Notes on Euclidean Geometry Kiran Kedlaya Based On
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Algebra/Geometry/Trigonometry App Samples
Algebra/Geometry/Trigonometry App Samples Holt McDougal Algebra 1 HMH Fuse: Algebra 1- HMH Fuse is the first core K-12 education solution developed exclusively for the iPad. The portability of a complete classroom course on an iPad enables students to learn in the classroom, on the bus, or at home—anytime, anywhere—with engaging content that provides an individually-tailored learning experience. Students and educators using HMH Fuse: will benefit from: •Instructional videos that teach or re-teach all key concepts •Math Motion is a step-by-step interactive demonstration that displays the process to solve complex equations •Homework Help provides at-home support for intricate problems by providing hints for each step in the solution •Vocabulary support throughout with links to a complete glossary that includes audio definitions •Tips, hints, and links that enable students to acquire the help they need to understand the lessons every step of the way •Quizzes that assess student’s skills before they begin a concept and at strategic points throughout the chapters. Instant, automatic grading of quizzes lets students know exactly how they have performed •Immediate assessment results sent to teachers so they can better differentiate instruction. Sample- Cost is Free, Complete App price- $59.99 Holt McDougal HMH Fuse: Geometry- Following our popular HMH Fuse: Algebra 1 app, HMH Fuse: Geometry is the newest offering in the HMH Fuse series. HMH Fuse: Geometry will allow you a sneak peek at the future of mobile geometry curriculum and includes a FREE sample chapter. HMH Fuse is the first core K-12 education solution developed exclusively for the iPad. -
Relationships Between Six Excircles
Sangaku Journal of Mathematics (SJM) c SJM ISSN 2534-9562 Volume 3 (2019), pp.73-90 Received 20 August 2019. Published on-line 30 September 2019 web: http://www.sangaku-journal.eu/ c The Author(s) This article is published with open access1. Relationships Between Six Excircles Stanley Rabinowitz 545 Elm St Unit 1, Milford, New Hampshire 03055, USA e-mail: [email protected] web: http://www.StanleyRabinowitz.com/ Abstract. If P is a point inside 4ABC, then the cevians through P divide 4ABC into smaller triangles of various sizes. We give theorems about the rela- tionship between the radii of certain excircles of some of these triangles. Keywords. Euclidean geometry, triangle geometry, excircles, exradii, cevians. Mathematics Subject Classification (2010). 51M04. 1. Introduction Let P be any point inside a triangle ABC. The cevians through P divide 4ABC into six smaller triangles. In a previous paper [5], we found relationships between the radii of the circles inscribed in these triangles. For example, if P is at the orthocenter H, as shown in Figure 1, then we found that r1r3r5 = r2r4r6, where the ri are radii of the incircles as shown in the figure. arXiv:1910.00418v1 [math.HO] 28 Sep 2019 Figure 1. r1r3r5 = r2r4r6 In this paper, we will find similar results using excircles instead of incircles. When the cevians through a point P interior to a triangle ABC are drawn, many smaller 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. -
LINEAR ALGEBRA METHODS in COMBINATORICS László Babai
LINEAR ALGEBRA METHODS IN COMBINATORICS L´aszl´oBabai and P´eterFrankl Version 2.1∗ March 2020 ||||| ∗ Slight update of Version 2, 1992. ||||||||||||||||||||||| 1 c L´aszl´oBabai and P´eterFrankl. 1988, 1992, 2020. Preface Due perhaps to a recognition of the wide applicability of their elementary concepts and techniques, both combinatorics and linear algebra have gained increased representation in college mathematics curricula in recent decades. The combinatorial nature of the determinant expansion (and the related difficulty in teaching it) may hint at the plausibility of some link between the two areas. A more profound connection, the use of determinants in combinatorial enumeration goes back at least to the work of Kirchhoff in the middle of the 19th century on counting spanning trees in an electrical network. It is much less known, however, that quite apart from the theory of determinants, the elements of the theory of linear spaces has found striking applications to the theory of families of finite sets. With a mere knowledge of the concept of linear independence, unexpected connections can be made between algebra and combinatorics, thus greatly enhancing the impact of each subject on the student's perception of beauty and sense of coherence in mathematics. If these adjectives seem inflated, the reader is kindly invited to open the first chapter of the book, read the first page to the point where the first result is stated (\No more than 32 clubs can be formed in Oddtown"), and try to prove it before reading on. (The effect would, of course, be magnified if the title of this volume did not give away where to look for clues.) What we have said so far may suggest that the best place to present this material is a mathematics enhancement program for motivated high school students. -
