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STUDENT MATHEMATICAL LIBRARY Volume 43

Elementary Geometry Ilka Agricola Thomas Friedric h

Translated by Philip G . Spain

^ S ^ f c j

#AMS AMERICAN MATHEMATICA L SOCIET Y

Providence, Rhode Islan d

http://dx.doi.org/10.1090/stml/043

Editor ia l B o a r d

Gerald B . Follan d Bra d G . Osgoo d Robin Forma n (Chair ) Michae l Starbir d

2000 Mathematics Subject Classification. Primar y 51M04 , 51M09 , 51M15 .

Originally publishe d i n Germa n b y Friedr . Viewe g & Soh n Verlag , 65189 Wiesbaden , Germany , a s "Ilk a Agricol a un d Thoma s Friedrich :

Elementargeometrie. 1 . Auflag e (1s t edition)" . © Friedr . Viewe g & Soh n Verlag/GW V Fachverlag e GmbH , Wiesbaden ,

2005

Translated b y Phili p G . Spai n

For additiona l informatio n an d update s o n thi s book , visi t www.ams.org/bookpages /s tml-43

Library o f Congres s Cataloging-in-Publicatio n D a t a

Agricola, Ilka , 1973-[Elementargeometrie. English ] Elementary geometr y / Ilk a Agricola , Thoma s Friedrich .

p. cm . — (Studen t mathematica l librar y ; v. 43) Includes bibliographica l reference s an d index . ISBN-13 : 978-0-8218-4347- 5 (alk . paper ) ISBN-10 : 0-8218-4347- 8 (alk . paper )

1. Geometry . I . Friedrich , Thomas , 1949 - II . Title .

QA453.A37 200 7 516—dc22 200706084 4

Copying an d reprinting . Individua l reader s o f this publication , an d nonprofi t libraries actin g fo r them , ar e permitted t o mak e fai r us e of the material , suc h a s to copy a chapte r fo r use in teachin g o r research . Permissio n i s granted t o quot e brie f passages fro m thi s publicatio n i n reviews, provide d th e customary acknowledgmen t of the sourc e i s given.

Republication, systemati c copying , o r multiple reproductio n o f any materia l i n this publication i s permitted onl y unde r licens e fro m th e American Mathematica l Society . Requests fo r suc h permissio n shoul d b e addresse d t o th e Acquisition s Department , American Mathematica l Society , 20 1 Charles Street , Providence , Rhod e Islan d 02904 -2294, USA . Request s ca n also be made b y e-mail t o reprint-permission@ams.org .

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10 9 8 7 6 5 4 3 2 1 1 3 12 11 10 09 0 8

Contents

Preface t o th e Englis h Editio n v

Preface t o th e Germa n Editio n vi i

Chapter 1 . Introduction : Euclidea n spac e 1

Exercises 6

Chapter 2 . Elementar y geometrica l figures an d thei r propertie s 9

§2.1. Th e lin e 9

§2.2. Th e triangl e 1 9

§2.3. Th e circl e 4 5

§2.4. Th e coni c section s 6 3

§2.5. Surface s an d bodie s 7 7

Exercises 8 9

Chapter 3 . Symmetrie s o f th e plan e an d o f spac e 9 9

§3.1. Affin e mapping s an d centroid s 9 9

§3.2. Projection s an d thei r propertie s 10 5

§3.3. Centra l dilation s an d translation s 10 8

§3.4. Plan e isometrie s an d similarit y transform s 11 4

§3.5. Comple x descriptio n o f plan e transformation s 12 7

§3.6. Elementar y transformation s o f th e spac e S 3 13 1

IV C o n t e n t s

§3.7. Discret e subgroup s o f th e plan e transformatio n grou p 13 9

§3.8. Finit e subgroup s o f th e spatia l transformatio n grou p 15 1

Exercises 15 6

Chapter 4 . Hyperboli c geometr y 16 7

§4.1. Th e axiomati c developmen t o f elementar y geometr y 16 7

§4.2. Th e Poincar e mode l 17 4

§4.3. Th e dis c mode l 18 3

§4.4. Selecte d propertie s o f th e hyperboli c plan e 18 5

§4.5. Thre e type s o f hyperboli c isometrie s 18 9

§4.6. Fuchsia n group s 19 4

Exercises 20 4

Chapter 5 . Spherica l geometr y 20 9

§5.1. Th e spac e § 2 20 9

§5.2. Grea t circle s i n S 2 21 1

§5.3. Th e isometr y grou p o f § 2 21 5

§5.4. Th e Mobiu s grou p o f § 2 21 6

§5.5. Selecte d topic s i n spherica l geometr y 21 8

Exercises 22 6

Bibliography 22 9

List o f Symbol s 23 5

Index 23 7

Preface t o th e Englis h Edition

We than k th e America n Mathematica l Society , an d Edwar d G . Dunn e

in particular , fo r commissionin g a translatio n o f our origina l boo k an d

for hi s willingnes s t o prin t th e whol e boo k i n four-color . W e seize d

the opportunit y t o mak e infinitel y man y infinitesima l correction s tha t

have bee n observe d sinc e th e first Germa n edition , al l o f whic h ar e

not wort h listin g here . W e than k al l ou r colleague s an d student s wh o

helped t o identif y them . Th e boo k i s intended fo r bot h undergraduat e

mathematics students , t o introduc e the m t o a n advance d poin t o f vie w

on geometry , an d fo r mathematic s teachers , a s a referenc e an d sourc e

book.

This edition , lik e th e original , ha s it s ow n homepage ,

h t tp : / /www.ams.org/bookpages/s tml-43

and an y furthe r correction s t o errors , mathematica l an d typographi -

cal, wil l be poste d ther e a s they com e t o ou r attention . W e als o inten d

to presen t ther e additiona l materia l an d a collectio n o f relate d We b

links tha t w e hop e th e reade r ma y find useful . W e wil l b e happ y t o

receive an d respon d t o an y comments . I n particular , an y studen t wh o

encounters difficultie s i n solvin g th e Exercise s i s invited t o outlin e th e

problem t o u s b y e-mail .

v

VI Preface t o t h e Engl i s h Ed i t i o n

For teacher s i n school s an d universitie s w e hav e prepare d a smal l

volume wit h hint s fo r solutions , availabl e fro m u s o n request .

