Pappus Theorem for a Conic and Mystic Hexagons. Ross Moore Macquarie University Sydney, Australia
|
|
|
- Erik Morgan
- 6 years ago
- Views:
Transcription
1 Pappus Theorem for a Conic and Mystic Hexagons Ross Moore Macquarie University Sydney, Australia
2 Pappus Theorem for a Conic and Mystic Hexagons Ross Moore Macquarie University Sydney, Australia Pappus Theorem is a well-known result for triples of points on two lines in the (projective) plane:
3 Theorem. (Pappus) 2
4 2 Theorem. (Pappus) Given lines l and m in the plane, l m
5 2 Theorem. (Pappus) Given lines l and m in the plane, three distinct points A,, C on l (but not on m), l A C m
6 2 Theorem. (Pappus) Given lines l and m in the plane, three distinct points A,, C on l (but not on m), and three distinct points D, E, F on m (but not on l), l A C m D E F
7 2 Theorem. (Pappus) Given lines l and m in the plane, three distinct points A,, C on l (but not on m), and three distinct points D, E, F on m (but not on l), construct the intersections L = F CE, l A C m D E F L
8 2 Theorem. (Pappus) Given lines l and m in the plane, three distinct points A,, C on l (but not on m), and three distinct points D, E, F on m (but not on l), construct the intersections L = F CE, M = CD AF l A C m D E F M L
9 2 Theorem. (Pappus) Given lines l and m in the plane, three distinct points A,, C on l (but not on m), and three distinct points D, E, F on m (but not on l), construct the intersections L = F CE, M = CD AF and N = AE D. l A C m D E F N M L
10 2 Theorem. (Pappus) Given lines l and m in the plane, three distinct points A,, C on l (but not on m), and three distinct points D, E, F on m (but not on l), construct the intersections L = F CE, M = CD AF and N = AE D. Then L, M, N are collinear. l A C m D E F N M L
11 3 Definition. y a Pappus configuration we mean a set of 6 points in the plane, arranged as a pair of triples of points (not necessarily collinear) (A,, C) and (D, E, F ), such that the intersections (L, M, N), of pairs of lines joining points taken as in Pappus Theorem, are collinear.
12 3 Definition. y a Pappus configuration we mean a set of 6 points in the plane, arranged as a pair of triples of points (not necessarily collinear) (A,, C) and (D, E, F ), such that the intersections (L, M, N), of pairs of lines joining points taken as in Pappus Theorem, are collinear. Not so well-known is that if the six points A,, C, D, E, F lie on a non-singular conic, then (A,, C), (D, E, F ) form a Pappus configuration.
13 3 Definition. y a Pappus configuration we mean a set of 6 points in the plane, arranged as a pair of triples of points (not necessarily collinear) (A,, C) and (D, E, F ), such that the intersections (L, M, N), of pairs of lines joining points taken as in Pappus Theorem, are collinear. Not so well-known is that if the six points A,, C, D, E, F lie on a non-singular conic, then (A,, C), (D, E, F ) form a Pappus configuration. A C M N L F D E
14 4 Remark. The usual Pappus Theorem is just the situation whereby the conic degenerates into a pair of lines.
15 4 Remark. The usual Pappus Theorem is just the situation whereby the conic degenerates into a pair of lines. Remark. When the six points are ordered as A, F,, D, C, F the resulting polygon is just Pascal s mystic hexagon. Alternatively, given a mystic hexagon, the Pappus configuration is obtained by taking the even and odd vertices for the groups of three.
16 4 Remark. The usual Pappus Theorem is just the situation whereby the conic degenerates into a pair of lines. Remark. When the six points are ordered as A, F,, D, C, F the resulting polygon is just Pascal s mystic hexagon. Alternatively, given a mystic hexagon, the Pappus configuration is obtained by taking the even and odd vertices for the groups of three. Remark. It is well known that a unique conic can be drawn through any 5 points in general position. Pappus configurations give a way to construct that conic, parametrised by the points on a line...