Projective Geometry: a Short Introduction
Projective Geometry: A Short Introduction Lecture Notes Edmond Boyer Master MOSIG Introduction to Projective Geometry Contents 1 Introduction 2 1.1 Objective . .2 1.2 Historical Background . .3 1.3 Bibliography . .4 2 Projective Spaces 5 2.1 Definitions . .5 2.2 Properties . .8 2.3 The hyperplane at infinity . 12 3 The projective line 13 3.1 Introduction . 13 3.2 Projective transformation of P1 ................... 14 3.3 The cross-ratio . 14 4 The projective plane 17 4.1 Points and lines . 17 4.2 Line at infinity . 18 4.3 Homographies . 19 4.4 Conics . 20 4.5 Affine transformations . 22 4.6 Euclidean transformations . 22 4.7 Particular transformations . 24 4.8 Transformation hierarchy . 25 Grenoble Universities 1 Master MOSIG Introduction to Projective Geometry Chapter 1 Introduction 1.1 Objective The objective of this course is to give basic notions and intuitions on projective geometry. The interest of projective geometry arises in several visual comput- ing domains, in particular computer vision modelling and computer graphics. It provides a mathematical formalism to describe the geometry of cameras and the associated transformations, hence enabling the design of computational ap- proaches that manipulates 2D projections of 3D objects. In that respect, a fundamental aspect is the fact that objects at infinity can be represented and manipulated with projective geometry and this in contrast to the Euclidean geometry. This allows perspective deformations to be represented as projective transformations. Figure 1.1: Example of perspective deformation or 2D projective transforma- tion. Another argument is that Euclidean geometry is sometimes difficult to use in algorithms, with particular cases arising from non-generic situations (e.g. -
CC Geometry H
CC Geometry H Aim #8: How do we find perimeter and area of polygons in the Cartesian Plane? Do Now: Find the area of the triangle below using decomposition and then using the shoelace formula (also known as Green's theorem). 1) Given rectangle ABCD, a. Identify the vertices. b. Find the exact perimeter. c. Find the area using the area formula. d. List the vertices starting with A moving counterclockwise and apply the shoelace formula. Does the method work for quadrilaterals? 2) Calculate the area using the shoelace formula and then find the perimeter (nearest hundredth). 3) Find the perimeter (to the nearest hundredth) and the area of the quadrilateral with vertices A(-3,4), B(4,6), C(2,-3), and D(-4,-4). 4) A textbook has a picture of a triangle with vertices (3,6) and (5,2). Something happened in printing the book and the coordinates of the third vertex are listed as (-1, ). The answers in the back of the book give the area of the triangle as 6 square units. What is the y-coordinate of this missing vertex? 5) Find the area of the pentagon with vertices A(5,8), B(4,-3), C(-1,-2), D(-2,4), and E(2,6). 6) Show that the shoelace formula used on the trapezoid shown confirms the traditional formula for the area of a trapezoid: 1 (b1 + b2)h 2 D (x3,y) C (x2,y) A B (x ,0) (0,0) 1 7) Find the area and perimeter (exact and nearest hundredth) of the hexagon shown. -
Definition Concurrent Lines Are Lines That Intersect in a Single Point. 1. Theorem 128: the Perpendicular Bisectors of the Sides
14.3 Notes Thursday, April 23, 2009 12:49 PM Definition 1. Concurrent lines are lines that intersect in a single point. j k m Theorem 128: The perpendicular bisectors of the sides of a triangle are concurrent at a point that is equidistant from the vertices of the triangle. This point is called the circumcenter of the triangle. D E F Theorem 129: The bisectors of the angles of a triangle are concurrent at a point that is equidistant from the sides of the triangle. This point is called the incenter of the triangle. A B Notes Page 1 C A B C Theorem 130: The lines containing the altitudes of a triangle are concurrent. This point is called the orthocenter of the triangle. A B C Theorem 131: The medians of a triangle are concurrent at a point that is 2/3 of the way from any vertex of the triangle to the midpoint of the opposite side. This point is called the centroid of the of the triangle. Example 1: Construct the incenter of ABC A B C Notes Page 2 14.4 Notes Friday, April 24, 2009 1:10 PM Examples 1-3 on page 670 1. Construct an angle whose measure is equal to 2A - B. A B 2. Construct the tangent to circle P at point A. P A 3. Construct a tangent to circle O from point P. Notes Page 3 3. Construct a tangent to circle O from point P. O P Notes Page 4 14.5 notes Tuesday, April 28, 2009 8:26 AM Constructions 9, 10, 11 Geometric mean Notes Page 5 14.6 Notes Tuesday, April 28, 2009 9:54 AM Construct: ABC, given {a, ha, B} a Ha B A b c B C a Notes Page 6 14.1 Notes Tuesday, April 28, 2009 10:01 AM Definition: A locus is a set consisting of all points, and only the points, that satisfy specific conditions. -