The Bibliograph y ha s bee n chose n t o signpos t othe r materia l whic h

may b e helpfu l t o ou r readers , an d t o whe t thei r appeti te s fo r geom -

etry an d it s ramifications . I t i s no t a lis t o f prerequisites .

We than k ou r translator , Dr . Phili p G . Spain , fo r aidin g u s i n makin g

our boo k availabl e t o a wide r public . I n it s presen t for m th e boo k

owes a lo t t o hi s expertise , an d w e ar e ver y muc h indebte d t o hi m fo r

an exceptionall y pleasan t an d intensiv e collaboration .

Berlin, Augus t 200 7

Ilka Agricol a

Thomas Friedric h

A c k n o w l e d g m e n t

The impression s o f th e ornament s o n page s 14 5 an d 16 5 ar e printe d

from Owe n Jones , Grammatik der Ornamente (unchange d reprin t

from th e Firs t Editio n o f 1856) , 1987 , b y graciou s permissio n o f

Greno Verlag , Nordlingen , Germany . Th e origina l Englis h editio n

appeared unde r th e titl e The Grammar of Ornament an d ha s ofte n

been reprinted .

Preface t o th e Germa n Edition

for Juliu s

Geometry occupie s a n extensiv e par t o f th e mathematica l syllabu s

in Germa n schools . I n middl e school , on e start s wit h propertie s o f

the elementar y plan e figures (line , triangle , circle) , elementar y trans -

formations o f th e plane , an d surface s an d bodie s i n space . I n hig h

school on e come s t o analyti c geometry , trigonometry , advance d trans -

formations, specia l curve s an d th e coni c sections . Element s o f non -

Euclidean geometr y ca n b e covere d i n furthe r optiona l courses . Al -

together w e hav e a broa d spectru m o f geometrica l theme s tha t th e

mathematics teache r ca n presen t t o hi s pupils . Durin g th e stud y fo r

the teacher' s diplom a a t universit y th e syllabu s star t s wit h lecture s

on linea r algebr a an d analyti c geometr y i n th e first year , followe d b y

lectures o n elementar y geometr y i n the secon d year . Thi s i s to presen t

these geometrica l theme s t o th e prospectiv e teache r i n a mathemat -

ically systemati c form . I f on e consider s th e universit y educatio n i n

Germany ove r a longe r tim e span , i t i s eas y t o recogniz e tha t i n th e

lectures o n linea r algebr a th e geometri c theme s ar e reduce d ste p b y

step, ofte n almos t completel y maske d out .

vii

V l l l Preface t o t h e G e r m a n Ed i t i o n

In all , on e gain s th e impressio n tha t th e cours e o n elementar y ge -

ometry, wit h it s clearl y define d contents , form s th e mai n par t o f th e

geometric educatio n fo r teacher s i n training .

This boo k aros e fro m a one-semeste r lectur e cours e o n "elementar y

geometry" fo r futur e teacher s i n thei r secon d yea r o f stud y a t th e

Humboldt-Universitat i n Berlin . Th e student s ha d alread y at tende d

the first-year course s o n linea r algebr a an d calculus ; i n th e first chap -

ter w e presen t a summar y o f som e aspect s o f thes e lectures . Ou r

t reatment o f elementar y geometr y assume s thi s fundamenta l knowl -

edge, althoug h i n a larg e par t o f th e tex t the y wil l hardl y b e needed .

Accordingly, thi s tex t i s intende d a s a companio n boo k t o suc h a

course an d seminars . Further , w e hop e tha t th e boo k wil l b e use d b y

working teacher s a s a compendiu m o f th e curriculu m fo r elementar y

geometry. Selecte d part s o f th e tex t ar e als o suitabl e fo r goo d hig h

school pupil s an d ideall y migh t b e use d a s a foundatio n fo r indepen -

dent stud y o r projects .

Chapter 2 i s devote d t o th e elementar y geometri c figures an d thei r

properties. W e begi n wit h th e incidenc e theorem s fo r line s an d the n

tu rn ou r at tentio n t o th e triangle . Afte r th e congruenc e an d similar -

ity theorems , w e appl y i n particula r th e theorem s o f Menelau s an d

Ceva i n orde r t o trea t th e intersectio n point s o f th e specia l line s i n a

triangle. Further , w e discus s th e incircle , circumcircl e an d excircl e o f

the triangle , it s area , an d it s relatio n t o th e radi i o f th e circle . W e

treat th e circl e similarly , an d discus s i n particula r th e Feuerbac h cir -

cle, an d th e Simso n an d Steine r lines . Wi t h a vie w t o th e underlyin g

hyperbolic geometr y i n Chapte r 4 w e alread y introduc e a sectio n o n

inversion i n th e circl e here . Th e coni c section s follow , wit h th e deriva -

tion o f thei r genera l equation , thei r eccentricit y an d parameters , a s

well a s th e determinatio n o f th e focu s an d directrix . Som e strikin g

properties o f th e coni c section s ar e prove d directl y i n th e text ; th e

reader wil l find som e othe r propertie s i n th e Exercise s a t th e en d o f

Chapter 2 . The n w e tur n t o surface s an d bodie s i n space . W e deriv e

the formula e fo r th e surfac e are a o f a surfac e o f revolutio n an d als o

the formul a fo r th e volum e o f a bod y o f revolution : w e prov e Euler' s

polyhedron theore m (fo r conve x polyhedra ) an d finish Chapte r 2 wit h

the classificatio n o f th e Platoni c bodies .

