Geomet for Colege Sdents

I. Martin Isaacs

THE BROOKS/COLE SERIES IN ADVANCED ADVAN CED MATHEMATCS Paul J. Say Jr. Jr. EDITR

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saacs, Martin, (date) Geomety for college students Martin saacs p cm ncludes index SBN 0-534-35179-4 (text) 1. Geometry Title 00-056461 QA445 63 2000 516--dc21

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Preface

his book is primariy intended for coege mathematics students who enjoyed high schoo geometry and who wish to ea more about the amaing properties of ines circles triange triangess and other geometric gures I n pparticuar articuar I hope that those who are are preparing preparing to become highschoo high schoo mathematics mathematics teachers teachers wi nd nd inspiration in spiration here here to hep them share their enthusiasm enthusia sm and enjoyment of geometry geometry with their own future future students utt why u why shoud sho ud anyone study geometry? geo metry? One reas reason on of course cou rse is that geometry geo metry and its descendant desce ndant trigo trigonometr nometry y are essentia ess entia toos in engineering architecture navigation and other other discipines hese practica appications however however surey sure y do not expain why it is that for for centurie centuriess geometry has been taught to amost every every student and why at east recenty a person who knew no geometry geometry was not considered to be propery edu until recenty cated I think there there are at east ea st two reaso reasons ns more important important than usefunes us efunesss that expain why geometry has been and shoud continue to be a part of the schoo schoo curricuum Since Eucid some 300 years ago geometry geometry has been taught as a deductive deductiv e science with theorems and proofs s a consequence, generations of geometry students have eaed how to draw vaid concusions from hypotheses and how to detect and avoid invaid invaid reasoning reasoning In other words by studying geometry geometry students students can c an e how to think are other subjects subjects that coud aso be used to teach deductive reasoning reasoning Of course there are but geometry is especiay eective because it seems to have the perfect baance of depth and concretenes s Many of the theorems that we prove prove in geometry are dee in the sense that that they they assert as sert something nonobvious nonobvious and sometimes even surprising hey are are concrete because students can easiy e asiy draw draw the appropr appropriat iatee trianges circes circes or whatever, whatever, and they they can see that what is aeged to happen actuay does appear appear to happen happe n Geometry is aso beautifu and some of its theorems are so amaing as to seem amost mracuou mracuouss In fact fact much the same s ame comment comment coud be made about most areas of mathematics but geometry is unique in that its miraces are visua so they can readiy be appreciated even by the uninitiated Surey one does not need a great dea of mathematica sophi s ophisti stication cation to marve marve at the fact fact that if a ine is drawn through each vertex vertex of any triange triange and if each of these these ines is i s perpendicuar to the opposite oppos ite side si de of the triange triange then these three ines a go through a common point One coud cou d argue that that the fundamenta fundamenta theorem theorem of cacuus cacuu s for exampe is equay equay amaing and beautifu beautifu but unfort unfortunatey unatey it is not possibe poss ibe to appreciate appreciate it without without rst studying cacuus cacu us ut ut the aesthetic vaue of geometry geo metry extends beyond beyon d the strikng statements of its theorems

PREACE

Many Many of the the proofs have their own subte and eegant beauty which whic h unfortunatey is a itte harder harder to perceive. Nevetheess Nevetheess we expect that that most readers of this this book wi ea to enoy the beauty of o f the the proofs proofs as a s we as that of the theorems theorems.. In shot we shoud continue to study and teach geometry because it is a highy attracti attractive ve subect and because b ecause we can c an ea from from it something about deductive reasoning and the nature of mathematica mathem atica proof. It seems se ems cear ce ar therefoe therefoe that as in the past students shoud shou d continue to see theorems theorems being proved in their their geometry geometry cass cas s they should sh ould be taught how to understand proofs and perhaps even more impotant they shoud shou d lea how to invent and write proofs. I have seected se ected for this book b ook some so me of the the more spectacuar theorems in pane ge ometry ometry and I have presented ustications usti cations of thes thesee facts facts using u sing a vaiety of diere dierent nt techniques of proof. Mosty ignored however are the kind of unsurprising theorem which whie required by mode standards of mathematica rigor can seem rather pointess to students students It requires considerabe sophistication to appreciate why why anyone woud want to prove a fact fact that that seems competey competey obvious obviou s . Even professiona professi ona mathe matcians, most of whom do understand the the signicance signi cance of these resuts often nd their their forma forma axiomatic proofs proo fs somewhat du earning proofs proofs shoud not be an unewarding chore instead we expect that students wi demand proofs proofs because becaus e the assertions being estabished are otherwis otherwisee so incredibe. is book was written as a text for oege Geometry which is a course that I have taught severa times at the niversit of Wisconsin Madison. The courses principa audience consi c onsists sts of o f sophomore and unior unior undergradua undergraduate te math maors who are speciaiing in secondary s econdary education. For them the course is required but there there is aso a so a substantia sub stantia minority who take the course eectivey eectivey Some S ome of these students s tudents simpy want to earn geometry whie others take oege Geometry because they nd it to be a more gente gente and accessibe acces sibe introduction to to mathematica proof proof than than a course cours e in abstract abstract agebra or advanced cacuus. cacuus . It is my beief bei ef that for some students s tudents the study of geometry geometry is i s an exceent preparation for these these more difcut difcut abstract courses . I have been dissatised with the avaiabe texts that might be used for suh a course. course . Many Many incude in cude a umber umber of interesting topics topics within a arge arge samping s amping of assorted geometric materia. ut they do not put the focus where I think it beongs: on the reay pretty theorems and their proofs proofs that in my opinion opini on shoud shou d be at the heat heat of a geometry course for coege students. so they often devote much more space than I think think is appopriate appopriate to fomaism fomaism and axiomatics Most Mo st students probaby do not nd this especiay esp eciay exciting excitin g and I share that opinion. here aso exist exi st some s ome wonderfu wonderfu books that that are ed with spectacuar s pectacuar theorems theorems and eegant proofs but none of these seems quite suitabe as a text for this geometry course either I have found that that most mos t students who register for for oege Geometry Ge ometry claim to remember remember very itte from from their one previous previous exposure to geometry in high schoo. schoo . For the sake of this maority some review and accimatiation are necessary. text is needed therefore that starts from a point cose to the beginning and introduces the notion of proof genty and then then gets to the the good stu" stu " as quicky as pos sibe. sibe . ecause ecause I coud not no t nd a book b ook that covered the right materia materia at the right pace pac e and an d that stated stated from the right pace pac e I taught the course cou rse many times without a text.

PREACE

Most of my students seemed to enjoy the course and many of them became very excited about about geometry. In this this book which is an expansion expansion of my course lecture notes I have have tried tried to reproduce as closely clos ely as pos sible the experience experience of the classro cla ssroom om and so so I hope that my readers readers will wi ll also als o nd that geometr geometryy is an enjoyable enjoyable and an d exciting sub s ubject ject Most of all I hope that those of my students and readers readers who wh o are or will be teaching highschoo high schooll mathematics will wil l convey some of that excitement to their their own students students I am grateful to the follo following wing reviewers for their their helpful comments comme nts:: Fred Flene, Notheaste llinois Univesity Aa Hoe Uivesity of Caifoia Ivne Kathyn Lenz, Uivesity of Minesota, Duuth David Poole, Tent Univesity Ron Soomo, Ohio State Univesity Lay Sowde, San Diego State Univesity Alex Turull Univesity of Florida, Gansville Jeane Wald Michigan State Uivesity

I Martin Isaacs

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