Information for More Experienced Geometers As the titles of the chapters and sections suggest,the book is organized mostly around concepts such as parallel lines and similarity and the related transformations rather than around shapes,such as triangles or circles.Each chapter includes some basic or elementary theorems but also a few less common results or more advanced topics.Proofs feature transformations more than most texts.You may wish to look around to see where certain favorite topics are located and how things are put together (and what seems essential and what may be optional in a course). Non-Euclidean geometry is a topic important for some readers or instructors.The first five axioms in this book are axioms valid for the hyperbolic non-Euclidean plane. The book is structured so that all of the first six chapters,with a couple of isolated and easily avoidable exceptions,use only the first five axioms,so they can be used as an introduction to neutral geometry.valid for both the euclidean and the non-Euclidean plane.(The principal exception is that Axiom 6 is used early to prove the angle sum theorem for triangles so that,after regular polygons are introduced,angles can be used in exercises:but this result is not used in the main text until the chapter on parallels.)
xvi Information for More Experienced Geometers As the titles of the chapters and sections suggest, the book is organized mostly around concepts such as parallel lines and similarity and the related transformations rather than around shapes, such as triangles or circles. Each chapter includes some basic or elementary theorems but also a few less common results or more advanced topics. Proofs feature transformations more than most texts. You may wish to look around to see where certain favorite topics are located and how things are put together (and what seems essential and what may be optional in a course). Non-Euclidean geometry is a topic important for some readers or instructors. The first five axioms in this book are axioms valid for the hyperbolic non-Euclidean plane. The book is structured so that all of the first six chapters, with a couple of isolated and easily avoidable exceptions, use only the first five axioms, so they can be used as an introduction to neutral geometry, valid for both the Euclidean and the non-Euclidean plane. (The principal exception is that Axiom 6 is used early to prove the angle sum theorem for triangles so that, after regular polygons are introduced, angles can be used in exercises; but this result is not used in the main text until the chapter on parallels.)
A Chapter Guide for Instructors and Others This chapter outine is offered in the hope that it will com ten and provide helpful information for instruetors and othethe boo and/or planning a course. The Basics:Chapters1to4.The first fourchapters begin with an introduction and tactonfS of ax th proofs of the triangleco gruence th ems.An ea to a e for futu Anyone using all or most of the uld not skip any section ort chapters Chapter 1:This is a mostly descriptive introduction to the goals of the book.Rigid motions and congruence are defined and some properties are proved in a very general context.But except for a preview example,there is no actual plane geom etry since there are as yet no axioms.For completeness,there is a formal proof of the equivalence properties of congruence,but students and instructors may find the justification by informal geometry adequate before moving on to Chapter 2. Chapter2:This chapter states all the axiom ler functions th ure distar e on lines and protractor unctio ciated pol rangles that measure angles are key topics.Many b are defined.Ib c geometry terms elieve that one could move briskly through this chapter and pick up a deeper understanding about rulers and protractors and terminology when encountering them in the geometrical proofs in Chapters 3 and 4 ortant chap concept to 么deheha9 ethe dmotio c but maj points on one proves tha ion n us sline refle cnon to prov re e properties of isosceles triangles(important for everything later)
A Chapter Guide for Instructors and Others This chapter outline is offered in the hope that it will complement the table of contents and provide helpful information for instructors and others in navigating the book and/or planning a course. The Basics: Chapters 1 to 4. The first four chapters begin with an introduction and statement of axioms and end with proofs of the triangle congruence theorems. An extract of this content could be adapted to a workshop or very short course for future or practicing teachers. Anyone using all or most of the book should not skip any sections of these short chapters. Chapter 1: This is a mostly descriptive introduction to the goals of the book. Rigid motions and congruence are defined and some properties are proved in a very general context. But except for a preview example, there is no actual plane geometry since there are as yet no axioms. For completeness, there is a formal proof of the equivalence properties of congruence, but students and instructors may find the justification by informal geometry