17 Given five points A,, C, D, E in the plane in sufficiently general position, 5 A C D E
18 Given five points A,, C, D, E in the plane in sufficiently general position, let N = AE D. 5 A C E N D
19 Given five points A,, C, D, E in the plane in sufficiently general position, let N = AE D. Pick P along CD. 5 A E P N D C
20 Given five points A,, C, D, E in the plane in sufficiently general position, let N = AE D. Pick P along CD. Let Q = NP CE. 5 A E P Q N D C
21 5 Given five points A,, C, D, E in the plane in sufficiently general position, let N = AE D. Pick P along CD. Let Q = NP CE. Then X = AP Q lies on the conic. A C D E N P Q X
22 5 Given five points A,, C, D, E in the plane in sufficiently general position, let N = AE D. Pick P along CD. Let Q = NP CE. Then X = AP Q lies on the conic. Vary P on CD for different points X on the conic: A C D E N P Q X
23 5 Given five points A,, C, D, E in the plane in sufficiently general position, let N = AE D. Pick P along CD. Let Q = NP CE. Then X = AP Q lies on the conic. Vary P on CD for different points X on the conic: A C D E N P Q X
24 5 Given five points A,, C, D, E in the plane in sufficiently general position, let N = AE D. Pick P along CD. Let Q = NP CE. Then X = AP Q lies on the conic. Vary P on CD for different points X on the conic: A C D E N P Q X
25 5 Given five points A,, C, D, E in the plane in sufficiently general position, let N = AE D. Pick P along CD. Let Q = NP CE. Then X = AP Q lies on the conic. Vary P on CD for different points X on the conic: A C D E N P Q X
26 5 Given five points A,, C, D, E in the plane in sufficiently general position, let N = AE D. Pick P along CD. Let Q = NP CE. Then X = AP Q lies on the conic. Vary P on CD for different points X on the conic: A C D E N P Q X
27 Given five points A,, C, D, E in the plane in sufficiently general position, let N = AE D. Pick P along CD. Let Q = NP CE. Then X = AP Q lies on the conic. Vary P on CD for different points X on the conic: A E P N Q D X C 5
Concurrency and Collinearity
Concurrency and Collinearity Victoria Krakovna vkrakovna@gmail.com 1 Elementary Tools Here are some tips for concurrency and collinearity questions: 1. You can often restate a concurrency question as a
Collinearity/Concurrence
Collinearity/Concurrence Ray Li (rayyli@stanford.edu) June 29, 2017 1 Introduction/Facts you should know 1. (Cevian Triangle) Let ABC be a triangle and P be a point. Let lines AP, BP, CP meet lines BC,
LECTURE 10, MONDAY MARCH 15, 2004
LECTURE 10, MONDAY MARCH 15, 2004 FRANZ LEMMERMEYER 1. Minimal Polynomials Let α and β be algebraic numbers, and let f and g denote their minimal polynomials. Consider the resultant R(X) of the polynomials
Institutionen för matematik, KTH.
Institutionen för matematik, KTH. Contents 4 Curves in the projective plane 1 4.1 Lines................................ 1 4.1.3 The dual projective plane (P 2 )............. 2 4.1.5 Automorphisms of P
Power Round: Geometry Revisited
Power Round: Geometry Revisited Stobaeus (one of Euclid s students): But what shall I get by learning these things? Euclid to his slave: Give him three pence, since he must make gain out of what he learns.
Warm Up Lesson Presentation Lesson Quiz. Holt McDougal Geometry
2-4 Warm Up Lesson Presentation Lesson Quiz Geometry Warm Up Write a conditional statement from each of the following. 1. The intersection of two lines is a point. If two lines intersect, then they intersect
ENGINEERING MECHANICS STATIC
Trusses Simple trusses The basic element of a truss is the triangle, three bars joined by pins at their ends, fig. a below, constitutes a rigid frame. The term rigid is used to mean noncollapsible and
Triangles III. Stewart s Theorem, Orthocenter, Euler Line. 23-Sept-2011 MA
Triangles III Stewart s Theorem, Orthocenter, Euler Line 23-Sept-2011 MA 341 001 1 Stewart s Theorem (1746) With the measurements given in the triangle below, the following relationship holds: a 2 n +
What Do Lattice Paths Have To Do With Matrices, And What Is Beyond Both?