Preface t o th e Germa n Editio n IX

Chapter 3 deals with the symmetrie s o f Euclidean space . W e describe afEne mapping s briefly , als o th e linea r mapping s correspondin g t o them, an d th e centroi d o f a finite weighte d poin t system . Paralle l projections ont o a plane along a line and ont o a line along a plane ar e the first example s o f affine mappings . The n w e treat centra l dilation s and translation s exhaustively . Firs t w e characteriz e the m throug h a commo n geometri c propert y an d deduc e tha t togethe r the y for m a nonabelia n grou p o f transformation s o f spac e t o itself . Then , fo r the plane , we determine i n detail their compositions an d discus s as an application th e dilation centers of two circles, with whose help one can construct th e common tangent s t o two circles. Nex t follow s th e stud y of isometrie s o f th e plane . Firs t com e example s o f axi s reflections , translations an d rotations , an d agai n w e stud y thei r compositions . Fixed point s ar e important : A n isometr y o f th e plan e wit h thre e noncollinear fixed point s i s the identity . Analogousl y w e characteriz e all isometrie s wit h exactl y tw o fixed points , wit h on e fixed point , and als o th e fixed poin t fre e isometries . Th e grou p generate d b y al l isometries an d centra l dilation s consist s o f th e similarit y transform s of the plane . I n a simila r wa y w e treat th e transformation s o f three -dimensional space . Firs t w e stud y th e compositio n o f distinc t suc h mappings and then turn again to the description of the fixed point set s of spatia l isometries . Thes e fixed poin t set s yiel d a classificatio n o f the isometrie s o f £3 . Th e las t tw o sections of this chapter ar e devote d to th e stud y o f th e discret e isometr y group s o f Euclidea n space . I n the cas e of the plan e w e treat th e cycli c rotation groups , th e dihedra l group an d lattice . W e deduc e a necessar y conditio n fo r th e poin t group o f a discret e isometr y grou p o f th e plan e an d finally obtai n a classification o f al l th e group s i n question . I n th e cas e o f spac e w e restrict ourselve s t o classifyin g th e finite isometr y groups . Thes e ar e the invarianc e group s o f th e Platoni c bodie s an d o f th e symmetr y group o f a pyramid o r o f a cylinder wit h regula r polygona l base . Th e tetrahedron group , th e cub e group , an d als o the dodecahedro n grou p are describe d completely .

We begin Chapte r 4 with th e axiomatic s o f elementary geometr y an d the significance o f the paralle l axiom . W e construct hyperboli c geom -etry i n the upper hal f plane , i n which lines are Euclidean circula r arc s or hal f lines . W e treat variou s expressions fo r th e hyperboli c distanc e

X Preface t o t h e G e r m a n E d i t i o n

of tw o points . Thi s ca n a s wel l b e represente d b y a cros s rati o a s b y

a direc t formula . I n particular , th e triangl e inequalit y holds , an d th e

hyperbolic plan e i s a metri c space . W e determin e it s isometr y grou p

and deriv e th e formula e fo r th e hyperboli c lengt h o f a curv e an d fo r

the hyperboli c are a o f a region . B y mean s o f th e Cayle y transfor m

we pas s t o th e dis c mode l o f hyperboli c geometry . W e the n trea t

selected propertie s o f geometri c figures i n th e hyperboli c plane . W e

compute th e perimete r o f a circle , it s hyperboli c area , an d deriv e th e

hyperbolic Pythagora s theore m a s wel l a s othe r formula e fro m trig -

onometry. Th e formul a fo r th e are a o f a triangl e an d it s angl e defec t

is prove d completely . I n th e Exercise s th e reade r wil l find a numbe r

of result s i n hyperboli c elementar y geometr y tha t ar e analogou s t o

those o f Euclidea n geometry . Thes e concer n pair s o f hyperboli c lines ,

triangles an d thei r notabl e points , th e incircl e an d circumcircl e o f a

triangle, an d als o th e horocycle . I n a furthe r sectio n w e presen t th e

classification o f th e isometrie s int o elliptic , paraboli c an d hyperboli c

transformations bot h b y mean s o f th e Jorda n norma l for m an d als o

through thei r fixed poin t sets . W e stud y i n detai l th e questio n o f

the typ e o f th e commutato r o f tw o isometries . Th e las t sectio n o f

this chapte r i s devote d t o Fuchsia n groups . Her e w e ar e dealin g wit h

discrete subgroup s o f th e isometr y grou p o f th e hyperboli c plane . A s

well a s a serie s o f examples o f such group s w e introduce thei r limi t set s

and prov e tha t thes e set s hav e eithe r 0 , 1 , 2 o r infinitel y man y points .

Fuchsian group s wit h n o mor e tha n tw o limi t point s ar e calle d ele -

mentary. W e classif y al l elementar y Fuchsia n groups .

Spherical geometr y i s t reate d i n th e las t chapte r i n imitatio n o f hy -

perbolic geometry . W e conside r th e se t o f al l point s o f th e two -

dimensional spher e § 2 . Th e grea t circle s pla y th e rol e o f spherica l

lines an d realiz e th e shortes t distanc e betwee n tw o point s i n spher -

ical space . W e determin e th e isometr y grou p an d als o th e grou p o f

all conforma l mapping s completely . T o conclud e w e prov e th e mos t

important formula e o f spherica l trigonometr y an d stud y th e pola r tri -

angle associate d t o eac h spherica l triangle . Fro m thi s w e obtai n th e

formulae fo r th e area s o f spherica l lune s an d triangle s an d variou s

inequalities betwee n th e sid e length s an d angles .

Preface t o t h e G e r m a n Edi t io n XI

At th e en d o f eac h chapte r th e reade r wil l find a selectio n o f Exer -

cises tha t hav e regularl y bee n assigne d t o ou r auditor s a s homework .