Special Isocubics in the Triangle Plane
Special Isocubics in the Triangle Plane Jean-Pierre Ehrmann and Bernard Gibert June 19, 2015 Special Isocubics in the Triangle Plane This paper is organized into five main parts : a reminder of poles and polars with respect to a cubic. • a study on central, oblique, axial isocubics i.e. invariant under a central, oblique, • axial (orthogonal) symmetry followed by a generalization with harmonic homolo- gies. a study on circular isocubics i.e. cubics passing through the circular points at • infinity. a study on equilateral isocubics i.e. cubics denoted 60 with three real distinct • K asymptotes making 60◦ angles with one another. a study on conico-pivotal isocubics i.e. such that the line through two isoconjugate • points envelopes a conic. A number of practical constructions is provided and many examples of “unusual” cubics appear. Most of these cubics (and many other) can be seen on the web-site : http://bernard.gibert.pagesperso-orange.fr where they are detailed and referenced under a catalogue number of the form Knnn. We sincerely thank Edward Brisse, Fred Lang, Wilson Stothers and Paul Yiu for their friendly support and help. Chapter 1 Preliminaries and definitions 1.1 Notations We will denote by the cubic curve with barycentric equation • K F (x,y,z) = 0 where F is a third degree homogeneous polynomial in x,y,z. Its partial derivatives ∂F ∂2F will be noted F ′ for and F ′′ for when no confusion is possible. x ∂x xy ∂x∂y Any cubic with three real distinct asymptotes making 60◦ angles with one another • will be called an equilateral cubic or a 60. -
Developing Creative Thinking in Mathematics: Trigonometry
Secondary Mathematics Developing creative thinking in mathematics: trigonometry Teacher Education through School-based Support in India www.TESS-India.edu.in http://creativecommons.org/licenses/ TESS-India (Teacher Education through School-based Support) aims to improve the classroom practices of elementary and secondary teachers in India through the provision of Open Educational Resources (OERs) to support teachers in developing student-centred, participatory approaches. The TESS-India OERs provide teachers with a companion to the school textbook. They offer activities for teachers to try out in their classrooms with their students, together with case studies showing how other teachers have taught the topic and linked resources to support teachers in developing their lesson plans and subject knowledge. TESS-India OERs have been collaboratively written by Indian and international authors to address Indian curriculum and contexts and are available for online and print use (http://www.tess-india.edu.in/). The OERs are available in several versions, appropriate for each participating Indian state and users are invited to adapt and localise the OERs further to meet local needs and contexts. TESS-India is led by The Open University UK and funded by UK aid from the UK government. Video resources Some of the activities in this unit are accompanied by the following icon: . This indicates that you will find it helpful to view the TESS-India video resources for the specified pedagogic theme. The TESS-India video resources illustrate key pedagogic techniques in a range of classroom contexts in India. We hope they will inspire you to experiment with similar practices. They are intended to complement and enhance your experience of working through the text-based units, but are not integral to them should you be unable to access them. -
Lesson 11: Perimeters and Areas of Polygonal Regions Defined by Systems of Inequalities
NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 11 M4 GEOMETRY Lesson 11: Perimeters and Areas of Polygonal Regions Defined by Systems of Inequalities Student Outcomes . Students find the perimeter of a triangle or quadrilateral in the coordinate plane given a description by inequalities. Students find the area of a triangle or quadrilateral in the coordinate plane given a description by inequalities by employing Green’s theorem. Lesson Notes In previous lessons, students found the area of polygons in the plane using the “shoelace” method. In this lesson, we give a name to this method—Green’s theorem. Students will draw polygons described by a system of inequalities, find the perimeter of the polygon, and use Green’s theorem to find the area. Classwork Opening Exercises (5 minutes) The opening exercises are designed to review key concepts of graphing inequalities. The teacher should assign them independently and circulate to assess understanding. Opening Exercises Graph the following: a. b. Lesson 11: Perimeters and Areas of Polygonal Regions Defined by Systems of Inequalities 135 Date: 8/28/14 This work is licensed under a © 2014 Common Core, Inc. Some rights reserved. commoncore.org Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 11 M4 GEOMETRY d. c. Example 1 (10 minutes) Example 1 A parallelogram with base of length and height can be situated in the coordinate plane as shown. Verify that the shoelace formula gives the area of the parallelogram as . What is the area of a parallelogram? Base height . The distance from the -axis to the top left vertex is some number . -
