# What is inversive geometry?

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 What is inversive geometry? Andrew Krieger July 18, 2013 Throughout, Greek letters (,,...) denote geometric objects like circles or lines; small Roman letters (a, b,... ) denote distances, and large Roman letters (A, B,... ) denote points.. The symbol AB denotes the line segment between points A and B and `(AB) denotes its length. The symbol AB denotes the ray with endpoint at A and passing through B. 1 Steiner s Porism Suppose we have two nonintersecting circles and in the plane. When is it possible to draw n circles 1,..., n such that each i is tangent to both and, as well as to i+1 and i 1 (indices taken mod n)? How many di erent such drawings are there for fixed circles and? Figure 1: A solution for n = 12 [Credit: WillowW / Wikimedia Commons / CC-BY-SA] Suppose, are concentric. Let a and b denote the radii of and respectively, and assume a > b. Clearly, such an arrangement of circles i exists only when the centers C 1,...,C n of the n circles 1,... n are the vertices of a regular n-gon, whose center is the common center O of and. Consider then a point of tangency T between two circles i and i+1, and the triangle 4OC i T. The edge OT is a tangent to i, and C i T is a radius, so OT? C i T. Furthermore, `(C i T )= a b 2 and `(OC i )= a+b 2, by straightforward geometry. So, we have sin = `(C it ) `(OC i ) = a b a + b = (a/b) 1 (a/b)+1 = 1 +1 where := a b. Rearranging the equation sin( /n) =( sin +sin = 1, sin 1)/( + 1) to solve for gives us: 1 = sin 1, = 1 + sin( /n) 1 sin( /n) = sin( /n) cos( /n) So, it is possible to draw n circles 1,..., n according to the rules of Steiner s Porism precisely when the ratio = a b satisfies =(sec( /n) + tan( /n))2 (after simplifying the trigonometric functions from the 1

2 / O 2 1 T C 1 Figure 2: Geometry of Steiner s Porism for concentric circles A and B previous equation). Furthermore, we can obtain uncountably many di erent drawings just by rotating about O through any angle. Direct analysis of this sort is significantly more di for another method. cult in the general (non-concentric) case. So, let s look 2 Inversive Geometry We will work in the Euclidean plane R 2, with an additional point at infinity (denoted 1) adjoined. We will consider a line in the plane to be a circle of infinite radius, which included the point at infinity. Any two lines intersect at 1. If the lines are parallel in the Euclidean sense, so that they meet only at 1, we consider them to be tangent. The first example of an inversion map: f(z) :C C defined by f(z) =z 1 for z 6= 0, f(0) = 1, f(1) = 0. This turns the unit circle {z 2 C : z = 1} inside-out, exchanging external points and internal points. We will generalize this to any circle in the Eucldean plane, and express inversion in purely geometric terms. Definition For any circle in the plane with center O and radius r, theinversion map through takes any point P 6= O to the (unique) point P 0 on the ray OP such that `(OP) `(OP 0 )=r 2. I will denote the inversion map through by T. Some properties: 1. T exchanges points inside with points outside. 2. The closer a point on the inside is to O, the farther away its image is. 3. The fixed points of T are precisely the points on the circle. 4. T is an involution, i.e. T T =id R2 [{1}. 5. If is a circle concentric with,thent ( ) is also a circle concentric with and. 6. If is a line running through the point O, thent ( )=. 7. If µ is any line not passing through O, thent (µ) is a circle containing O. 8. If is any circle which included the point O, thent ( ) is a line not passing through O. The first 6 properties are straightforward to prove. The last two are only slightly more involved, and they follow from elementary geometry: Suppose µ is a line not running through O as in Figure 3. We want to show that the image of µ under T is a circle containing O. If we draw the perpendicular OA to µ, we can find the image A 0 = T (A). Then, we consider the circle with diameter OA 0 and show that any point P on µ maps to this circle. Let P 0 be the point where OP intersects the circle with diameter OA 0. By similar triangles, `(OP)/`(OA) =`(OA 0 )/`(OP 0 ), so `(OP) `(OP 0 )=`(OA) `(OA 0 )=r 2,soP 0 = T (P ). Thus, T (µ) is the circle with diameter OA 0. 2