What Do Lattice Paths Have To Do With Matrices, And What Is Beyond Both? Joseph E. Bonin The George Washington University These slides are available at blogs.gwu.edu/jbonin Some ideas in this talk were
[BIT Ranchi 1992] (a) 2 (b) 3 (c) 4 (d) 5. (d) None of these. then the direction cosine of AB along y-axis is [MNR 1989]
![[BIT Ranchi 1992] (a) 2 (b) 3 (c) 4 (d) 5. (d) None of these. then the direction cosine of AB along y-axis is [MNR 1989]](/thumbs/95/126116659.jpg "[BIT Ranchi 1992] (a) 2 (b) 3 (c) 4 (d) 5. (d) None of these. then the direction cosine of AB along y-axis is [MNR 1989]")
VECTOR ALGEBRA o. Let a i be a vector which makes an angle of 0 with a unit vector b. Then the unit vector ( a b) is [MP PET 99]. The perimeter of the triangle whose vertices have the position vectors
37th United States of America Mathematical Olympiad
37th United States of America Mathematical Olympiad 1. Prove that for each positive integer n, there are pairwise relatively prime integers k 0, k 1,..., k n, all strictly greater than 1, such that k 0
SMT Power Round Solutions : Poles and Polars
SMT Power Round Solutions : Poles and Polars February 18, 011 1 Definition and Basic Properties 1 Note that the unit circles are not necessary in the solutions. They just make the graphs look nicer. (1).0
MATH 243 Winter 2008 Geometry II: Transformation Geometry Solutions to Problem Set 1 Completion Date: Monday January 21, 2008
MTH 4 Winter 008 Geometry II: Transformation Geometry Solutions to Problem Set 1 ompletion Date: Monday January 1, 008 Department of Mathematical Statistical Sciences University of lberta Question 1. Let
The Alberta High School Mathematics Competition Solution to Part I, 2014.
The Alberta High School Mathematics Competition Solution to Part I, 2014. Question 1. When the repeating decimal 0.6 is divided by the repeating decimal 0.3, the quotient is (a) 0.2 (b) 2 (c) 0.5 (d) 0.5
Chapter (Circle) * Circle - circle is locus of such points which are at equidistant from a fixed point in
Chapter - 10 (Circle) Key Concept * Circle - circle is locus of such points which are at equidistant from a fixed point in a plane. * Concentric circle - Circle having same centre called concentric circle.
Recognise the Equation of a Circle. Solve Problems about Circles Centred at O. Co-Ordinate Geometry of the Circle - Outcomes
1 Co-Ordinate Geometry of the Circle - Outcomes Recognise the equation of a circle. Solve problems about circles centred at the origin. Solve problems about circles not centred at the origin. Determine
Calgary Math Circles: Triangles, Concurrency and Quadrilaterals 1
Calgary Math Circles: Triangles, Concurrency and Quadrilaterals 1 1 Triangles: Basics This section will cover all the basic properties you need to know about triangles and the important points of a triangle.
The geometry of projective space
Chapter 1 The geometry of projective space 1.1 Projective spaces Definition. A vector subspace of a vector space V is a non-empty subset U V which is closed under addition and scalar multiplication. In
2013 IMO Shortlist. a b c d 1
IMO Shortlist 2013 Algebra A1 A2 Letnbeapositiveintegerandleta 1,...,a n 1 bearbitraryrealnumbers. Define the sequences u 0,...,u n and v 0,...,v n inductively by u 0 = u 1 = v 0 = v 1 = 1, and u k+1 =
An analytic proof of the theorems of Pappus and Desargues
Note di Matematica 22, n. 1, 2003, 99 106. An analtic proof of the theorems of Pappus and Desargues Erwin Kleinfeld and Tuong Ton-That Department of Mathematics, The Universit of Iowa, Iowa Cit, IA 52242,
5. Introduction to Euclid s Geometry
5. Introduction to Euclid s Geometry Multiple Choice Questions CBSE TREND SETTER PAPER _ 0 EXERCISE 5.. If the point P lies in between M and N and C is mid-point of MP, then : (A) MC + PN = MN (B) MP +
Vandermonde Matrices for Intersection Points of Curves
This preprint is in final form. It appeared in: Jaén J. Approx. (009) 1, 67 81 Vandermonde Matrices for Intersection Points of Curves Hakop Hakopian, Kurt Jetter and Georg Zimmermann Abstract We reconsider
Chapter 2 Review. Short Answer Determine whether the biconditional statement about the diagram is true or false.
Chapter 2 Review Short Answer Determine whether the biconditional statement about the diagram is true or false. 1. are supplementary if and only if they form a linear pair. 2. are congruent if and only
Spring, 2006 CIS 610. Advanced Geometric Methods in Computer Science Jean Gallier Homework 1
Spring, 2006 CIS 610 Advanced Geometric Methods in Computer Science Jean Gallier Homework 1 January 23, 2006; Due February 8, 2006 A problems are for practice only, and should not be turned in. Problem
(A) 2S + 3 (B) 3S + 2 (C) 3S + 6 (D) 2S + 6 (E) 2S + 12
1 The sum of two numbers is S Suppose 3 is added to each number and then each of the resulting numbers is doubled What is the sum of the final two numbers? (A) S + 3 (B) 3S + (C) 3S + 6 (D) S + 6 (E) S
Conic Sections Session 2: Ellipse
Conic Sections Session 2: Ellipse Toh Pee Choon NIE Oct 2017 Toh Pee Choon (NIE) Session 2: Ellipse Oct 2017 1 / 24 Introduction Problem 2.1 Let A, F 1 and F 2 be three points that form a triangle F 2
Conics. Chapter 6. When you do this Exercise, you will see that the intersections are of the form. ax 2 + bxy + cy 2 + ex + fy + g =0, (1)
Chapter 6 Conics We have a pretty good understanding of lines/linear equations in R 2 and RP 2. Let s spend some time studying quadratic equations in these spaces. These curves are called conics for the
This section will help you revise previous learning which is required in this topic.