On the Standard Lengths of Angle Bisectors and the Angle Bisector Theorem
Global Journal of Advanced Research on Classical and Modern Geometries ISSN: 2284-5569, pp.15-27 ON THE STANDARD LENGTHS OF ANGLE BISECTORS AND THE ANGLE BISECTOR THEOREM G.W INDIKA SHAMEERA AMARASINGHE ABSTRACT. In this paper the author unveils several alternative proofs for the standard lengths of Angle Bisectors and Angle Bisector Theorem in any triangle, along with some new useful derivatives of them. 2010 Mathematical Subject Classification: 97G40 Keywords and phrases: Angle Bisector theorem, Parallel lines, Pythagoras Theorem, Similar triangles. 1. INTRODUCTION In this paper the author introduces alternative proofs for the standard length of An- gle Bisectors and the Angle Bisector Theorem in classical Euclidean Plane Geometry, on a concise elementary format while promoting the significance of them by acquainting some prominent generalized side length ratios within any two distinct triangles existed with some certain correlations of their corresponding angles, as new lemmas. Within this paper 8 new alternative proofs are exposed by the author on the angle bisection, 3 new proofs each for the lengths of the Angle Bisectors by various perspectives with also 5 new proofs for the Angle Bisector Theorem. 1.1. The Standard Length of the Angle Bisector Date: 1 February 2012 . 15 G.W Indika Shameera Amarasinghe The length of the angle bisector of a standard triangle such as AD in figure 1.1 is AD2 = AB · AC − BD · DC, or AD2 = bc 1 − (a2/(b + c)2) according to the standard notation of a triangle as it was initially proved by an extension of the angle bisector up to the circumcircle of the triangle. -
Circles in Areals
Circles in areals 23 December 2015 This paper has been accepted for publication in the Mathematical Gazette, and copyright is assigned to the Mathematical Association (UK). This preprint may be read or downloaded, and used for any non-profit making educational or research purpose. Introduction August M¨obiusintroduced the system of barycentric or areal co-ordinates in 1827[1, 2]. The idea is that one may attach weights to points, and that a system of weights determines a centre of mass. Given a triangle ABC, one obtains a co-ordinate system for the plane by placing weights x; y and z at the vertices (with x + y + z = 1) to describe the point which is the centre of mass. The vertices have co-ordinates A = (1; 0; 0), B = (0; 1; 0) and C = (0; 0; 1). By scaling so that triangle ABC has area [ABC] = 1, we can take the co-ordinates of P to be ([P BC]; [PCA]; [P AB]), provided that we take area to be signed. Our convention is that anticlockwise triangles have positive area. Points inside the triangle have strictly positive co-ordinates, and points outside the triangle must have at least one negative co-ordinate (we are allowed negative masses). The equation of a line looks like the equation of a plane in Cartesian co-ordinates, but note that the equation lx + ly + lz = 0 (with l 6= 0) is not satisfied by any point (x; y; z) of the Euclidean plane. We use such equations to describe the line at infinity when doing projective geometry. -
Some Curves and the Lengths of Their Arcs Amelia Carolina Sparavigna
Some Curves and the Lengths of their Arcs Amelia Carolina Sparavigna To cite this version: Amelia Carolina Sparavigna. Some Curves and the Lengths of their Arcs. 2021. hal-03236909 HAL Id: hal-03236909 https://hal.archives-ouvertes.fr/hal-03236909 Preprint submitted on 26 May 2021 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Some Curves and the Lengths of their Arcs Amelia Carolina Sparavigna Department of Applied Science and Technology Politecnico di Torino Here we consider some problems from the Finkel's solution book, concerning the length of curves. The curves are Cissoid of Diocles, Conchoid of Nicomedes, Lemniscate of Bernoulli, Versiera of Agnesi, Limaçon, Quadratrix, Spiral of Archimedes, Reciprocal or Hyperbolic spiral, the Lituus, Logarithmic spiral, Curve of Pursuit, a curve on the cone and the Loxodrome. The Versiera will be discussed in detail and the link of its name to the Versine function. Torino, 2 May 2021, DOI: 10.5281/zenodo.4732881 Here we consider some of the problems propose in the Finkel's solution book, having the full title: A mathematical solution book containing systematic solutions of many of the most difficult problems, Taken from the Leading Authors on Arithmetic and Algebra, Many Problems and Solutions from Geometry, Trigonometry and Calculus, Many Problems and Solutions from the Leading Mathematical Journals of the United States, and Many Original Problems and Solutions.