3 ' ' Figure 3: Proof that a line not through O inverts to a circle Since T = T 1, it follows that the circle with diameter OA 0 is mapped to a line not containing O, and perpendicular to the ray OA 0. This holds as we move A 0 throughout the plane, i.e. any circle through O inverts to a line not passing through O. 3 Concyclic points and separation Definition If four points A, B, C, D lie on the same circle (or line, since we count lines as circles of infinite radius), we say following [1] that the pair AC separates BD written AC//BD if either arc from A to C intersects either B or D. In other words, AC//BD if and only if the points lie in the cyclic order A, B, C, D along the circle (up to reversal of the order). We will extend the definition of separation to four points in general position after a theorem. Theorem (Proof from [1]) When A, B, C, D are not concyclic (or colinear), there are two non-intersecting circles, one through A and C, and another though B and D. Remark This was a Putnam exam problem in Proof. Consider the perpendicular bisectors and of AC and BD respectively. If these bisectors coincide, then A, B, C, D are concyclic. So, these bisectors either intersect once in R 2 (and again at 1, which intersection we will ignore) or are parallel. If the bisectors interesect at the (finite) point O, then there are two circles and with center O, such that A and C lie on, and B and D lie on. If A, B, C, D are not concycic, these are two distinct concentric circles, which competes this case of the proof. Otherwise, and are parallel. This forces AC and BD to be parallel as well, so these four lines form a rectangle. Consider the points P and Q, the midpoints of the sides of this rectangle which coincide with and respectively. There is a circle through A, P, and C and another through B, Q, and D, and because of the separation between lines AC and BD, these two circles cannot intersect. This completes the proof. Figure 4: Diagram for proof of theorem [Credit: Reproduction of figure in [1]] 3

4 With this fact in mind, we can redefine the symbol AC//BD to mean that every circle through A and C intersects or coincides with every circle through B and D. Then by the theorem, AC//BD only if A, B, C, D are concyclic, in the concyclic case, this new definition agrees with the old definition of separation. Remark In [1], a third characterization of AC//BD is presented. distinct points A, B, C, D, The authors prove that for any four `(AB) `(CD)+`(BC) `(AD) `(AC) `(BD) and then prove that equality holds if and only if AC//BD. I will use this fact without proof, due to time constraints. Now, we can give a new method of specifyng circles. For any three points A, B, and C, the circle though A, B, and C is the union of {A, B, C} and the set of all points X such that BC//AX, CA//BX, or AB//CX. Clearly, any point X satisfying any of these three separation conditions lies on the same circle as A, B, and C (by the Theorem above), and conversely any point X on the circle is either A, B, or C, or satisfies one of the separation conditions. An important result is that inversion through a circle preserves separation. The following short Lemmas are due to [1]: Lemma If is any circle with center O and radius r, and A, B are any two points with T (A) =A 0 and T (B) =B 0,then `(A 0 B 0 )= r 2`(AB) `(OA) `(OB) Proof. The triangles 4OAB and 4OB 0 A 0 are similar, so we have: `(A 0 B 0 ) `(AB) = `(OA0 ) `(OB) = `(OA) `(OA0 ) `(OA) `(OB) = r 2 `(OA) `(OB) Definition The cross ratio between two pairs of points AC and BD is {AB, CD} =(`(AC) `(BD))/(`(AD) `(BC)). In particular, AC//BD if and only if {AB, CD} = 1. Lemma Inversion through circle Proof. {A 0 B 0,C 0 D 0 } = `(A0 C 0 ) `(B 0 D 0 ) `(A 0 D 0 ) `(B 0 C 0 ) = with center O and radius r preserves cross ratios. r 2`(AC) `(OA) `(OC) r 2`(AD) `(OA) `(OD) r 2`(BD) `(OB) `(OD) r 2`(BC) `(OB) `(OC) = `(AC) `(BD) = {AB, CD} `(AD) `(BC) Combining this with the previous results, we see that inversion also preserves separation. This leads us to the following theorem: Theorem Any inversion map takes circles to circles (recall that lines are considered circles of infinite radius). 4

