Topics in Elementary Geometry
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Angle Chasing
Angle Chasing Ray Li June 12, 2017 1 Facts you should know 1. Let ABC be a triangle and extend BC past C to D: Show that \ACD = \BAC + \ABC: 2. Let ABC be a triangle with \C = 90: Show that the circumcenter is the midpoint of AB: 3. Let ABC be a triangle with orthocenter H and feet of the altitudes D; E; F . Prove that H is the incenter of 4DEF . 4. Let ABC be a triangle with orthocenter H and feet of the altitudes D; E; F . Prove (i) that A; E; F; H lie on a circle diameter AH and (ii) that B; E; F; C lie on a circle with diameter BC. 5. Let ABC be a triangle with circumcenter O and orthocenter H: Show that \BAH = \CAO: 6. Let ABC be a triangle with circumcenter O and orthocenter H and let AH and AO meet the circumcircle at D and E, respectively. Show (i) that H and D are symmetric with respect to BC; and (ii) that H and E are symmetric with respect to the midpoint BC: 7. Let ABC be a triangle with altitudes AD; BE; and CF: Let M be the midpoint of side BC. Show that ME and MF are tangent to the circumcircle of AEF: 8. Let ABC be a triangle with incenter I, A-excenter Ia, and D the midpoint of arc BC not containing A on the circumcircle. Show that DI = DIa = DB = DC: 9. Let ABC be a triangle with incenter I and D the midpoint of arc BC not containing A on the circumcircle. -
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). -
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, -
Mathematical Gems
AMS / MAA DOLCIANI MATHEMATICAL EXPOSITIONS VOL 1 Mathematical Gems ROSS HONSBERGER MATHEMATICAL GEMS FROM ELEMENTARY COMBINATORICS, NUMBER THEORY, AND GEOMETRY By ROSS HONSBERGER THE DOLCIANI MATHEMATICAL EXPOSITIONS Published by THE MArrHEMATICAL ASSOCIATION OF AMERICA Committee on Publications EDWIN F. BECKENBACH, Chairman 10.1090/dol/001 The Dolciani Mathematical Expositions NUMBER ONE MATHEMATICAL GEMS FROM ELEMENTARY COMBINATORICS, NUMBER THEORY, AND GEOMETRY By ROSS HONSBERGER University of Waterloo Published and Distributed by THE MATHEMATICAL ASSOCIATION OF AMERICA © 1978 by The Mathematical Association of America (Incorporated) Library of Congress Catalog Card Number 73-89661 Complete Set ISBN 0-88385-300-0 Vol. 1 ISBN 0-88385-301-9 Printed in the United States of Arnerica Current printing (last digit): 10 9 8 7 6 5 4 3 2 1 FOREWORD The DOLCIANI MATHEMATICAL EXPOSITIONS serIes of the Mathematical Association of America came into being through a fortuitous conjunction of circumstances. Professor-Mary P. Dolciani, of Hunter College of the City Uni versity of New York, herself an exceptionally talented and en thusiastic teacher and writer, had been contemplating ways of furthering the ideal of excellence in mathematical exposition. At the same time, the Association had come into possession of the manuscript for the present volume, a collection of essays which seemed not to fit precisely into any of the existing Associa tion series, and yet which obviously merited publication because of its interesting content and lucid expository style. It was only natural, then, that Professor Dolciani should elect to implement her objective by establishing a revolving fund to initiate this series of MATHEMATICAL EXPOSITIONS. -
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. -
Cyclic Quadrilaterals — the Big Picture Yufei Zhao [email protected]
Winter Camp 2009 Cyclic Quadrilaterals Yufei Zhao Cyclic Quadrilaterals | The Big Picture Yufei Zhao [email protected] An important skill of an olympiad geometer is being able to recognize known configurations. Indeed, many geometry problems are built on a few common themes. In this lecture, we will explore one such configuration. 1 What Do These Problems Have in Common? 1. (IMO 1985) A circle with center O passes through the vertices A and C of triangle ABC and intersects segments AB and BC again at distinct points K and N, respectively. The circumcircles of triangles ABC and KBN intersects at exactly two distinct points B and M. ◦ Prove that \OMB = 90 . B M N K O A C 2. (Russia 1995; Romanian TST 1996; Iran 1997) Consider a circle with diameter AB and center O, and let C and D be two points on this circle. The line CD meets the line AB at a point M satisfying MB < MA and MD < MC. Let K be the point of intersection (different from ◦ O) of the circumcircles of triangles AOC and DOB. Show that \MKO = 90 . C D K M A O B 3. (USA TST 2007) Triangle ABC is inscribed in circle !. The tangent lines to ! at B and C meet at T . Point S lies on ray BC such that AS ? AT . Points B1 and C1 lies on ray ST (with C1 in between B1 and S) such that B1T = BT = C1T . Prove that triangles ABC and AB1C1 are similar to each other. 1 Winter Camp 2009 Cyclic Quadrilaterals Yufei Zhao A B S C C1 B1 T Although these geometric configurations may seem very different at first sight, they are actually very related. -
The Role of Euclidean Geometry in High School