Higher Portfolio Circle Higher 10. Circle Revision Section This section will help you revise previous learning which is required in this topic. R1 I can use the distance formula to find the distance between
Chapter. Triangles. Copyright Cengage Learning. All rights reserved.
Chapter 3 Triangles Copyright Cengage Learning. All rights reserved. 3.5 Inequalities in a Triangle Copyright Cengage Learning. All rights reserved. Inequalities in a Triangle Important inequality relationships
CN#4 Biconditional Statements and Definitions
CN#4 s and Definitions OBJECTIVES: STUDENTS WILL BE ABLE TO WRITE AND ANALYZE BICONDITIONAL STATEMENTS. Vocabulary biconditional statement definition polygon triangle quadrilateral When you combine a conditional
MODEL ANSWERS TO HWK #3
MODEL ANSWERS TO HWK #3 1. Suppose that the point p = [v] and that the plane H corresponds to W V. Then a line l containing p, contained in H is spanned by the vector v and a vector w W, so that as a point
LESSON 2 5 CHAPTER 2 OBJECTIVES
LESSON 2 5 CHAPTER 2 OBJECTIVES POSTULATE a statement that describes a fundamental relationship between the basic terms of geometry. THEOREM a statement that can be proved true. PROOF a logical argument
The sum x 1 + x 2 + x 3 is (A): 4 (B): 6 (C): 8 (D): 14 (E): None of the above. How many pairs of positive integers (x, y) are there, those satisfy
Important: Answer to all 15 questions. Write your answers on the answer sheets provided. For the multiple choice questions, stick only the letters (A, B, C, D or E) of your choice. No calculator is allowed.
(II.B) Basis and dimension
(II.B) Basis and dimension How would you explain that a plane has two dimensions? Well, you can go in two independent directions, and no more. To make this idea precise, we formulate the DEFINITION 1.
HANOI OPEN MATHEMATICAL COMPETITON PROBLEMS
HANOI MATHEMATICAL SOCIETY NGUYEN VAN MAU HANOI OPEN MATHEMATICAL COMPETITON PROBLEMS HANOI - 2013 Contents 1 Hanoi Open Mathematical Competition 3 1.1 Hanoi Open Mathematical Competition 2006... 3 1.1.1
SMT 2018 Geometry Test Solutions February 17, 2018
SMT 018 Geometry Test Solutions February 17, 018 1. Consider a semi-circle with diameter AB. Let points C and D be on diameter AB such that CD forms the base of a square inscribed in the semicircle. Given
The City School. Comprehensive Worksheet (2 nd Term) Mathematics Class 8. Branch/Campus:
The City School Comprehensive Worksheet (2 nd Term) 2017-2018 Mathematics Class 8 Index No: Branch/Campus: Maximum Marks: 100 Section: Date: Time Allowed:2 hours INSTRUCTIONS Write your index number, section,
THE THEOREM OF DESARGUES
THE THEOREM OF DESARGUES TIMOTHY VIS 1. Introduction In this worksheet we will be exploring some proofs surrounding the Theorem of Desargues. This theorem plays an extremely important role in projective
1 Hanoi Open Mathematical Competition 2017
1 Hanoi Open Mathematical Competition 017 1.1 Junior Section Question 1. Suppose x 1, x, x 3 are the roots of polynomial P (x) = x 3 6x + 5x + 1. The sum x 1 + x + x 3 is (A): 4 (B): 6 (C): 8 (D): 14 (E):
1987 IMO Shortlist. IMO Shortlist 1987
IMO Shortlist 1987 1 Let f be a function that satisfies the following conditions: (i) If x > y and f(y) y v f(x) x, then f(z) = v + z, for some number z between x and y. (ii) The equation f(x) = 0 has
Classical Theorems in Plane Geometry 1
BERKELEY MATH CIRCLE 1999 2000 Classical Theorems in Plane Geometry 1 Zvezdelina Stankova-Frenkel UC Berkeley and Mills College Note: All objects in this handout are planar - i.e. they lie in the usual
Lecture 3: Geometry. Sadov s theorem. Morley s miracle. The butterfly theorem. E-320: Teaching Math with a Historical Perspective Oliver Knill, 2013
E-320: Teaching Math with a Historical Perspective Oliver Knill, 2013 Lecture 3: Geometry Sadov s theorem Here is an example of a geometric theorem which has been found only 10 years ago: The quadrilateral
Finite affine planes in projective spaces
Finite affine planes in projective spaces J. A.Thas H. Van Maldeghem Ghent University, Belgium {jat,hvm}@cage.ugent.be Abstract We classify all representations of an arbitrary affine plane A of order q
arxiv:math/ v1 [math.ag] 3 Mar 2002
![arxiv:math/ v1 [math.ag] 3 Mar 2002](/thumbs/81/83196187.jpg "arxiv:math/ v1 [math.ag] 3 Mar 2002")
How to sum up triangles arxiv:math/0203022v1 [math.ag] 3 Mar 2002 Bakharev F. Kokhas K. Petrov F. June 2001 Abstract We prove configuration theorems that generalize the Desargues, Pascal, and Pappus theorems.