5 Proof. Suppose that we have a circle through points A, B, and C. Recall that this circle is = {X : BC//AX, AC//BX, or AB//CX} [ {A, B, C}. If we invert through some circle, we get A 0 = T (A), B 0 = T (B), and C 0 = T (C). Furthermore, for any other point X on the circle, if (say) BC//AX, then B 0 C 0 //A 0 X 0,whereX 0 = T (X). Thus, T ( ) ={X 0 : X 0 = T (X) and BC//AX, AC//BX, or AB//CX} [ {A 0,B 0,C 0 } is also a circle. = {X 0 : B 0 C 0 //A 0 X 0,A 0 C 0 //B 0 X 0, or A 0 B 0 //C 0 X 0 } [ {A 0,B 0,C 0 } Equally important, inversion preserves the number of intersections of any two circles, so tangent circles invert to other tangent circles, etc. Finally, we shall consider some results about orthogonal circles. Definition Two circles are called orthogonal if they intersect twice at right angles. Lemma Inversion through a circle with center O preserves the angle of intersection (namely, the angle between the two tangents at the point of intersection) of any two intersectings circles. Proof. First, Let, be two intersecting circles. Let be tangent to and µ tangent to. Under the inversion map, maps to a circle 0 which touches the center O of the inversion, and whose tangent at O is parallel to. Likewise, the µ 0 has a tangent at O parallel to. By well-known facts about parallel lines, the angle between the tangents to 0 and µ 0 is congruent to the angle between and µ. Now, if and were tangent to and µ respectively at the point of intersection of the two lines (call it P ) then 0 and 0 are tangent to 0 and µ 0 at P 0, which forces their angle of intersection to equal that of 0 and µ 0, which is equal to the angle between and µ, which is the angle of intersection of and. Corollary Inversion maps orthogonal circles to orthogonal circles. Theorem Given any two non-intersecting circles, it is possible to invert them into two concentric circles. Proof. Let, be two non-intersecting circles. By geometry (see [1], Section 5.7), there is a line, called the radical axis of and, which is perpendicular to the line of the centers of and. For many points P on this line (precisely, those points for which the quantity `(O P ) 2 r 2 = `(O P ) 2 r 2 > 0, where O is the center of and r its radius), the length of the tangents from P to either A or B are equal. A circle drawn with center P and radius equal to this common length will be orthogonal to both and. We can draw two such circles and, which will intersect at two points, say O inside of, and P inside of. We then invert through any circle with center O. Let P 0 be the image of P under this inversion, 0 the image of, and so on. Then, 0 and 0 are two lines which meet at O, since and were circles touching O. Since was orthogonal to both and,wehave 0 orthogonal to 0 and 0.Since 0 is a line, it must be a diameter of 0 ;likewise 0 is another diameter. these diameters meet at P 0,soP 0 must be the center of 0. Likewise, P 0 is the center of 0, and we are done. 4 Finishing the proof of Steiner s Porism... is now easy. I leave it as an exercise to the reader in applying the prior theorem. 5

6 ' ' ' Not shown: ', ' (centered around ') Figure 5: Proof that you can invert two non-intersecting circles into two concentric circles [Credit: This author, based on a proof and figure from [1]] 5 Bibliography Coxeter, H.S.M. and Greitzer, S.L. Geometry Revisited. New York: Random House, Print. 6 Image Credits Figure 1: WillowW / Wikimedia Commons / CC-BY-SA Figure 2: Created by the author of these notes Figure 3: Created by the author of these notes; similar to part of Figure 5.3B (p. 109) in [1] Figure 4: Reproduction by the author of Figure 5.1 from [1] Figure 5: Created by the author of these notes; based upon Figure 5.7A of [1] 6

### Theorem 3.1. If two circles meet at P and Q, then the magnitude of the angles between the circles is the same at P and Q.

3 rthogonal circles Theorem 3.1. If two circles meet at and, then the magnitude of the angles between the circles is the same at and. roof. Referring to the figure on the right, we have A B AB (by SSS),

### The measure of an arc is the measure of the central angle that intercepts it Therefore, the intercepted arc measures

8.1 Name (print first and last) Per Date: 3/24 due 3/25 8.1 Circles: Arcs and Central Angles Geometry Regents 2013-2014 Ms. Lomac SLO: I can use definitions & theorems about points, lines, and planes to

### INVERSION WITH RESPECT TO A CIRCLE

INVERSION WITH RESPECT TO A CIRCLE OLGA RADKO (DEPARTMENT OF MATHEMATICS, UCLA) 1. Introduction: transformations of the plane Transformation of the plane is a rule which specifies where each of the points

### Inversion in a Circle

Inversion in a Circle Tom Davis tomrdavis@earthlink.net http://www.geometer.org/mathcircles March 23, 2011 Abstract This article will describe the geometric tool of inversion in a circle, and will demonstrate

### Inversion. Chapter 7. 7.1 Constructing The Inverse of a Point: If P is inside the circle of inversion: (See Figure 7.1)

Chapter 7 Inversion Goal: In this chapter we define inversion, give constructions for inverses of points both inside and outside the circle of inversion, and show how inversion could be done using Geometer

### 13 Groups of Isometries

13 Groups of Isometries For any nonempty set X, the set S X of all one-to-one mappings from X onto X is a group under the composition of mappings (Example 71(d)) In particular, if X happens to be the Euclidean

### Chapter 6 Notes: Circles

Chapter 6 Notes: Circles IMPORTANT TERMS AND DEFINITIONS A circle is the set of all points in a plane that are at a fixed distance from a given point known as the center of the circle. Any line segment

### 4. Prove the above theorem. 5. Prove the above theorem. 9. Prove the above corollary. 10. Prove the above theorem.

14 Perpendicularity and Angle Congruence Definition (acute angle, right angle, obtuse angle, supplementary angles, complementary angles) An acute angle is an angle whose measure is less than 90. A right

### Let s Talk About Symmedians!

Let s Talk bout Symmedians! Sammy Luo and osmin Pohoata bstract We will introduce symmedians from scratch and prove an entire collection of interconnected results that characterize them. Symmedians represent

### CK-12 Geometry: Parts of Circles and Tangent Lines

CK-12 Geometry: Parts of Circles and Tangent Lines Learning Objectives Define circle, center, radius, diameter, chord, tangent, and secant of a circle. Explore the properties of tangent lines and circles.