Any studen t wh o encounter s difficultie s i n solvin g thes e Exercise s

is warml y invite d t o outlin e hi s proble m t o u s b y e-mail . W e wil l

endeavor t o help . Fo r teacher s i n school s an d universitie s w e hav e

prepared a smal l volum e wit h hint s fo r solutions , whic h i s availabl e

from u s o n request . Moreover , th e Germa n editio n o f th e boo k ha s

its ow n Interne t page ,

http:/ /www-irm.mathematik.hu-berl in.de/^agricola/elemgeo.html

One wil l find ther e a lis t o f al l know n typographica l errors , an d pdf -

files o f al l th e page s o n whic h picture s appea r tha t ar e multicolore d

in th e origina l bu t ar e printe d her e i n black-and-whit e o n ground s o f

cost. Ther e i s als o a collectio n o f www-link s o n elementar y geometry ,

though i t make s n o clai m t o completeness .

We than k th e participant s i n ou r seminar s fo r numerou s suggestion s

tha t hav e le d t o extendin g an d improvin g th e text . Dr . sc. Huber t

Gollek an d Dr . Chris t of Puhl e hav e rea d throug h th e whol e man -

uscript an d hav e indicate d necessar y correction s i n man y chapters .

Not las t w e than k Fra u Schmickler-Hirzebruc h o f Viewe g Verla g fo r

her willingnes s t o prin t som e page s o f thi s boo k i n two-tone . W e ar e

aware tha t thi s i s a rar e (i f als o muc h desired ) privilege . W e hop e tha t

this wil l no t remai n a n isolate d cas e i n th e mathematica l li teratur e

and tha t th e reade r wil l appreciat e an d enjo y thi s enrichmen t o f th e

text, whic h wa s no t t o b e take n fo r granted .

Berlin, Decembe r 200 4

Ilka Agricol a

Thomas Friedric h

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Bibliography

Bibliography fo r Chapte r 1

I. Agricola , T . Friedrich , Global Analysis, Graduat e Studie s i n Math -ematics, vol . 52 , America n Mathematica l Society , Providence , RI , 2002.

Anthony W. Knapp , Basic Real Analysis, Cornerstones , Birkhause r Boston, Inc. , Boston , MA , 2005 .

Serge Lang , Undergraduate Analysis, Springe r Verlag , Ne w York , 1997.

John McCleary , Geometry from a Differentiable Viewpoint, Cam -bridge Universit y Press , Cambridge , 1994 .

Frank Morgan , Real Analysis, America n Mathematica l Society , Prov -idence, RI , 2005 .

M. Helena Noronha , Euclidean and Non-Euclidean Geometries, Pren -tice Hall , Inc. , Ne w Jersey , 2002 .

MurrayH. Protter , CharlesB . Morrey , Jr. , A First Course in Real Analysis, 2nd Edition, Springe r Verlag , Ne w York, 1997 .

Walter Rudin , Principles of Mathematical Analysis, Third Edition, International Serie s i n Pur e an d Applie d Mathematics , McGraw-Hil l Book Co. , New York-Auckland-Diisseldorf , 1976 .

229

230 Bibliography

Bibliography fo r Chapte r 2

N. Altshiller-Court , College Geometry: An Introduction to the Mod-ern Geometry of the Triangle and the Circle (2n d edition) , Barne s & Noble, Ne w York , 1952 .

H. F. Baker , An Introduction to Plane Geometry, with Many Exam-ples, Reprin t o f 194 3 firs t edition , Chelse a Publishin g Co. , Bronx , NY, 1971.

Allan Berele , Jerr y Goldman , Geometry: Theorems and Construc-tions, Prentic e Hall , Inc. , Ne w Jersey , 2001.

George Davi d Birkhoff , Ralp h Beatley , Basic Geometry: Third Edi-tion, AM S Chelse a Publications , America n Mathematica l Society , Providence, RI , 1959 .

H. S.M. Coxeter , S.L . Greitzer , Geometry Revisited, Th e Mathemat -ical Associatio n o f America , Washington , DC , 1967 .

Robin Hartshorne , Geometry: Euclid and Beyond, Undergraduat e Texts i n Mathematics , Springe r Verlag , Ne w York , 2000 .

H. Knorrer, Geometric, Viewe g Verlag, Braunschweig and Wiesbaden , 1996.

Sebastian Montie l an d Antoni o Ros , Curves and Surfaces, Graduat e Studies i n Mathematics , vol . 69 , America n Mathematica l Society , Providence, RI , an d Rea l Socieda d Matematic a Espahola , Madrid , 2005.

S. Muller-Philipp , H.-J . Gorski , Leitfaden Geometric, Viewe g Verlag , Braunschweig an d Wiesbaden , 2004 .

H. Scheid , Elemente der Geometric, Spektru m Akademische r Verlag , Heidelberg-Berlin, 2001 .

H. Weyl, Symmetry, Princeto n Universit y Press , Princeton , NJ , 1952 .

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Bibliography fo r Chapte r 3

M. A. Armstrong, Groups and Symmetry, Springe r Verlag, Berlin-Hei -delberg-New York , 1988 .

Michael Artin , Algebra, Prentice-Hall , Englewoo d Cliffs , NJ , 1991 .

Gustave Choquet , Geometry in a Modern Setting, Herman n Publish -ers i n Art s an d Science , Paris , 1969 .

David W. Farmer , Groups and Symmetry: A Guide to Discovering Mathematics, Mathematica l World , vol . 5 , America n Mathematica l Society, Providence , RI , 1996 .

M. Klemm , Symmetrien von Ornamenten und Kristallen, Springe r Verlag, Berlin-Heidelberg-Ne w York , 1982 .

H. Scheid , Elemente der Geometric, Spektru m Akademische r Verlag , Heidelberg-Berlin, 2001 .

Bibliography fo r Chapte r 4

R. Baldus , Nichteuklidische Geometric, Sammlun g Goschen , Ban d 970 (2 . Aufl.) , Walte r d e Gruyte r & Co. , Berlin , 1944 .

A. F. Beardon , The Geometry of Discrete Groups, Springe r Verlag , Berlin-Heidelberg-New York , 1983 .

Allan Berele , Jerr y Goldman , Geometry: Theorems and Construc-tions, Prentic e Hall , Inc. , Ne w Jersey , 2001.

