(Gceooniceltir y off Coolie s

This page intentionally left blank

N\/if A\TiunFiM[A\Tir<rA W A\A\/opi r in <> w/oi f if m/ifIF ixa

(Gceeiraiicettr y e f (Cerate s A.VAlkopya e A.A.Zaslavslk y

>AMS AMERICAN MATHEMATICA L SOCIETY

I www.ams.or g I

http://dx.doi.org/10.1090/mawrld/026

TEOMETPHHECKHE CBOHCTBA KPHBblX BTOPOrO nOPHUKA A. B . AKOIIHH , A . A . 3acjiaBCKH H

MIIHMO, MOCKBA , 200 7

This wor k wa s originall y publishe d i n Russia n b y Mosco w Cente r fo r Continuou s

Mathematical Educatio n unde r th e titl e 'TeoMeTpiraecKH e CBOHCTB a KPHBBI X BTopor o

nopn^Ka" © 2007 . Th e presen t translatio n wa s create d unde r licens e fo r th e America n

Mathematical Societ y an d i s published b y permission .

Translated b y Ale x Martsinkovsk y

Cover ar t create d b y Iva n Velichk o

2000 Mathematics Subject Classification. Primar y 51-02 , 51M04 .

For additiona l informatio n an d update s o n thi s book , visi t

www.ams.org/bookpages/mawrld-26

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

Akopyan, A . V. (Arsen y V.) , 1984-Geometry o f conies / A . V. Akopyan, A . A. Zaslavsky .

p. cm . — (Mathematica l worl d ; v. 26) Includes bibliographica l reference s an d index. ISBN 978-08218-4323- 9 (alk . paper) 1. Geometry , Analytic—Plane . 2 . Curves, Algebraic . I . Zaslavskii, A . A. (Alexey A.) , 1960 -

II. Title .

QA552.A46 200 7 516.2'152—dc22 200706084 1

Copying an d reprinting . Individua l reader s o f thi s publication , an d nonprofi t librarie s acting fo r them, ar e permitted t o make 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 grante d t o quot e brie f passage s fro m thi s publicatio n i n reviews, provide d th e customary acknowledgmen t o f the source i s given.

Republication, systemati c copying , or multiple reproductio n o f any materia l in this publicatio n is permitte d onl y unde r licens e fro m th e America n Mathematica l Society . Request s fo r suc h permission shoul d b e addressed t o the Acquisitions Department , America n Mathematica l Society , 201 Charle s Street , Providence , Rhod e Islan d 02904-2294 , USA . Requests ca n also b e mad e b y e-mail t o reprint-permission@ams.org .

© 200 7 by the American Mathematica l Society . Al l rights reserved . The America n Mathematica l Societ y retain s al l right s

except thos e grante d t o the United State s Government . Printed i n the United State s o f America .

@ Th e paper use d i n this boo k i s acid-free an d falls withi n th e guideline s established t o ensure permanenc e an d durability .

Visit th e AMS hom e pag e a t ht tp: / /www.ams.org /

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

Contents

Preface vn

Chapter 1 . Elementar y Propertie s o f Curve s o f Secon d Degre e 1 1.1. Definition s 1 1.2. Analyti c definitio n an d classificatio n o f curve s

of second degre e 5 1.3. Th e optica l propert y 6 1.4. Th e isogona l propert y o f conies 1 0 1.5. Curve s o f second degre e a s projections o f the circl e 1 5 1.6. Th e eccentricit y an d ye t anothe r definitio n o f conies 1 7 1.7. Som e remarkable propertie s o f the parabol a 1 9

Chapter 2 . Som e Result s fro m Classica l Geometr y 2 7 2.1. Inversio n an d Feuerbach' s theore m 2 7 2.2. Basi c fact s abou t projectiv e transformation s 2 9 2.3. Som e fact s fro m th e geometr y o f the triangl e 3 8 2.4. Radica l axe s an d pencil s o f circles 5 7

Chapter 3 . Projectiv e Propertie s o f Conie s 6 5 3.1. Th e cross-rati o o f four point s on a curve. Parametrization . Th e

converses o f Pascal' s an d Brianchon' s theorem s 6 5 3.2. Pola r correspondence . Th e dualit y principl e 6 8 3.3. Pencil s o f curves . Poncelet' s theore m 7 8