### Chapters 6 and 7 Notes: Circles, Locus and Concurrence

Chapters 6 and 7 Notes: Circles, Locus and Concurrence IMPORTANT TERMS AND DEFINITIONS A circle is the set of all points in a plane that are at a fixed distance from a given point known as the center of

### Selected practice exam solutions (part 5, item 2) (MAT 360)

Selected practice exam solutions (part 5, item ) (MAT 360) Harder 8,91,9,94(smaller should be replaced by greater )95,103,109,140,160,(178,179,180,181 this is really one problem),188,193,194,195 8. On

### DEFINITIONS. Perpendicular Two lines are called perpendicular if they form a right angle.

DEFINITIONS Degree A degree is the 1 th part of a straight angle. 180 Right Angle A 90 angle is called a right angle. Perpendicular Two lines are called perpendicular if they form a right angle. Congruent

### Lesson 2: Circles, Chords, Diameters, and Their Relationships

Circles, Chords, Diameters, and Their Relationships Student Outcomes Identify the relationships between the diameters of a circle and other chords of the circle. Lesson Notes Students are asked to construct

### Geometry: Euclidean. Through a given external point there is at most one line parallel to a

Geometry: Euclidean MATH 3120, Spring 2016 The proofs of theorems below can be proven using the SMSG postulates and the neutral geometry theorems provided in the previous section. In the SMSG axiom list,

### The Inversion Transformation

The Inversion Transformation A non-linear transformation The transformations of the Euclidean plane that we have studied so far have all had the property that lines have been mapped to lines. Transformations

### Factoring Patterns in the Gaussian Plane

Factoring Patterns in the Gaussian Plane Steve Phelps Introduction This paper describes discoveries made at the Park City Mathematics Institute, 00, as well as some proofs. Before the summer I understood

### HYPERBOLIC TRIGONOMETRY

HYPERBOLIC TRIGONOMETRY STANISLAV JABUKA Consider a hyperbolic triangle ABC with angle measures A) = α, B) = β and C) = γ. The purpose of this note is to develop formulae that allow for a computation of

### 1. A student followed the given steps below to complete a construction. Which type of construction is best represented by the steps given above?

1. A student followed the given steps below to complete a construction. Step 1: Place the compass on one endpoint of the line segment. Step 2: Extend the compass from the chosen endpoint so that the width

### Circle Name: Radius: Diameter: Chord: Secant:

12.1: Tangent Lines Congruent Circles: circles that have the same radius length Diagram of Examples Center of Circle: Circle Name: Radius: Diameter: Chord: Secant: Tangent to A Circle: a line in the plane

### Projective Geometry - Part 2

Projective Geometry - Part 2 Alexander Remorov alexanderrem@gmail.com Review Four collinear points A, B, C, D form a harmonic bundle (A, C; B, D) when CA : DA CB DB = 1. A pencil P (A, B, C, D) is the

### Contents. 2 Lines and Circles 3 2.1 Cartesian Coordinates... 3 2.2 Distance and Midpoint Formulas... 3 2.3 Lines... 3 2.4 Circles...

Contents Lines and Circles 3.1 Cartesian Coordinates.......................... 3. Distance and Midpoint Formulas.................... 3.3 Lines.................................. 3.4 Circles..................................

### Geometry Unit 5: Circles Part 1 Chords, Secants, and Tangents

Geometry Unit 5: Circles Part 1 Chords, Secants, and Tangents Name Chords and Circles: A chord is a segment that joins two points of the circle. A diameter is a chord that contains the center of the circle.

### Grade 7/8 Math Circles Greek Constructions - Solutions October 6/7, 2015

Faculty of Mathematics Waterloo, Ontario N2L 3G1 Centre for Education in Mathematics and Computing Grade 7/8 Math Circles Greek Constructions - Solutions October 6/7, 2015 Mathematics Without Numbers The

### CSU Fresno Problem Solving Session. Geometry, 17 March 2012

CSU Fresno Problem Solving Session Problem Solving Sessions website: http://zimmer.csufresno.edu/ mnogin/mfd-prep.html Math Field Day date: Saturday, April 21, 2012 Math Field Day website: http://www.csufresno.edu/math/news

### Higher Geometry Problems

Higher Geometry Problems ( Look up Eucidean Geometry on Wikipedia, and write down the English translation given of each of the first four postulates of Euclid. Rewrite each postulate as a clear statement

### Chapter 12. The Straight Line

302 Chapter 12 (Plane Analytic Geometry) 12.1 Introduction: Analytic- geometry was introduced by Rene Descartes (1596 1650) in his La Geometric published in 1637. Accordingly, after the name of its founder,