Geometry Honors Review for Midterm Exam
Geometry Honors Review for Midterm Exam Format of Midterm Exam: Scantron Sheet: Always/Sometimes/Never and Multiple Choice 40 Questions @ 1 point each = 40 pts. Free Response: Show all work and write answers
Conjugate conics and closed chains of Poncelet polygons
Mitt. Math. Ges. Hamburg?? (????), 1 0 Conjugate conics and closed chains of Poncelet polygons Lorenz Halbeisen Department of Mathematics, ETH Zentrum, Rämistrasse 101, 8092 Zürich, Switzerland Norbert
An Exploration of the Group Law on an Elliptic Curve. Tanuj Nayak
An Exploration of the Group Law on an Elliptic Curve Tanuj Nayak Abstract Given its abstract nature, group theory is a branch of mathematics that has been found to have many important applications. One
Honors Geometry Mid-Term Exam Review
Class: Date: Honors Geometry Mid-Term Exam Review Multiple Choice Identify the letter of the choice that best completes the statement or answers the question. 1. Classify the triangle by its sides. The
On the Coincidences of Pascal Lines
Jaydeep Chipalkatti October 19, 2014 Halifax, Nova Scotia. Pascal s Theorem Pascal s Theorem The points AE BF,AD CF,BD CE are collinear. Pascal line { A B C F E D { A B C F E D, { A D C F B D, { A C E
Triangles. Example: In the given figure, S and T are points on PQ and PR respectively of PQR such that ST QR. Determine the length of PR.
Triangles Two geometric figures having the same shape and size are said to be congruent figures. Two geometric figures having the same shape, but not necessarily the same size, are called similar figures.
number. However, unlike , three of the digits of N are 3, 4 and 5, and N is a multiple of 6.
C1. The positive integer N has six digits in increasing order. For example, 124 689 is such a number. However, unlike 124 689, three of the digits of N are 3, 4 and 5, and N is a multiple of 6. How many
2011 Olympiad Solutions
011 Olympiad Problem 1 Let A 0, A 1, A,..., A n be nonnegative numbers such that Prove that A 0 A 1 A A n. A i 1 n A n. Note: x means the greatest integer that is less than or equal to x.) Solution: We
It is known that the length of the tangents drawn from an external point to a circle is equal.
CBSE -MATHS-SET 1-2014 Q1. The first three terms of an AP are 3y-1, 3y+5 and 5y+1, respectively. We need to find the value of y. We know that if a, b and c are in AP, then: b a = c b 2b = a + c 2 (3y+5)
HANOI OPEN MATHEMATICS COMPETITON PROBLEMS
HANOI MATHEMATICAL SOCIETY NGUYEN VAN MAU HANOI OPEN MATHEMATICS COMPETITON PROBLEMS 2006-2013 HANOI - 2013 Contents 1 Hanoi Open Mathematics Competition 3 1.1 Hanoi Open Mathematics Competition 2006...