H. S. M. Coxeter , Non-Euclidean Geometry, 6t h Edition , Mathemati -cal Associatio n o f America , Washington , DC , 1998 .

R. Fricke , F . Klein , Vorlesungen iiber die Theorie automorpher Funk-tionen, Teil I und Teil II, Teubne r Verlag , Leipzig , 189 7 and 1912 .

D. Hilbert , Foundations of Geometry, 2n d edition , Ope n Cour t Pub -lishing Company , L a Salle , IL , 1971.

232 Bibliography

D. Hilbert , S . Cohn-Vossen , Geometry and the Imagination, AM S Chelsea Publishing , America n Mathematica l Society , Providence , RI , (reprinted) 1999 .

W. Klingenberg , Grundlagen der Geometrie, Bibliographische s Insti -tut AG , Mannheim , 1971 .

J. Lehner , Discontinous Groups and Automorphic Functions, Ameri -can Mathematica l Society , Providence , RI , 1964 .

George E. Martin, The Foundations of Geometry and the Non-Euclid-ean Plane, correcte d fourt h printing , Undergraduat e Text s i n Math -ematics, Springe r Verlag , Berlin-Heidelberg-Ne w York , 1998 .

D. Mumford , C . Series , D . Wright , Indra ps Pearls - The Vision of Felix Klein, Cambridg e Universit y Press , Ne w York , 2002 .

M. Helena Noronha , Euclidean and Non-Euclidean Geometries, Pren -tice Hall , Inc. , Ne w Jersey , 2002 .

J. G. Ratcliffe , Foundations of Hyperbolic Manifolds, Springe r Verlag , Berlin-Heidelberg-New York , 1994 .

Bibliography fo r Chapte r 5

Allan Berele , Jerr y Goldman , Geometry: Theorems and Construc-tions, Prentic e Hall , Inc. , Ne w Jersey , 2001.

D.A. Brannan , M . F. Esplen , J .J . Gray , Geometry, Cambridg e Uni -versity Press , Cambridge , 1999 .

Julian Lowel l Coolidge, A Treatise on the Circle and the Sphere, AM S Chelsea Publishing , America n Mathematica l Society , Providence , RI , 2004.

R. Fenn , Geometry, Springe r Undergraduat e Mathematic s Series , Springer Verlag , Berlin-Heidelberg-Ne w York , 2001.

Marvin Ja y Greenberg , Euclidean and Non-Euclidean Geometry: De-velopment and History, W . H . Freeman , Ne w York, 1993 .

Bibliography 233

G. A. Jennings , Modern Geometry with Applications, Springe r Uni -versitext, Springe r Verlag , Berlin-Heidelberg-Ne w York , 1994 .

M. Helena Noronha , Euclidean and Non-Euclidean Geometries, Pren -tice Hall , Inc. , Ne w Jersey , 2002 .

B. A. Rosenfeld , Die Grundbegriffe der sphdrischen Geometrie und Trigonometric, i n "Enzyklopadi e der Elementarmathematik, Ban d IV: Geometrie", VEB Deutscher Verla g der Wissenschaften, Berlin , 1969 .

Further readin g

Arkady L. Onishchik, Rol f Sulanke , Projective and Cayley-Klein Geo-metries, Springe r Monograph s i n Mathematics , Springe r Verlag , Ber -lin, 2006 .

M. Berger, Geometry I, Springe r Verlag, Berlin-Heidelberg-New York , 1994; Geometry II, Springe r Verlag , Berlin-Heidelberg-Ne w York , 1996.

F. Klein , Lectures on the Icosahedron and the Solution of the Fifth Degree, 2nd revise d edition , Dove r Publications , Ne w York , 2003 .

C. J. Scriba , P . Schreiber , 5000 Jahre Geometrie, Springe r Verlag , Berlin-Heidelberg-New York , 2003 .

Marvin Ja y Greenberg , Euclidean and Non-Euclidean Geometry: De-velopment and History, W . H . Freeman , Ne w York , 1993 .

Audim Holme , Geometry: Our Cultural Heritage, Springe r Verlag , Berlin-Heidelberg-New York , 2002 .