### Student Name: Teacher: Date: District: Miami-Dade County Public Schools. Assessment: 9_12 Mathematics Geometry Exam 1

Student Name: Teacher: Date: District: Miami-Dade County Public Schools Assessment: 9_12 Mathematics Geometry Exam 1 Description: GEO Topic 1 Test: Tools of Geometry Form: 201 1. A student followed the

### Angle Bisectors in a Triangle

346/ ANGLE BISECTORS IN A TRIANGLE Angle Bisectors in a Triangle I. F. Sharygin In this article, we have collected some geometric facts which are directly or tangentially related to the angle bisectors

### Three Lemmas in Geometry

Winter amp 2010 Three Lemmas in Geometry Yufei Zhao Three Lemmas in Geometry Yufei Zhao Massachusetts Institute of Technology yufei.zhao@gmail.com 1 iameter of incircle T Lemma 1. Let the incircle of triangle

### STRAIGHT LINES. , y 1. tan. and m 2. 1 mm. If we take the acute angle between two lines, then tan θ = = 1. x h x x. x 1. ) (x 2

STRAIGHT LINES Chapter 10 10.1 Overview 10.1.1 Slope of a line If θ is the angle made by a line with positive direction of x-axis in anticlockwise direction, then the value of tan θ is called the slope

### Elementary triangle geometry

Elementary triangle geometry Dennis Westra March 26, 2010 bstract In this short note we discuss some fundamental properties of triangles up to the construction of the Euler line. ontents ngle bisectors

### Topics Covered on Geometry Placement Exam

Topics Covered on Geometry Placement Exam - Use segments and congruence - Use midpoint and distance formulas - Measure and classify angles - Describe angle pair relationships - Use parallel lines and transversals

### Education Resources. This section is designed to provide examples which develop routine skills necessary for completion of this section.

Education Resources Circle Higher Mathematics Supplementary Resources Section A This section is designed to provide examples which develop routine skills necessary for completion of this section. R1 I

Forum Geometricorum Volume 10 (2010) 79 91. FORUM GEOM ISSN 1534-1178 Orthic Quadrilaterals of a Convex Quadrilateral Maria Flavia Mammana, Biagio Micale, and Mario Pennisi Abstract. We introduce the orthic

### Tangent Properties. Line m is a tangent to circle O. Point T is the point of tangency.

CONDENSED LESSON 6.1 Tangent Properties In this lesson you will Review terms associated with circles Discover how a tangent to a circle and the radius to the point of tangency are related Make a conjecture

### 206 MATHEMATICS CIRCLES

206 MATHEMATICS 10.1 Introduction 10 You have studied in Class IX that a circle is a collection of all points in a plane which are at a constant distance (radius) from a fixed point (centre). You have

### Circle geometry theorems

Circle geometry theorems http://topdrawer.aamt.edu.au/geometric-reasoning/big-ideas/circlegeometry/angle-and-chord-properties Theorem Suggested abbreviation Diagram 1. When two circles intersect, the line

### Radius, diameter, circumference, π (Pi), central angles, Pythagorean relationship. about CIRCLES

Grade 9 Math Unit 8 : CIRCLE GEOMETRY NOTES 1 Chapter 8 in textbook (p. 384 420) 5/50 or 10% on 2011 CRT: 5 Multiple Choice WHAT YOU SHOULD ALREADY KNOW: Radius, diameter, circumference, π (Pi), central

### Geometry in a Nutshell

Geometry in a Nutshell Henry Liu, 26 November 2007 This short handout is a list of some of the very basic ideas and results in pure geometry. Draw your own diagrams with a pencil, ruler and compass where

### Centroid: The point of intersection of the three medians of a triangle. Centroid

Vocabulary Words Acute Triangles: A triangle with all acute angles. Examples 80 50 50 Angle: A figure formed by two noncollinear rays that have a common endpoint and are not opposite rays. Angle Bisector:

### Plane transformations and isometries

Plane transformations and isometries We have observed that Euclid developed the notion of congruence in the plane by moving one figure on top of the other so that corresponding parts coincided. This notion

### Conjectures. Chapter 2. Chapter 3

Conjectures Chapter 2 C-1 Linear Pair Conjecture If two angles form a linear pair, then the measures of the angles add up to 180. (Lesson 2.5) C-2 Vertical Angles Conjecture If two angles are vertical

### Lesson 19: Equations for Tangent Lines to Circles

Classwork Opening Exercise A circle of radius 5 passes through points ( 3, 3) and (3, 1). a. What is the special name for segment? b. How many circles can be drawn that meet the given criteria? Explain

### Projective Geometry. Stoienescu Paul. International Computer High School Of Bucharest,

Projective Geometry Stoienescu Paul International Computer High School Of Bucharest, paul98stoienescu@gmail.com Abstract. In this note, I will present some olympiad problems which can be solved using projective