KENDRIYA VIDYALAYA SANGATHAN, HYDERABAD REGION
KENDRIYA VIDYALAYA SANGATHAN, HYDERABAD REGION SAMPLE PAPER 03 (2017-18) SUBJECT: MATHEMATICS(041) BLUE PRINT : CLASS X Unit Chapter VSA (1 mark) SA I (2 marks) SA II (3 marks) LA (4 marks) Total Unit
9th Bay Area Mathematical Olympiad
9th Bay rea Mathematical Olympiad February 27, 2007 Problems with Solutions 1 15-inch-long stick has four marks on it, dividing it into five segments of length 1,2,3,4, and 5 inches (although not neccessarily
Unofficial Solutions
Canadian Open Mathematics Challenge 2016 Unofficial Solutions COMC exams from other years, with or without the solutions included, are free to download online. Please visit http://comc.math.ca/2016/practice.html
FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 48
FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 48 RAVI VAKIL CONTENTS 1. A little more about cubic plane curves 1 2. Line bundles of degree 4, and Poncelet s Porism 1 3. Fun counterexamples using elliptic curves
x = y +z +2, y = z +x+1, and z = x+y +4. C R
1. [5] Solve for x in the equation 20 14+x = 20+14 x. 2. [5] Find the area of a triangle with side lengths 14, 48, and 50. 3. [5] Victoria wants to order at least 550 donuts from Dunkin Donuts for the
Identifying second degree equations
Chapter 7 Identifing second degree equations 71 The eigenvalue method In this section we appl eigenvalue methods to determine the geometrical nature of the second degree equation a 2 + 2h + b 2 + 2g +
Hanoi Open Mathematical Competition 2017
Hanoi Open Mathematical Competition 2017 Junior Section Saturday, 4 March 2017 08h30-11h30 Important: Answer to all 15 questions. Write your answers on the answer sheets provided. For the multiple choice
10. Circles. Q 5 O is the centre of a circle of radius 5 cm. OP AB and OQ CD, AB CD, AB = 6 cm and CD = 8 cm. Determine PQ. Marks (2) Marks (2)
10. Circles Q 1 True or False: It is possible to draw two circles passing through three given non-collinear points. Mark (1) Q 2 State the following statement as true or false. Give reasons also.the perpendicular
SM2H Unit 6 Circle Notes
Name: Period: SM2H Unit 6 Circle Notes 6.1 Circle Vocabulary, Arc and Angle Measures Circle: All points in a plane that are the same distance from a given point, called the center of the circle. Chord:
SRSD 2093: Engineering Mechanics 2SRRI SECTION 19 ROOM 7, LEVEL 14, MENARA RAZAK
SRSD 2093: Engineering Mechanics 2SRRI SECTION 19 ROOM 7, LEVEL 14, MENARA RAZAK SIMPLE TRUSSES, THE METHOD OF JOINTS, & ZERO-FORCE MEMBERS Today s Objectives: Students will be able to: a) Define a simple
Conics and perspectivities
10 Conics and perspectivities Some people have got a mental horizon of radius zero and call it their point of view. Attributed to David Hilbert In the last chapter we treated conics more or less as isolated
MAS345 Algebraic Geometry of Curves: Exercises
MAS345 Algebraic Geometry of Curves: Exercises AJ Duncan, September 22, 2003 1 Drawing Curves In this section of the exercises some methods of visualising curves in R 2 are investigated. All the curves
1998 IMO Shortlist BM BN BA BC AB BC CD DE EF BC CA AE EF FD
IMO Shortlist 1998 Geometry 1 A convex quadrilateral ABCD has perpendicular diagonals. The perpendicular bisectors of the sides AB and CD meet at a unique point P inside ABCD. the quadrilateral ABCD is
The CENTRE for EDUCATION in MATHEMATICS and COMPUTING cemc.uwaterloo.ca Euclid Contest. Wednesday, April 15, 2015
The CENTRE for EDUCATION in MATHEMATICS and COMPUTING cemc.uwaterloo.ca 015 Euclid Contest Wednesday, April 15, 015 (in North America and South America) Thursday, April 16, 015 (outside of North America
VECTOR NAME OF THE CHAPTER. By, Srinivasamurthy s.v. Lecturer in mathematics. K.P.C.L.P.U.College. jogfalls PART-B TWO MARKS QUESTIONS
NAME OF THE CHAPTER VECTOR PART-A ONE MARKS PART-B TWO MARKS PART-C FIVE MARKS PART-D SIX OR FOUR MARKS PART-E TWO OR FOUR 1 1 1 1 1 16 TOTAL MARKS ALLOTED APPROXIMATELY By, Srinivasamurthy s.v Lecturer
NEW YORK CITY INTERSCHOLASTIC MATHEMATICS LEAGUE Senior A Division CONTEST NUMBER 1
Senior A Division CONTEST NUMBER 1 PART I FALL 2011 CONTEST 1 TIME: 10 MINUTES F11A1 Larry selects a 13-digit number while David selects a 10-digit number. Let be the number of digits in the product of
The details of the derivation of the equations of conics are com-
Part 6 Conic sections Introduction Consider the double cone shown in the diagram, joined at the verte. These cones are right circular cones in the sense that slicing the double cones with planes at right-angles
Pascal s Hexagon Theorem implies a Butterfly Theorem in the Complex Projective Plane
arxiv:0910.4792v1 [math.gm] 26 Oct 2009 Pascal s Hexagon Theorem implies a Butterfly Theorem in the Complex Projective Plane 1 Introduction Greg Markowsky August 20, 2018 Some time ago I attempted to prove
CO-ORDINATE GEOMETRY. 1. Find the points on the y axis whose distances from the points (6, 7) and (4,-3) are in the. ratio 1:2.
