoted by |A1A2|, ..., |AnA1| or a1, ..., an. The interior angle at the vertex Ai will be denoted by αi or ∠Ai, i.e. ∠Ai = ∠An−1+iAiAi+1,i=1, ..., n (0 <αi <π). (1) Of course, indices are calculated modulo n. A polygon A = A1...An is a tangential polygon if there exists a circle C such that each side of A is on a tangent line of C. Definition 1. Let A = A1...An be a tangential polygon, and let k be a positive ≤bn−1 c ≤ n−1 ≤ n−2 integer such that k 2 , that is, k 2 if n is odd and k 2 if n is even. Then the polygon A will be called a k-tangential polygon if any two of its consecutive sides have only one point in common, and if there holds π β + ···+ β =(n − 2k) , (2) 1 n 2 where 2βi = ∠Ai, i =1, ..., n. Consequently, a tangential polygon A is k-tangential if n X ϕi =2kπ, i=1 where ϕi = ∠AiCAi+1 and C is the centre of the circle inscribed into the polygon A. -
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. -
Pythagorean Triples Before and After Pythagoras
computation Review Pythagorean Triples before and after Pythagoras Ravi P. Agarwal Department of Mathematics, Texas A&M University-Kingsville, 700 University Blvd., Kingsville, TX 78363, USA; [email protected] Received: 5 June 2020; Accepted: 30 June 2020; Published: 7 July 2020 Abstract: Following the corrected chronology of ancient Hindu scientists/mathematicians, in this article, a sincere effortPythagorean is made to report Triples the origin Before of Pythagorean and After triples. Pythagoras We shall account for the development of these triples from the period of their origin and list some known astonishing directions. Although for researchers in this field, there is not much that is new in this article, we genuinely hope students and teachers of mathematicsRavi P Agarwal will enjoy this article and search for new directions/patterns. Department of Mathematics, Texas A&M University-Kingsville 700 University Blvd., Kingsville, TX, USA Keywords: Pythagorean triples; [email protected] and patterns; extensions; history; problems AMS Subject Classification: 01A16; 0A25; 0A32; 11-02; 11-03; 11D09 Abstract: Following the corrected chronology of ancient Hindu scientists/mathematicians by Lakshmikan- tham, et. al. [27], in this article a sincere effort has been made to report the origin of Pythagorean triples. We shall account the development of these triples from the period of its origin, and list some known aston- ishing directions. Although, for researchers in this field there is not much new in this article, we genuinely 1. Introductionhope -
Arxiv:1712.00299V1 [Math.GT]
POLYGONS WITH PRESCRIBED EDGE SLOPES: CONFIGURATION SPACE AND EXTREMAL POINTS OF PERIMETER JOSEPH GORDON, GAIANE PANINA, YANA TEPLITSKAYA Abstract. We describe the configuration space S of polygons with pre- scribed edge slopes, and study the perimeter as a Morse function on S. We characterize critical points of (these areP tangential polygons) and compute their Morse indices. ThisP setup is motivated by a number of re- sults about critical points and Morse indices of the oriented area function defined on the configuration space of polygons with prescribed edge lengths (flexible polygons). As a by-product, we present an independent computa- tion of the Morse index of the area function (obtained earlier by G. Panina and A. Zhukova). 1. Introduction Consider the space L of planar polygons with prescribed edge lengths1 and the oriented area as a Morse function defined on it. It is known that gener- ically: A L is a smooth closed manifold whose diffeomorphic type depends on • the edge lengths [1, 2]. The oriented area is a Morse function whose critical points are cyclic • configurations (thatA is, polygons with all the vertices lying on a cir- cle), whose Morse indices are known, see Theorem 2, [3, 5, 6]. The Morse index depends not only on the combinatorics of a cyclic poly- arXiv:1712.00299v1 [math.GT] 1 Dec 2017 gon, but also on some metric data. Direct computations of the Morse index proved to be quite involved, so the existing proof comes from bifurcation analysis combined with a number of combinatorial tricks. Bifurcations of are captured by cyclic polygons P whose dual poly- • gons P ∗ have zeroA perimeter [5]; see also Lemma 4. -
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. -
(Theorem 1) Is Proved
Rad HAZU Volume 503 (2009), 41–54 ABOUT ONE RELATION CONCERNING TWO CIRCLES MIRKO RADIC´ AND ZORAN KALIMAN Abstract. This article can be considered as an appendix to the article [1]. Here the article [1] is extended to the cases when one circle is outside of the other and when circles are intersecting. 1. Preliminaries In [1] the following theorem (Theorem 1) is proved: Let C1 and C2 be any given two circles such that C1 is inside of the C2 and let A1 , A2 , A3 be any given three different points on C2 such that there are points T1 and T2 on C1 with properties |A1A2| = t1 +t2, |A2A3| = t2 +t3, (1) where t1 = |A1T1|, t2 = |T1A2|, t3 = |T2A3|.Then 2rR |A A | =(t +t ) , (2) 1 3 1 3 R2 − d2 where r = radius of C1 , R = radius of C2 , d = |IO|, I is the center of C1 , O is center of C2 .(SeeFigure1.) In short about the proof of this theorem. First the following lemma is proved. It t1 is given then t2 can be calculated using the expression √ t (R2 − d2) ± D (t ) = 1 1 (3a) 2 1,2 2 + 2 r t1 where = 2( 2 − 2)2 +( 2 + 2) 2 2 − 2 2 − ( 2 + 2 − 2)2 . D1 t1 R d r t1 4R d r t1 R d r (3b) The values (t2)1,2 given by (3) are solutions of the equation ( 2 + 2) 2 − ( 2 − 2)+ 2 2 − 2 2 +( 2 + 2 − 2)2 = . r t1 t2 2t1t2 R d r t1 4R d R d r 0 (4) Mathematics subject classification (2000): 51M04.