### SMT 2014 Geometry Test Solutions February 15, 2014

SMT 014 Geometry Test Solutions February 15, 014 1. The coordinates of three vertices of a parallelogram are A(1, 1), B(, 4), and C( 5, 1). Compute the area of the parallelogram. Answer: 18 Solution: Note

### The Protractor Postulate and the SAS Axiom. Chapter The Axioms of Plane Geometry

The Protractor Postulate and the SAS Axiom Chapter 3.4-3.7 The Axioms of Plane Geometry The Protractor Postulate and Angle Measure The Protractor Postulate (p51) defines the measure of an angle (denoted

### CIRCLE COORDINATE GEOMETRY

CIRCLE COORDINATE GEOMETRY (EXAM QUESTIONS) Question 1 (**) A circle has equation x + y = 2x + 8 Determine the radius and the coordinates of the centre of the circle. r = 3, ( 1,0 ) Question 2 (**) A circle

### CIRCLE DEFINITIONS AND THEOREMS

DEFINITIONS Circle- The set of points in a plane equidistant from a given point(the center of the circle). Radius- A segment from the center of the circle to a point on the circle(the distance from the

### d(f(g(p )), f(g(q))) = d(g(p ), g(q)) = d(p, Q).

10.2. (continued) As we did in Example 5, we may compose any two isometries f and g to obtain new mappings f g: E E, (f g)(p ) = f(g(p )), g f: E E, (g f)(p ) = g(f(p )). These are both isometries, the

### 3. Lengths and areas associated with the circle including such questions as: (i) What happens to the circumference if the radius length is doubled?

1.06 Circle Connections Plan The first two pages of this document show a suggested sequence of teaching to emphasise the connections between synthetic geometry, co-ordinate geometry (which connects algebra

### Casey s Theorem and its Applications

Casey s Theorem and its Applications Luis González Maracaibo. Venezuela July 011 Abstract. We present a proof of the generalized Ptolemy s theorem, also known as Casey s theorem and its applications in

### GEOMETRY. Constructions OBJECTIVE #: G.CO.12

GEOMETRY Constructions OBJECTIVE #: G.CO.12 OBJECTIVE Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic

### 5 Hyperbolic Triangle Geometry

5 Hyperbolic Triangle Geometry 5.1 Hyperbolic trigonometry Figure 5.1: The trigonometry of the right triangle. Theorem 5.1 (The hyperbolic right triangle). The sides and angles of any hyperbolic right

### A note on the geometry of three circles

A note on the geometry of three circles R. Pacheco, F. Pinheiro and R. Portugal Departamento de Matemática, Universidade da Beira Interior, Rua Marquês d Ávila e Bolama, 6201-001, Covilhã - Portugal. email:

PYTHAGOREAN TRIPLES KEITH CONRAD 1. Introduction A Pythagorean triple is a triple of positive integers (a, b, c) where a + b = c. Examples include (3, 4, 5), (5, 1, 13), and (8, 15, 17). Below is an ancient

### 1. Determine all real numbers a, b, c, d that satisfy the following system of equations.

altic Way 1999 Reykjavík, November 6, 1999 Problems 1. etermine all real numbers a, b, c, d that satisfy the following system of equations. abc + ab + bc + ca + a + b + c = 1 bcd + bc + cd + db + b + c

### Practical Geometry. Chapter Introduction

Practical Geometry Chapter 14 14.1 Introduction We see a number of shapes with which we are familiar. We also make a lot of pictures. These pictures include different shapes. We have learnt about some

### Transversals. 1, 3, 5, 7, 9, 11, 13, 15 are all congruent by vertical angles, corresponding angles,

Transversals In the following explanation and drawing, an example of the angles created by two parallel lines and two transversals are shown and explained: 1, 3, 5, 7, 9, 11, 13, 15 are all congruent by

### Lesson 19: Equations for Tangent Lines to Circles

Student Outcomes Given a circle, students find the equations of two lines tangent to the circle with specified slopes. Given a circle and a point outside the circle, students find the equation of the line

### Section 9-1. Basic Terms: Tangents, Arcs and Chords Homework Pages 330-331: 1-18

Chapter 9 Circles Objectives A. Recognize and apply terms relating to circles. B. Properly use and interpret the symbols for the terms and concepts in this chapter. C. Appropriately apply the postulates,

### alternate interior angles

alternate interior angles two non-adjacent angles that lie on the opposite sides of a transversal between two lines that the transversal intersects (a description of the location of the angles); alternate

### BC AB = AB. The first proportion is derived from similarity of the triangles BDA and ADC. These triangles are similar because

150 hapter 3. SIMILRITY 397. onstruct a triangle, given the ratio of its altitude to the base, the angle at the vertex, and the median drawn to one of its lateral sides 398. Into a given disk segment,

### Solutions to Practice Problems

Higher Geometry Final Exam Tues Dec 11, 5-7:30 pm Practice Problems (1) Know the following definitions, statements of theorems, properties from the notes: congruent, triangle, quadrilateral, isosceles

### Poincaré s Disk Model for Hyperbolic Geometry

Chapter 9 oincaré s Disk Model for Hyperbolic Geometry 9.1 Saccheri s Work Recall that Saccheri introduced a certain family of quadrilaterals. Look again at Section 7.3 to remind yourself of the properties

### Circle Theorems. This circle shown is described an OT. As always, when we introduce a new topic we have to define the things we wish to talk about.