UNIT- CO-ORDINATE GEOMETRY Mathematics is the tool specially suited for dealing with abstract concepts of any ind and there is no limit to its power in this field.. Find the points on the y axis whose
0113ge. Geometry Regents Exam In the diagram below, under which transformation is A B C the image of ABC?
0113ge 1 If MNP VWX and PM is the shortest side of MNP, what is the shortest side of VWX? 1) XV ) WX 3) VW 4) NP 4 In the diagram below, under which transformation is A B C the image of ABC? In circle
Which number listed below belongs to the interval 0,7; 0,8? c) 6 7. a) 3 5. b) 7 9. d) 8 9
Problem 1 Which number listed below belongs to the interval 0,7; 0,8? a) 3 5 b) 7 9 c) 6 7 d) 8 9 2 Problem 2 What is the greatest common divisor of the numbers 3 2 3 5 and 2 3 3 5? a) 6 b) 15 c) 30 d)
Part (1) Second : Trigonometry. Tan
Part (1) Second : Trigonometry (1) Complete the following table : The angle Ratio 42 12 \ Sin 0.3214 Cas 0.5321 Tan 2.0625 (2) Complete the following : 1) 46 36 \ 24 \\ =. In degrees. 2) 44.125 = in degrees,
Screening Test Gauss Contest NMTC at PRIMARY LEVEL V & VI Standards Saturday, 27th August, 2016
THE ASSOCIATION OF MATHEMATICS TEACHERS OF INDIA Screening Test Gauss Contest NMTC at PRIMARY LEVEL V & VI Standards Saturday, 7th August, 06 :. Fill in the response sheet with your Name, Class and the
Sample Question Paper Mathematics First Term (SA - I) Class X. Time: 3 to 3 ½ hours
Sample Question Paper Mathematics First Term (SA - I) Class X Time: 3 to 3 ½ hours M.M.:90 General Instructions (i) All questions are compulsory. (ii) The question paper consists of 34 questions divided
ENGR-1100 Introduction to Engineering Analysis. Lecture 19
ENGR-1100 Introduction to Engineering Analysis Lecture 19 SIMPLE TRUSSES, THE METHOD OF JOINTS, & ZERO-FORCE MEMBERS Today s Objectives: Students will be able to: In-Class Activities: a) Define a simple
KENDRIYA VIDYALAYA GACHIBOWLI, GPRA CAMPUS, HYD 32
KENDRIYA VIDYALAYA GACHIBOWLI, GPRA CAMPUS, HYD 32 SAMPLE PAPER 03 (2018-19) SUBJECT: MATHEMATICS(041) BLUE PRINT : CLASS X Unit Chapter VSA (1 mark) SA I (2 marks) SA II (3 marks) LA (4 marks) Total Unit
Extra Test Review. 3. Use the following graph to find the area of a rectangle with vertices of ( 2, 4), ( 2, 4), (1, 4), and (1, 4).
Name: Date: 1. The sides of the outer square are about 14 inches. The sides of the inner square about 10 inches. What is a logical estimate for the circumference of the circle? 3. Use the following graph
2017 Harvard-MIT Mathematics Tournament
Team Round 1 Let P(x), Q(x) be nonconstant polynomials with real number coefficients. Prove that if P(y) = Q(y) for all real numbers y, then P(x) = Q(x) for all real numbers x. 2 Does there exist a two-variable
Georgia Tech High School Math Competition
Georgia Tech High School Math Competition Multiple Choice Test February 28, 2015 Each correct answer is worth one point; there is no deduction for incorrect answers. Make sure to enter your ID number on
40th International Mathematical Olympiad
40th International Mathematical Olympiad Bucharest, Romania, July 999 Determine all finite sets S of at least three points in the plane which satisfy the following condition: for any two distinct points
Geometry. Unit 2- Reasoning and Proof. Name:
Geometry Unit 2- Reasoning and Proof Name: 1 Geometry Chapter 2 Reasoning and Proof ***In order to get full credit for your assignments they must me done on time and you must SHOW ALL WORK. *** 1. (2-1)
div(f ) = D and deg(d) = deg(f ) = d i deg(f i ) (compare this with the definitions for smooth curves). Let:
Algebraic Curves/Fall 015 Aaron Bertram 4. Projective Plane Curves are hypersurfaces in the plane CP. When nonsingular, they are Riemann surfaces, but we will also consider plane curves with singularities.