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List o f Symbol s

A4 - tetrahedro n group , 15 1

A5 - dodecahedro n group , 15 3

\AB\ - lengt h o f a segment , 1

AB : BC - divisio n ratio , 1 0

a,b,c - triangl e sides , 1 9

a,/3,7 - triangl e angles , 1 9

a x b - vecto r product , 5

B - grou p o f transformations , 108

CS(ri, r2 , h) - frustu m o f a cone, 79

C(Z1r) - circle , 4 5

C - extende d complex numbers, 59

Cn - cycli c group , 14 0

Conf0(S2) - Mobiu s group, 217

D2, 18 3

Dn - dihedra l group , 14 0

A(A,B,C) -triangle , 1 9

£n - affln e space , 1

F i ,F 2 -foci , 6 7

F(C) - surfac e o f revolution, 7 7

Fix(7) - fixed poin t set , 19 0

G - centroid , 10 3

[7,71] _ commutator , 19 1

H - hal f line , 11 3

MS2 - hemisphere , 22 4

H2 - hyperboli c plane , 17 4

H2 - extende d hyperbolic plane, 189

fyft,fc) ~ central dilation , 10 8

235

d(A, B) - distanc e between points, 1

236 List o f Symbol s

X - grou p o f isometries , 11 4

X+ - positiv e isometries , 12 3

X~ - negativ e isometries , 12 3

XQ - isometrie s with fixed point, 116

XQ - positiv e isometrie s wit h fixed point , 12 0

XQ - negativ e isometrie s wit h fixed point , 12 0

K(C) - bod y o f revolution , 7 7

L(A, B) - lin e through A and B, 6

L(C) - curv e length , 3

L(f) - vectoria l mapping , 10 1

A(r) - limi t set , 19 6

m(A) - area , 4

^A - poin t masses , 10 1

Afn - pol e o f grea t circle , 21 2

0(V(£n)) - orthogona l group , 115

Q ~ center o f dilation , 10 8

Pc - centroid , 3 8

Pcc - circumcenter , 3 8

Pic - incenter , 3 5

Poc - orthocenter , 3 8

7r, II - plane , 63 , 105

II(r) - paraboli c points , 19 6

PSL(2,C), 6 0

PSL(2,R), 17 7

PSL(2,Z), 19 4

R - extende d rea l line , 17 6

Rn - Euclidea n space , 1 r(Q,fc) ~ rotation, 11 8

S - similarit y transformatio n group, 12 4

<S+ - positiv e similarities , 12 4

S~ - negativ e similarities , 12 4

£4 - octahedro n group , 15 2

8L - reflectio n i n line , 11 7

sn - reflectio n i n plane , 13 2

Sc - circl e reflection , 5 9

SL(2,C), 6 0

SL(2,R), 17 7

§2 - spherica l space , 20 9

T - translatio n group , 11 5

t$ - translation , 11 0

vol(A) - volume , 4

V(Sn) - Euclidea n vector space, 1

ipA - exterio r angl e in triangle , 19

(zi : Z2 ; Z3 : Z4) - cros s ratio , 62

Index

affine mapping , 9 9 alternate angle , 2 2 altitude, 19 , 3 3

common, 20 5 in hyperboli c triangle , 20 6 theorem, 2 5

angle central, 5 1 inscribed, 5 1 peripheral, 5 1

angle bisectio n theore m fo r ellipse , 71

angle bisector s in Euclidea n triangle , 3 5 in hyperboli c triangle , 20 6

angle cosin e theorem , 22 0 angle su m

in Euclidea n triangle , 2 1 in hyperboli c triangle , 18 8 in spherica l triangle , 22 4

antipodal points , 21 1 antiprism, 8 7 Archimedean bodies , 8 7 area

Euclidean triangle , 4 0 hyperbolic, 18 2 hyperbolic triangle , 18 8

ASA theorem , 2 6 asymptotes o f th e hyperbola , 7 2 asymptotic triangle , 18 7

autopolar triangle , 22 6 axis reflection , 11 7

barycenter, 10 3 Beltrami, E . (1835-1899) , 17 3 Berger, Marce l (1927 - ) , 8 6 betweenness relation , 16 8 body o f revolution , 7 7

volume of , 7 8 Bolyai, J . (1802-1860) , 17 2 boundary circle , 18 7 boundary paralle l lines , 20 4

Caspar, Donal d L . D. (1927 - ) , 16 4 Cassini curve , 7 5 cathetus, 2 4

theorem, 2 4 Cavalieri's principle , 4 , 79 , 81 , 82 Cay ley transform , 18 3 central angle , 5 1 central dilation , 10 8 centroid, 34 , 38 , 91 , 101 , 10 4

in hyperboli c triangle , 20 6 weighted, 102 , 11 5

Ceva's theorem , 30 , 32 , 9 2 chord o f a circle , 4 6 chord theorem , 4 8 chordal quadrilateral , 5 6 circle, 4 5

chord theorem , 4 8

237

238 Index

Feuerbach, 40 , 5 3 generalized, 57 , 17 5 hyperbolic, 18 5 inversion in , 5 8 reflection in , 5 8 secant theorem , 4 7 spherical, 21 8 tangent theorem , 4 6

circular cone , 7 8 circumcenter

of Euclidea n triangle , 3 6 circumcircle

of Euclidea n triangle , 36 , 4 3 of hyperboli c triangle , 20 6

collinear points , 6 commutator, 19 1 complex numbers , 5 cone, 8 1

volume, 8 1 conformal

group, 21 7 mapping, 60 , 11 4

congruence theorems , 26 , 22 6 congruent sets , 3 conic section , 6 3

directrix, 6 7 eccentricity, 6 5 focus, 6 7 parameter, 6 5 polar equation , 6 5

conjugate matrices , 18 9 coordinates

polar, 6 3 spherical, 21 0

cosine la w Euclidean, 2 0 hyperbolic, 20 5 spherical, 22 0

cotangent formula , 22 1 cross ratio , 61 , 17 6 cube (hexahedron) , 86 , 15 2 cube group , 15 6 cuboctahedron, 88 , 9 7

large, 8 8 cyclic group , 140 , 15 6 cylinder, 8 2

volume of , 8 2

Delambre's equations , 22 6 Desargues' theorem , 1 8 diametrically opposit e points , 21 1 dihedral group , 140 , 15 6 dilation cente r (o f tw o circles) , 11 2 dilation factor , 10 8 direction vector , 6 directrix, 6 7 disc model , 18 3 distance

Euclidean, 1 hyperbolic, 17 6 in geometri c plane , 17 0 spherical, 21 4

divergent lines , 20 5 division ratio , 1 0 dodecahedron, 86 , 15 3

group, 153 , 15 6 truncated, 8 7

eccentricity, 6 5 Einstein, Alber t (1879-1955) , 17 3 ellipse, 6 6

angle bisectio n theorem , 7 1 length, 7 1 surface, 7 1

ellipsoid, 7 8 elliptic

isometry group , 21 5 line, 21 1 space, 20 9 transformation, 19 0

Euclidean cosine law , 2 0 distance, 1 isometry group , 2 length, 3 line, 6 plane, 5 sine law , 4 0 space, 1 triangle, 1 9

angle bisectors , 3 5 area, 4 0 circumcenter, 3 6 circumcircle, 4 3 excircle, 4 4 incenter, 3 6

Index 239

incircle, 36 , 4 3 Mollweide's equations , 9 1 Napier's equations , 9 1 orthocenter, 3 3 perpendicular bisectors , 3 6 side bisectors , 3 4