adequate before moving on to Chapter 2. Chapter 2: This chapter states all the axioms and goes into the first four in detail. Ruler functions that measure distance on lines and protractor functions and associated polar angles that measure angles are key topics. Many basic geometry terms are defined. I believe that one could move briskly through this chapter and pick up a deeper understanding about rulers and protractors and terminology when encountering them in the geometrical proofs in Chapters 3 and 4. Chapter 3: This important chapter provides the first chance to use the rigid motion concept to prove basic but major theorems. Beginning with an existence axiom for a rigid motion that fixes points on a line, one proves that this rigid motion is the usual line reflection and then uses line reflection to prove theorems about perpendiculars, the properties of isosceles triangles (important for everything later), and other results. xvii
xviij A Chapter Guide for Instructors and Others Chapter 4:This is where all the triangle congruence theorems are proved using compositions of reflections.An important feature of these theorems is that the rigid motion defining congruence from AABC to DEF is unique.In addition, in the last section,the angle sum theorem for triangles is proved using the dila- tion axiom.If one wants to defer this axiom,equivalent to the Euclidean Parallel Postulate,until later,this result is only used for regular polygons until Chapter7. Rotations,Transations,and Dilations:.By theed ofthse topic an i ne geor work ons. rigid motions are introduce Chapter 5:The first part of this cha st of the r is 00 .le of r on c an be defined on the plane by asingle tria gle.Ori r signed angle me ation-preserving s critic applied tangent construct ns an incirc solution to Fagnano's problem of finding the triangle of minimum perimeter in- scribed in a given triangle. Chapter 7:Beginning with a proof of the Euclidean Parallel Postulate as a theo rem using the dilation axiom,this chapter develops a number of theorems about parallelograms and rectangles,with half-turns as a main tool.This culminates in the very important definition of translations as products of half-turns and also as products of reflections in parallel lines,with proofs of their properties.At the end of the chapter,translations are applied to define orientation and polar an- gles on the whole plane,concluding with a brief introduction to vectors and a (nonstandard)vector notation.Many parts of this chapter are important for later developments. Chapter 8:Similarity is the topic of this long chapter.The early parts of the chapter contain basic material about similarity of triangles and other polygons(including the Pythagorean Theorem and the Golden Ratio)as well as ratios and signed ratios from transversals.The middle of the chapter applies dilations to solve problems about triangles and circles.The last part of the chapter is devoted to theorems concerning circles,including inscribed angles.Some topics near the end,such as Apollonian circles and the radical axis of circles,are interesting geometry topics not found in every beginning course
xviii A Chapter Guide for Instructors and Others Chapter 4: This is where all the triangle congruence theorems are proved using compositions of reflections. An important feature of these theorems is that the rigid motion defining congruence from △𝐴𝐵𝐶 to △𝐷𝐸𝐹 is unique. In addition, in the last section, the angle sum theorem for triangles is proved using the dilation axiom. If one wants to defer this axiom, equivalent to the Euclidean Parallel Postulate, until later, this result is only used for regular polygons until Chapter 7. Rotations, Translations, and Dilations: Chapters 5 to 8. By the end of these chapters, one will have covered all or most of the topics about lines, angle, polygons, and circles that may be thought of as the essential core of an introduction to Euclidean plane geometry plus experience working with rotations, translations, and dilations in solving problems and proving theorems. New geometrical theorems are proven as new rigid motions are introduced. Chapter 5: The first part of this chapter introduces rotations as double line reflections, an essential topic for the rest of the book. This theory is applied to rotational and dihedral symmetry groups at a point, leading to a definition of regular polygons, with some details left to the exercises. The last section proves that an orientation can be defined on the plane by the order of vertices of a single triangle. Orientation is important for signed angle measure and orientation-preserving rigid motions, but some details of the proofs may be less critical. Chapter 6: This chapter features half-turns, inequalities within triangles, and the triangle inequality. The concept of distance from a point to a line is applied to tangent constructions and incircles, and reflected light paths culminate in the solution to Fagnano’s problem of finding the triangle of minimum perimeter inscribed in a given triangle. Chapter 7: Beginning with a proof of the Euclidean Parallel Postulate as a theorem