Circle s circle is a set of points in a plane that are a given distance from a given point, called the center. The center is often used to name the circle. T This circle shown is described an OT. s always,

### IMO Training 2008 Circles Yufei Zhao. Circles. Yufei Zhao.

ircles Yufei Zhao yufeiz@mit.edu 1 Warm up problems 1. Let and be two segments, and let lines and meet at X. Let the circumcircles of X and X meet again at O. Prove that triangles O and O are similar.

### 1 Solution of Homework

Math 3181 Dr. Franz Rothe February 4, 2011 Name: 1 Solution of Homework 10 Problem 1.1 (Common tangents of two circles). How many common tangents do two circles have. Informally draw all different cases,

### ABC is the triangle with vertices at points A, B and C

Euclidean Geometry Review This is a brief review of Plane Euclidean Geometry - symbols, definitions, and theorems. Part I: The following are symbols commonly used in geometry: AB is the segment from the

### 55 questions (multiple choice, check all that apply, and fill in the blank) The exam is worth 220 points.

Geometry Core Semester 1 Semester Exam Preparation Look back at the unit quizzes and diagnostics. Use the unit quizzes and diagnostics to determine which topics you need to review most carefully. The unit

### Theorem 5. The composition of any two symmetries in a point is a translation. More precisely, S B S A = T 2

Isometries. Congruence mappings as isometries. The notion of isometry is a general notion commonly accepted in mathematics. It means a mapping which preserves distances. The word metric is a synonym to

### 7. The Gauss-Bonnet theorem

7. The Gauss-Bonnet theorem 7. Hyperbolic polygons In Euclidean geometry, an n-sided polygon is a subset of the Euclidean plane bounded by n straight lines. Thus the edges of a Euclidean polygon are formed

### The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Wednesday, January 29, :15 a.m. SAMPLE RESPONSE SET

The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Wednesday, January 29, 2014 9:15 a.m. SAMPLE RESPONSE SET Table of Contents Question 29................... 2 Question 30...................

### Geometry Chapter 1 Vocabulary. coordinate - The real number that corresponds to a point on a line.

Chapter 1 Vocabulary coordinate - The real number that corresponds to a point on a line. point - Has no dimension. It is usually represented by a small dot. bisect - To divide into two congruent parts.

### If a question asks you to find all or list all and you think there are none, write None.

If a question asks you to find all or list all and you think there are none, write None 1 Simplify 1/( 1 3 1 4 ) 2 The price of an item increases by 10% and then by another 10% What is the overall price

### Lesson 13: Proofs in Geometry

211 Lesson 13: Proofs in Geometry Beginning with this lesson and continuing for the next few lessons, we will explore the role of proofs and counterexamples in geometry. To begin, recall the Pythagorean

### Magic circles in the arbelos

by Christer Bergsten LiTH-MAT-R-006-1 Matematiska institutionen Department of Mathematics Linköpings universitet Linköpings universitet 58183 Linköping SE-58183 Linköping, Sweden Magic circles in the arbelos

### 14. Dirichlet polygons: the construction

14. Dirichlet polygons: the construction 14.1 Recap Let Γ be a Fuchsian group. Recall that a Fuchsian group is a discrete subgroup of the group Möb(H) of all Möbius transformations. In the last lecture,

### ON THE SIMSON WALLACE THEOREM

South Bohemia Mathematical Letters Volume 21, (2013), No. 1, 59 66. ON THE SIMSON WALLACE THEOREM PAVEL PECH 1, EMIL SKŘÍŠOVSKÝ2 Abstract. The paper deals with the well-known Simson Wallace theorem and

### MATHEMATICS Grade 12 EUCLIDEAN GEOMETRY: CIRCLES 02 JULY 2014

EUCLIDEAN GEOMETRY: CIRCLES 02 JULY 2014 Checklist Make sure you learn proofs of the following theorems: The line drawn from the centre of a circle perpendicular to a chord bisects the chord The angle

### Sec 1.1 CC Geometry - Constructions Name: 1. [COPY SEGMENT] Construct a segment with an endpoint of C and congruent to the segment AB.