Euler line , 3 8 Euler's polyhedro n formula , 8 2 exact sequence , 11 6 excircle o f Euclidea n triangle , 4 4 extended

complex plane , 17 6 complex plane , 59 , 21 6 real line , 17 6

exterior angle , 2 0

Fermat problem , 2 4 Feuerbach circle , 40 , 5 3 first Steine r line , 5 5 five elemen t formula , 22 0 fixed poin t

of hyperboli c isometry , 19 0 focus, 6 7 focus-directrix pair , 6 7 football, 8 7 fractional linea r transformation , 6 0 frieze, 14 4

group, 146 , 16 4 Fuchsian group , 19 4

elementary, 20 1 first kind , 20 4 nonelementary, 20 1 second kind , 20 4

Fuller, Buckminste r (1895-1983) , 95

function (fractiona l linear) , 6 0

Gauss, C.F . (1777-1855) , 17 2 generalized circle , 57 , 17 5 geometric plane , 16 8 glide reflection , 11 8 golden

ratio, 5 0 rectangle, 89 , 15 4 section, 49 , 89 , 15 4

great circle , 21 1 pole, 21 2

group

cyclic, 140 , 15 6 dihedral, 140 , 15 6 dodecahedron, 153 , 15 6 Fuchsian, 19 4

elementary, 20 1 first kind , 20 4 nonelementary, 20 1 second kind , 20 4

octahedron, 152 , 15 6 orthogonal, 11 5 tetrahedron, 151 , 15 6

Haeckel, Erns t (1834-1919) , 8 6 half space , 8 2 hemisphere, 21 2 Heron's formula , 4 1 hexagonal lattice , 14 6 hexahedron (cube) , 86 , 15 2

snub, 8 8 truncated, 8 7

Hippocrates, lune s of , 9 0 homothety, 10 8 horocycle, 20 6 hyperbola, 6 6

asymptotes, 7 2 hyperbolic

area, 18 2 boundary circle , 18 7 circle, 18 5 cosine law , 20 5 distance, 17 6 isometry group , 18 0 length, 18 1 line, 17 5 lines

boundary parallel , 20 4 divergent, 20 5

plane, 17 4 disc model , 18 3 Poincare model , 17 4

sine law , 20 6 transformation, 19 0 triangle

altitude, 20 6 angle bisectors , 20 6 area, 18 8 asymptotic, 18 7 circumcircle, 20 6

240 Index

incircle, 20 6 orthocenter, 20 6 perpendicular bisector , 20 6 side bisector , 20 6

hyperboloid one-sheeted, 7 8 two-sheeted, 7 8

icosadeltahedron, 95 , 16 3 icosahedron, 86 , 15 4

truncated, 8 7 icosidodecahedron, 8 8

large, 8 8 incenter, 35 , 3 6 incidence theorem , 9

converse, 1 3 in space , 15 , 10 7 oriented, 11 , 31

incircle of Euclidea n triangle , 36 , 4 3 of hyperboli c triangle , 20 6

inscribed angle , 5 1 inversion i n circle , 5 8 isobarycenter, 10 3 isoceles triangle , 2 0 isometry, 1 , 114 , 17 1

group of ellipti c space , 21 5 of Euclidea n space , 2 of hyperboli c plane , 18 0

negatively/positively oriented , 123

isoperimetric problem , 4 2

Jordan measure , 4 Jordan norma l form , 19 0

Klein, Feli x (1849-1925) , 144 , 17 3 Klein fou r group , 14 4 Klug, Aaro n (1926 - ) , 16 4

Lambert projection , 21 0 lattice, 14 6 leg, 2 4 lemniscate, 7 6 length

Euclidean, 3 hyperbolic, 18 1 spherical, 21 9

lever law , 10 1 l'Huilier's equation , 22 7 limit circle , 20 6 limit set , 19 6 line

elliptic, 21 1 Euclidean, 6 Euler, 3 8 first Steiner , 5 5 hyperbolic, 17 5 second Steiner , 55 , 126 , 15 9 Simson, 54 , 126 , 15 9

lines hyperbolic

boundary parallel , 20 4 divergent, 20 5

parallel, 6 , 16 9 skew, 6

Lobachevsky, N.I . (1793-1856) , 17 2 lune, 21 2

of Hippocrates , 9 0

Mobius group , 21 7 mapping

affine, 9 9 angle-preserving ( = conformal) ,

60, 11 4 orientation-preserving/reversing,

123 vectorial, 10 1

mass, 10 2 median theorem , 34 , 10 4 Menelaus' theorem , 3 0 midtriangle, 3 8 modular group , 19 4 Mollweide's equations , 9 1 Morley's theorem , 9 1

Nagel point , 9 2 Napier's equation s

for Euclidea n triangle , 9 1 for spherica l triangle , 22 7

Napoleon's theorem , 13 0 nine poin t circle , 4 0

octahedron, 86 , 15 2 group, 152 , 15 6 truncated, 8 7

octant, 21 9

Index 241

one-sheeted hyperboloid , 7 8 order axioms , 16 8 orientation-preserving/reversing

mapping, 12 3 oriented

angle, 11 7 incidence theorem , 11 , 31

ornament group , 14 6 orthocenter, 9 1

in Euclidea n triangle , 3 3 in hyperboli c triangle , 20 6

orthogonal group , 11 5 orthogonal projection , 10 5

Pappus' theorem , 1 3 parabola, 6 6 parabolic

mirror, 9 4 point, 19 6 transformation, 19 0

parallel axiom , 17 2 parallelepiped, 5 parameter o f coni c section , 6 5 Pasch's Axiom , 16 9 pentagon, 50 , 16 1 perfect set , 20 4 peripheral angle , 5 1 perpendicular bisecto r

in Euclidea n triangle , 3 6 in hyperboli c triangle , 20 6 planes, 13 2

picornavirus, 16 3 plane

Euclidean, 5 geometric, 16 8 hyperbolic, 17 4

disc model , 18 3 Poincare model , 17 4

reflection, 13 2 Platonic body , 8 5 Poincare, Henr i (1854-1912) , 17 3 Poincare model , 17 4 point

group, 14 4 projection, 10 5 reflection, 108 , 13 6 space, 1 symmetry group , 14 4

polar, 21 2 coordinates, 63 , 18 5 equation o f coni c section , 6 5 triangle, 22 2

pole, 21 2 polyhedron, 8 2

dual, 9 6 polytope, 8 2

regular, quasiregular , 8 5 primitive pythagorea n triple , 2 3 principal congruenc e group , 19 4 prism, 8 7 problem