using the dilation axiom, this chapter develops a number of theorems about parallelograms and rectangles, with half-turns as a main tool. This culminates in the very important definition of translations as products of half-turns and also as products of reflections in parallel lines, with proofs of their properties. At the end of the chapter, translations are applied to define orientation and polar angles on the whole plane, concluding with a brief introduction to vectors and a (nonstandard) vector notation. Many parts of this chapter are important for later developments. Chapter 8: Similarity is the topic of this long chapter. The early parts of the chapter contain basic material about similarity of triangles and other polygons (including the Pythagorean Theorem and the Golden Ratio) as well as ratios and signed ratios from transversals. The middle of the chapter applies dilations to solve problems about triangles and circles. The last part of the chapter is devoted to theorems concerning circles, including inscribed angles. Some topics near the end, such as Apollonian circles and the radical axis of circles, are interesting geometry topics not found in every beginning course
A Chapter Guide for Instructors and Others xix Further Topics in the Plane:Chapters 9to 11.The last chapters include one on area,one on products of rigid motions applied to symmetry,and one introducing coordinates. one could put together a short course (e.g.a one-quarter course)that includes the topics essential to a basic geometry course by covering most of the first eight chapters and then as tim e time Chapter 9:This short chapter about area includes the usual elen nentary area for- mulas,a discussion of scaling and area,area proofs of the Pythagorean Theorem. and the formulas for circle area and perimeter(without explicitly estimatingπ)】 The chapter r points out the effect of similarity transformations on area.ending with affine the orems about ratios and area and the introduction of shear trans formation eT a introdt of area r quires addin some as tions ab tie tha based. other Chapter 10:This long chapter is organized around the topic of composition of rigid motions and how the relationships from composition restrict possible symme- tries.It begins by showing how to compute the composition of two rotations with different centers.or of a rotation and a translation.and applving this to a de tailed examination of the symmetries of some wallpaper patterns,starting with a leisurely and detailed examination of a tile pattern with 90-degree rotational symmetry.Wallpaper groups are introduced and some other examples are given. but there is no attempt to look at every crystallographic group.The composition rules are then applied to prove Napoleon's Theorem,the existence of the Fermat Point and the erties of several interesting tessellations.A complete proof that p成shanamakianingr5saiclotnme2tdsnwiosad there are sever es of frieze ning facts about glide reflections completing the picture of the four kinds of rigid motions. Chapter 11:This chapter begins by constructing affine coordinates on the plane. proves formulas for half-turns and translations,and derives familiar equations for lines from dilations.Next comes the case of Cartesian coordinates and met- ric formulas:the distance formulas and dot products.There is a short section on why the(x,y)-plane for graphs is not the Euclidean plane.There are rotation and reflection formulas and an introduction to the use of complex numbers to express these formulas.The latter part of the chapter introduces barycentric coordinates, proves Ceva's Theorem,and explores affine transformations and the interpreta- tion of determinant as signed area.The last topic returns to the very beginning of the book and proves that the Cartesian plane is a mathematical model satisfying the six axioms in Chapter 2
A Chapter Guide for Instructors and Others xix Further Topics in the Plane: Chapters 9 to 11. The last chapters include one on area, one on products of rigid motions applied to symmetry, and one introducing coordinates. One could put together a short course (e.g., a one-quarter course) that includes the topics essential to a basic geometry course by covering most of the first eight chapters and then, as time permits, including topics from these chapters. With more time, each chapter offers a number of valuable geometric ideas. Chapter 9: This short chapter about area includes the usual elementary area formulas, a discussion of scaling and area, area proofs of the Pythagorean Theorem, and the formulas for circle area and perimeter (without explicitly estimating 𝜋). The chapter points out the effect of similarity transformations on area, ending with affine theorems about ratios and area and the introduction of shear transformations. The introduction of area requires adding some assumptions about the properties of area that were not part of the six axioms on which all the other proofs are based. Chapter 10: This long chapter is organized around the topic of composition of rigid motions and how the relationships from composition restrict possible symmetries. It