JOURNAL OF MATHEMATICAL BEHAVIOR 15, 221-237 (1996) The Role of Euclidean Geometry in High School HUNG-HSI Wu University of California When I was a high school student long ago, the need to study Euclidean geome- try was taken for granted. In fact, the novelty of learning to prove something was so overwhelming to me and some of my friends that, years later, we would look back at Euclidean geometry as the high point of our mathematics education. In recent years, the value of Euclid has depreciated considerably. A vigorous debate is now going on about how much, if any, of his work is still relevant. In this article, I would like to give my perspective on this subject, and in so doing, I will be taking a philosophical tour, so to speak, of a small portion of mathematics. This may not be so surprising because after all, in life as in mathematics, one does need a little philosophy for guidance from time to time. The bone of contention in the geometry curriculum is of course how many of the traditional “two-column proofs” should be retained. Let me first briefly indicate the range of available options in this regard: 1. A traditional text in use in a local high school in Berkeley not too long ago begins on page 1 with the undefined terms of the axioms, and formal proofs start on page 22. There is no motivation or explanation of the whys and hows of an axiomatic system. 2. A recently published text does experimental geometry (i.e., “hands-on ge- ometry” with no proofs) all the way until the last 130 pages of its 700 pages of exposition. -
A Conic Through Six Triangle Centers
Forum Geometricorum b Volume 2 (2002) 89–92. bbb FORUM GEOM ISSN 1534-1178 A Conic Through Six Triangle Centers Lawrence S. Evans Abstract. We show that there is a conic through the two Fermat points, the two Napoleon points, and the two isodynamic points of a triangle. 1. Introduction It is always interesting when several significant triangle points lie on some sort of familiar curve. One recently found example is June Lester’s circle, which passes through the circumcenter, nine-point center, and inner and outer Fermat (isogonic) points. See [8], also [6]. The purpose of this note is to demonstrate that there is a conic, apparently not previously known, which passes through six classical triangle centers. Clark Kimberling’s book [6] lists 400 centers and innumerable collineations among them as well as many conic sections and cubic curves passing through them. The list of centers has been vastly expanded and is now accessible on the internet [7]. Kimberling’s definition of triangle center involves trilinear coordinates, and a full explanation would take us far afield. It is discussed both in his book and jour- nal publications, which are readily available [4, 5, 6, 7]. Definitions of the Fermat (isogonic) points, isodynamic points, and Napoleon points, while generally known, are also found in the same references. For an easy construction of centers used in this note, we refer the reader to Evans [3]. Here we shall only require knowledge of certain collinearities involving these points. When points X, Y , Z, . are collinear we write L(X,Y,Z,...) to indicate this and to denote their common line. -
Prove It Classroom
How to Prove it ClassRoom SHAILESH SHIRALI Euler’s formula for the area of a pedal triangle Given a triangle ABC and a point P in the plane of ABC In this episode of “How (note that P does not have to lie within the triangle), the To Prove It”, we prove pedal triangle of P with respect to ABC is the triangle △ a beautiful and striking whose vertices are the feet of the perpendiculars drawn from formula first found by P to the sides of ABC. See Figure 1. The pedal triangle relates Leonhard Euler; it gives the area of the pedal triangle in a natural way to the parent triangle, and we may wonder of a point with reference whether there is a convenient formula giving the area of the to another triangle. pedal triangle in terms of the parameters of the parent triangle. The great 18th-century mathematician Euler found just such a formula (given in Box 1). It is a compact and pleasing result, and it expresses the area of the pedal triangle in terms of the radius R of the circumcircle of ABC and the △ distance between P and the centre O of the circumcircle. A O: circumcentre of ABC E △ P: arbitrary point Area ( DEF) 1 OP2 F △ = 1 P Area ( ABC) 4 − R2 O △ ( ) B D C Figure 1. Euler's formula for the area of the pedal triangle of an arbitrary point Keywords: Circle theorem, pedal triangle, power of a point, Euler, sine rule, extended sine rule, Wallace-Simson theorem Azim Premji University At Right Angles, July 2018 95 1 A E F P O B D C Figure 2. -
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. -
Isodynamic Points of the Triangle
Isodynamic points of the triangle A file of the Geometrikon gallery by Paris Pamfilos It is in the nature of an hypothesis, when once a man has conceived it, that it assimilates every thing to itself, as proper nourishment; and, from the first moment of your begetting it, it generally grows the stronger by every thing you see, hear, read, or understand. L. Sterne, Tristram Shandy, II, 19 Contents (Last update: 09‑03‑2021) 1 Apollonian circles of a triangle 1 2 Isodynamic points, Brocard and Lemoine axes 3 3 Given Apollonian circles 6 4 Given isodynamic points 8 5 Trilinear coordinates of the isodynamic points 11 6 Same isodynamic points and same circumcircle 12 7 A matter of uniqueness 14 1 Apollonian circles of a triangle Denote by {푎 = |퐵퐶|, 푏 = |퐶퐴|, 푐 = |퐴퐵|} the lengths of the sides of triangle 퐴퐵퐶. The “Apollonian circle of the triangle relative to side 퐵퐶” is the locus of points 푋 which satisfy A X D' BCD Figure 1: The Apollonian circle relative to the side 퐵퐶 the condition |푋퐵|/|푋퐶| = 푐/푏. By its definition, this circle passes through the vertex 퐴 and the traces {퐷, 퐷′} of the bisectors of 퐴̂ on 퐵퐶 (see figure 1). Since the bisectors are orthogonal, 퐷퐷′ is a diameter of this circle. Analogous is the definition of the Apollo‑ nian circles on sides 퐶퐴 and 퐴퐵. The following theorem ([Joh60, p.295]) gives another characterization of these circles in terms of Pedal triangles. 1 Apollonian circles of a triangle 2 Theorem 1. The Apollonian circle 훼, passing through 퐴, is the locus of points 푋, such that their pedal triangles are isosceli relative to the vertex lying on 퐵퐶 (see figure 2).