Sec 1.1 CC Geometry - Constructions Name: 1. [COPY SEGMENT] Construct a segment with an endpoint of C and congruent to the segment AB. A B C **Using a ruler measure the two lengths to make sure they have

### Chapter 1: Points, Lines, Planes, and Angles

Chapter 1: Points, Lines, Planes, and Angles (page 1) 1-1: A Game and Some Geometry (page 1) In the figure below, you see five points: A,B,C,D, and E. Use a centimeter ruler to find the requested distances.

### 4.1 Euclidean Parallelism, Existence of Rectangles

Chapter 4 Euclidean Geometry Based on previous 15 axioms, The parallel postulate for Euclidean geometry is added in this chapter. 4.1 Euclidean Parallelism, Existence of Rectangles Definition 4.1 Two distinct

### NCERT. Choose the correct answer from the given four options:

CIRCLES (A) Main Concepts and Results The meaning of a tangent and its point of contact on a circle. Tangent is perpendicular to the radius through the point of contact. Only two tangents can be drawn

### Content Area: GEOMETRY Grade 9 th Quarter 1 st Curso Serie Unidade

Content Area: GEOMETRY Grade 9 th Quarter 1 st Curso Serie Unidade Standards/Content Padrões / Conteúdo Learning Objectives Objetivos de Aprendizado Vocabulary Vocabulário Assessments Avaliações Resources

### Hansen s Right Triangle Theorem, Its Converse and a Generalization

Forum Geometricorum Volume 6 (2006) 335 342. FORUM GEO ISSN 1534-1178 M Hansen s Right Triangle Theorem, Its onverse and a Generalization my ell bstract. We generalize D. W. Hansen s theorem relating the

### Incenter Circumcenter

TRIANGLE: Centers: Incenter Incenter is the center of the inscribed circle (incircle) of the triangle, it is the point of intersection of the angle bisectors of the triangle. The radius of incircle is

### 11 th Annual Harvard-MIT Mathematics Tournament

11 th nnual Harvard-MIT Mathematics Tournament Saturday February 008 Individual Round: Geometry Test 1. [] How many different values can take, where,, are distinct vertices of a cube? nswer: 5. In a unit

### Tangent circles in the hyperbolic disk

Rose- Hulman Undergraduate Mathematics Journal Tangent circles in the hyperbolic disk Megan Ternes a Volume 14, No. 1, Spring 2013 Sponsored by Rose-Hulman Institute of Technology Department of Mathematics

### Power of a Point Solutions

ower of a oint Solutions Yufei Zhao Trinity ollege, ambridge yufei.zhao@gmail.com pril 2011 ractice problems: 1. Let Γ 1 and Γ 2 be two intersecting circles. Let a common tangent to Γ 1 and Γ 2 touch Γ

### San Jose Math Circle April 25 - May 2, 2009 ANGLE BISECTORS

San Jose Math Circle April 25 - May 2, 2009 ANGLE BISECTORS Recall that the bisector of an angle is the ray that divides the angle into two congruent angles. The most important results about angle bisectors

### Geometry Unit 7 (Textbook Chapter 9) Solving a right triangle: Find all missing sides and all missing angles

Geometry Unit 7 (Textbook Chapter 9) Name Objective 1: Right Triangles and Pythagorean Theorem In many geometry problems, it is necessary to find a missing side or a missing angle of a right triangle.

### Chapter 1: Essentials of Geometry

Section Section Title 1.1 Identify Points, Lines, and Planes 1.2 Use Segments and Congruence 1.3 Use Midpoint and Distance Formulas Chapter 1: Essentials of Geometry Learning Targets I Can 1. Identify,

### STRAIGHT LINE COORDINATE GEOMETRY

STRAIGHT LINE COORDINATE GEOMETRY (EXAM QUESTIONS) Question 1 (**) The points P and Q have coordinates ( 7,3 ) and ( 5,0), respectively. a) Determine an equation for the straight line PQ, giving the answer

### Co-ordinate Geometry THE EQUATION OF STRAIGHT LINES

Co-ordinate Geometry THE EQUATION OF STRAIGHT LINES This section refers to the properties of straight lines and curves using rules found by the use of cartesian co-ordinates. The Gradient of a Line. As

### Chapter 6 Quiz. Section 6.1 Circles and Related Segments and Angles

Chapter 6 Quiz Section 6.1 Circles and Related Segments and Angles (1.) TRUE or FALSE: The center of a circle lies in the interior of the circle. For exercises 2 4, use the figure provided. (2.) In O,

### Math 311 Test III, Spring 2013 (with solutions)

Math 311 Test III, Spring 2013 (with solutions) Dr Holmes April 25, 2013 It is extremely likely that there are mistakes in the solutions given! Please call them to my attention if you find them. This exam

### Definitions, Postulates and Theorems

Definitions, s and s Name: Definitions Complementary Angles Two angles whose measures have a sum of 90 o Supplementary Angles Two angles whose measures have a sum of 180 o A statement that can be proven