begins by showing how to compute the composition of two rotations with different centers, or of a rotation and a translation, and applying this to a detailed examination of the symmetries of some wallpaper patterns, starting with a leisurely and detailed examination of a tile pattern with 90-degree rotational symmetry. Wallpaper groups are introduced and some other examples are given, but there is no attempt to look at every crystallographic group. The composition rules are then applied to prove Napoleon’s Theorem, the existence of the Fermat Point and the properties of several interesting tessellations. A complete proof that there are seven types of frieze symmetry is carried out in great detail, with lots of pictures. Then a final section fills in some remaining facts about glide reflections, completing the picture of the four kinds of rigid motions. Chapter 11: This chapter begins by constructing affine coordinates on the plane, proves formulas for half-turns and translations, and derives familiar equations for lines from dilations. Next comes the case of Cartesian coordinates and metric formulas: the distance formulas and dot products. There is a short section on why the (𝑥, 𝑦)-plane for graphs is not the Euclidean plane. There are rotation and reflection formulas and an introduction to the use of complex numbers to express these formulas. The latter part of the chapter introduces barycentric coordinates, proves Ceva’s Theorem, and explores affine transformations and the interpretation of determinant as signed area. The last topic returns to the very beginning of the book and proves that the Cartesian plane is a mathematical model satisfying the six axioms in Chapter 2
Acknowledgments I have many people to thank for help with this book and also for deepening my appre ciation of geometry and supporting my interest in sharing it with others. This book was Park City Mathemati en as ar er Leadership Program at th this material nity to develop as part of its overall activities.For their support an en couragement, wish to thank PCMI and its institutional sponsor,the Institute for Advanced Study. and also the other public and private financial sponsors of PCMI,including Math for America. Ihave benefited greatly from my decades-long ssociation with the pcmir for teachers.I ha ed m nd have had t time f Herb clen tute my and he rill is vant to on th supp ort of Brian Hopkins m I dto s my apprec ion fo Other valuable geometrical experiences include years of teaching geometry for future teachers at the University of Washington and many summers of geometry in Seattle with our local PCMI offshoot called Northwest Mathematics Interaction (NWMD.Among the many NwMI people to thank.special mention goes to Jovce and Joe Frost,Art Mabbott,and Clint Chan,who persist in keeping our group going. The spiration for this book was a desire to exp and upon ideas presented PCMI online geometry course developed with PCMIcolleagueGabriel Rosenberg.aided by two working groups of PCMI teachers,including Irene Espritu,Alice Hsaio,Tamuka Hwami,Robert Janes,Wendy Menard,Jennifer Tate,Robert Garber,Jen Katz,Daniel Kerns,Elizabeth McGrath,Vince Muccioli,and Brian Shay. 网
Acknowledgments I have many people to thank for help with this book and also for deepening my appreciation of geometry and supporting my interest in sharing it with others. This book was written as an outgrowth of the Teacher Leadership Program at the Park City Mathematics Institute (PCMI), which provided the opportunity to develop this material as part of its overall activities. For their support and encouragement, I wish to thank PCMI and its institutional sponsor, the Institute for Advanced Study, and also the other public and private financial sponsors of PCMI, including Math for America. I have benefited greatly from my decades-long association with the PCMI program for teachers. I have learned much and have had a great time, starting in 1991 with the visionary leadership of Herb Clemens and Naomi Fisher, when our new program was officially a geometry institute. My years of collaboration with Carol Hattan and Gail Burrill is something I treasure. I also want to mention the support of Brian Hopkins, whom I lured to PCMI as a grad student, and to express my appreciation for our current leaders, Rafe Mazzeo, Peg Cagle, and Monica Tienda. Other valuable geometrical experiences include years of teaching geometry for future teachers at the University of Washington and many summers of geometry in Seattle with our local PCMI offshoot called Northwest Mathematics Interaction (NWMI). Among the many NWMI people to thank, special mention goes to Joyce and Joe Frost, Art Mabbott, and Clint Chan, who persist in keeping our group going. The inspiration for this book was a desire to expand upon ideas presented in a PCMI online geometry course developed with PCMI colleague Gabriel Rosenberg, aided by two working groups of PCMI teachers, including Irene Espritu, Alice Hsaio, Tamuka Hwami, Robert Janes, Wendy Menard, Jennifer Tate, Robert Garber, Jen Katz, Daniel Kerns, Elizabeth McGrath, Vince Muccioli, and Brian Shay. xxi