Fermat, 2 4 isoperimetric, 4 2

projection, 10 5 Lambert, 21 0 stereographic, 21 0

Ptolemy's theorem , 6 2 Pythagoras' theore m

Euclidean, 2 2 hyperbolic, 18 6 spherical, 22 1

Pythagoras triple , 2 3

quadratic lattice , 14 6

Radiolaria, 8 6 rectangular fac e centere d lattice ,

146 rectangular lattice , 14 6 reflection

glide, 11 8 in circle , 5 8 in grea t circle , 21 6 in hyperboli c line , 18 0 in plane , 13 2 in point , 108 , 136 , 21 6

relation (betweenness) , 16 8 revolution

ellipsoid of , 7 8 hyperboloid of , 7 8

rhombic lattice , 14 6 Riemann, Bernhar d (1826-1866) ,

173 rotation, 13 5

half, 13 3 rotation-dilation, 12 4

242 Index

SAS theorem , 2 6 secant o f circle , 4 5 secant theorem , 4 7 second Steine r line , 55 , 126 , 15 9 segment (i n absolut e geometry) ,

169 semiperimeter, 4 1 semiregular polyhedron , 8 7 sets

congruent, 3 , 17 1 Jordan measurable , 4 perfect, 20 4 similar, 3

side bisecto r in Euclidea n triangle , 34 , 10 4 in hyperboli c triangle , 20 6

side cosin e law , 22 0 similar sets , 3 similarity transform , 12 4 Simson line , 5 4 sine law , 4 0

hyperbolic, 20 6 spherical, 22 0

skew lattice , 14 6 skew reflection , 11 4 snub

hexahedron, 8 8 space, elliptic , 20 9 spherical

angle cosin e theorem , 22 0 circle, 21 8 coordinates, 21 0 cosine law , 22 0 cotangent formula , 22 1 distance, 21 4 excess, 22 6 five elemen t formula , 22 0 lune, 21 2 octant, 21 9 polar triangle , 22 2 sine law , 22 0 triangle, 22 0

area, 22 4 autopolar, 22 6 Delambre's equations , 22 6 l'Huilier's equation , 22 7 Napier's equations , 22 7

trigonometry, 22 0

star polygon , 9 6 Steiner, Jaco b (1796-1863) , 12 6 Steiner lin e

first, 5 5 second, 55 , 126 , 15 9

stereographic projection , 21 0 supplementary angle , 21 3 surface o f revolution , 7 7

surface are a of , 8 0 Sylvester an d Galla i theorem , 9 0 symmetry point , 20 5

tangent quadrilateral, 5 6 theorem, 4 6 to circle , 4 5 to tw o circles , 11 3

tetrahedron, 86 , 104 , 151 , 15 7 group, 151 , 15 6 truncated, 8 7

Thales' theorem , 15 , 50 , 10 7 theorem

alternate angle , 2 2 angle bisectors , 3 5 angle cosine , 22 0 central angle , 5 1 Ceva, 30 , 32 , 9 2 congruence, 22 6 Desargues, 1 8 ellipse

angle bisection , 7 1 exterior angle , 2 0 incidence, 9

in space , 15 , 10 7 median, 34 , 10 4 Menelaus, 3 0 Morley, 9 1 Napoleon, 13 0 oriented incidence , 1 1 Pappus, 1 3 Ptolemy, 6 2 Pythagoras

Euclidean, 2 2 hyperbolic, 18 6 spherical, 22 1

side cosine , 22 0 spherical

angle cosine , 22 0

Index 243

side cosine , 22 0 Sylvester an d Gallai , 9 0 Thales, 15 , 5 0

tiling, 14 4 torus, 7 8 transformation, 10 8

complex, 12 8 elliptic, 19 0 hyperbolic, 19 0 parabolic, 19 0

translation, 1 , 10 8 domain, 14 4 group,115 subgroup, 14 4

transversal, 3 0 vertex, 3 2

triangle asymptotic, 18 7 congruence theorems , 2 6 Euclidean, 1 9

angle bisectors , 3 5 angle sum , 2 1 area, 4 0 centroid, 3 8 circumcenter, 3 6 circumcircle, 4 3 excircle, 4 4 incenter, 35 , 3 6 incircle, 36 , 4 3 Mollweide's equations , 9 1 Napier's equations , 9 1 orthocenter, 3 3 perpendicular bisectors , 3 6 side bisectors , 3 4

hyperbolic angle bisectors , 20 6 area, 18 8 circumcircle, 20 6 incircle, 20 6 orthocenter, 20 6 perpendicular bisector , 20 6 side bisector , 20 6

isoceles, 2 0 leg, 2 0 right-angled

altitude theorem , 2 5 cathetus theorem , 2 4 Pythagoras' theorem , 2 2

semiperimeter, 4 1 similarity theorem , 3 0 spherical, 22 0

area, 22 4 autopolar, 22 6 Delambre's equations , 22 6 excess, 22 6 l'Huilier's equation , 22 7 Napier's equations , 22 7 polar, 22 2 surface, 22 0

transversal, 3 0 trigonometry (spherical) , 22 0 truncated

dodecahedron, 8 7 hexahedron, 8 7 icosahedron, 8 7 octahedron, 8 7 tetrahedron, 8 7

two-sheeted hyperboloid , 7 8

vector mapping, 10 1 product, 5 projection, 10 5 triple product , 5

vectorial isometry, 11 5 mapping, 10 1

vertex angle, 21 3 transversal, 3 2

weighted centroid , 102 , 11 5 Weyl, Herman n (1885-1955) , 8 6 Wiles, Andre w (1953 - ) , 2 4