HIGHER RANK NUMERICAL RANGES OF NORMAL MATRICES
HIGHER RANK NUMERICAL RANGES OF NORMAL MATRICES HWA-LONG GAU, CHI-KWONG LI, YIU-TUNG POON, AND NUNG-SING SZE Abstract. The higher rank numerical range is closely connected to the construction of quantum
On the Torsion Subgroup of an Elliptic Curve
S.U.R.E. Presentation October 15, 2010 Linear Equations Consider line ax + by = c with a, b, c Z Integer points exist iff gcd(a, b) c If two points are rational, line connecting them has rational slope.
(A) 50 (B) 40 (C) 90 (D) 75. Circles. Circles <1M> 1.It is possible to draw a circle which passes through three collinear points (T/F)
Circles 1.It is possible to draw a circle which passes through three collinear points (T/F) 2.The perpendicular bisector of two chords intersect at centre of circle (T/F) 3.If two arcs of a circle
Conics and their duals
9 Conics and their duals You always admire what you really don t understand. Blaise Pascal So far we dealt almost exclusively with situations in which only points and lines were involved. Geometry would
Two applications of the theorem of Carnot
Annales Mathematicae et Informaticae 40 (2012) pp. 135 144 http://ami.ektf.hu Two applications of the theorem of Carnot Zoltán Szilasi Institute of Mathematics, MTA-DE Research Group Equations, Functions
Mathematics 2260H Geometry I: Euclidean geometry Trent University, Fall 2016 Solutions to the Quizzes
Mathematics 2260H Geometry I: Euclidean geometry Trent University, Fall 2016 Solutions to the Quizzes Quiz #1. Wednesday, 13 September. [10 minutes] 1. Suppose you are given a line (segment) AB. Using
right angle an angle whose measure is exactly 90ᴼ
right angle an angle whose measure is exactly 90ᴼ m B = 90ᴼ B two angles that share a common ray A D C B Vertical Angles A D C B E two angles that are opposite of each other and share a common vertex two
12 th Annual Johns Hopkins Math Tournament Saturday, February 19, 2011 Power Round-Poles and Polars
1 th Annual Johns Hopkins Math Tournaent Saturday, February 19, 011 Power Round-Poles and Polars 1. Definition and Basic Properties 1. Note that the unit circles are not necessary in the solutions. They
PROJECTIVE GEOMETRY. Contents
PROJECTIVE GEOMETRY Abstract. Projective geometry Contents 5. Geometric constructions, projective transformations, transitivity on triples, projective pl 1. Ceva, Menelaus 1 2. Desargues theorem, Pappus
KENDRIYA VIDYALAYA SANGATHAN, HYDERABAD REGION
KENDRIYA VIDYALAYA SANGATHAN, HYDERABAD REGION SAMPLE PAPER 08 (2017-18) SUBJECT: MATHEMATICS(041) BLUE PRINT : CLASS X Unit Chapter VSA (1 mark) SA I (2 marks) SA II (3 marks) LA (4 marks) Total Unit
2016 State Mathematics Contest Geometry Test
2016 State Mathematics Contest Geometry Test In each of the following, choose the BEST answer and record your choice on the answer sheet provided. To ensure correct scoring, be sure to make all erasures
10.1 circles and circumference 2017 ink.notebook. March 13, Page 91. Page 92. Page 90. Ch 10 Circles Circles and Circumference.
Page 90 Page 91 Page 92 Ch 10 Circles 10.1 Circles and Circumference Lesson Objectives Page 93 Standards Lesson Notes Page 94 10.1 Circles and Circumference Press the tabs to view details. 1 Lesson Objectives
f(x + y) + f(x y) = 10.
Math Field Day 202 Mad Hatter A A Suppose that for all real numbers x and y, Then f(y x) =? f(x + y) + f(x y) = 0. A2 Find the sum + 2 + 4 + 5 + 7 + 8 + 0 + + +49 + 50 + 52 + 53 + 55 + 56 + 58 + 59. A3
CONCURRENT LINES- PROPERTIES RELATED TO A TRIANGLE THEOREM The medians of a triangle are concurrent. Proof: Let A(x 1, y 1 ), B(x, y ), C(x 3, y 3 ) be the vertices of the triangle A(x 1, y 1 ) F E B(x,