Chapter 4 . Euclidea n Propertie s o f Curve s o f Secon d Degre e 9 9 4.1. Specia l propertie s o f equilatera l hyperbola s 9 9 4.2. Inscribe d conie s 10 5 4.3. Normal s t o conies . Joachimstahl' s circl e 11 3 4.4. Th e Poncele t theore m fo r confoca l ellipse s 11 5

vi CONTENT S

Chapter 5 . Solution s t o th e Problem s 11 9

Bibliography 13 1

Index 13 3

Preface

Curves o f secon d degree , o r conies , ar e traditionall y viewe d a s object s per -taining t o analyti c geometr y an d ar e studie d i n lower-leve l course s i n engi -neering colleges. A t best , only the optical properties of conies are mentione d among thei r geometri c properties . Bu t thos e curve s als o posses s a numbe r of othe r nic e properties , a majorit y o f whic h ca n b e establishe d b y meth -ods o f elementar y geometr y wel l withi n th e reac h o f hig h schoo l students . Moreover, conie s hel p solv e som e geometri c problem s seemingl y unrelate d to conies . I n thi s boo k th e reade r wil l find the mos t interestin g fact s abou t curves o f order two , includin g thos e prove d recently .

Chapter 1 deals wit h th e elementar y propertie s o f conies . Mos t o f th e facts mentione d there are well known. Th e remaining materia l i s also rathe r simple, s o tha t th e entir e chapte r doe s no t impos e an y prerequisite s o n the reade r beyon d th e standar d hig h schoo l curriculum . Som e simpl e bu t important result s ar e offere d a s exercises . W e recommen d tha t th e reade r try to solve them before reaching for the solutions . Thi s should facilitate th e understanding o f the materia l late r on . Chapte r 2 is of an auxiliar y nature . It contain s som e fact s fro m classica l geometr y neede d fo r understandin g the remainin g chapters , whic h ar e no t usuall y studie d i n hig h school . I n Chapter 3 w e mentio n projectiv e propertie s commo n t o al l conies . Som e of them , suc h a s th e theore m o n pencil s o f conies , ar e quit e complicated . Finally, Chapte r 4 is devoted t o metri c properties . A s a rule , they concer n only specia l kind s o f conies . Thi s i s th e mos t complicate d chapte r o f th e book, whic h require s a goo d understandin g o f th e materia l i n th e previou s chapters.

The author s ar e gratefu l t o I . I . Bogdano v an d E . Yu . Bun'kov a fo r valuable comments .

This page intentionally left blank

This page intentionally left blank

Bibliography

[1] P . S . Aleksandrov , Lectures on analytic geometry. Nauka , Moscow , 1968 . (Russian ) [2] M . Berger , Geometry. I . Springer-Verlag , Berlin , 1994 . [3] , Geometry. II . Springer-Verlag , Berlin , 1987 . [4] H . S . M . Coxeter , The real projective plane. Springer-Verlag , Ne w York , 1993 . [5] H . S . M . Coxete r an d S . L . Greitzer , Geometry revisited. MAA , Ne w York , 1967 . [6] J.-P . Ehrmann an d F . van Lamoen, A projective generalization of the Droz-Farny line

theorem. Foru m Geom . 4 (2004) , 225-22 7 (electronic) . [7] L . A . Emelyano v an d T . L . Emelyanova , Feuerbach's family. Matematichesko e

prosveshchenie, Serie s 3 , 6 (2004) , 78-92 . (Russian ) [8] D . Hilber t an d S . Cohn-Vossen , Geometry and the imagination. Chelse a Publishin g

Company, Ne w York , Amer . Math . Soc , Providence , RI , 1952 . [9] J . Lemaire , L'hyperbole equilatere. Vuibert , Paris .

[10] I . M . Yaglom , Geometric transformations. Rando m House , Ne w York , 1962 . [11] , Geometric transformations. II . Rando m House , Ne w York , 1968 . [12] A . A . Zaslavsky , Geometric transformations. MCCME , Moscow , 2003 . (Russian )

131

This page intentionally left blank

Index

isotomic conjugation , 8 8

asymptotes o f a hyperbola , 2 axis

of a hyperbol a imaginary, 2 real, 2

of a parabola , 2 of a n ellips e

major, 1 minor, 1

radical, 5 7

Brocard angle , 4 9

circle Apollonius, 4 1 Ceva, 3 9 Euler, 2 8 Fermat-Apollonius, 1 3 Joachimstahl's, 11 5 nine-point, 2 8 pedal, 3 8

coaxial circles , 5 8 complete quadrilateral , 10 9 conic, 2 , 1 7 conic section , 1 7 correspondence

polar, 6 8 cross-ratio, 31 , 32 curve o f secon d degree , 5 curve o f secon d degree , 2

Dandelin spheres , 1 7 directrix, 1 8 duality principle , 3 6

eccentricity, 1 8

ellipse, 1 Brocard, 4 8 Steiner

circumscribed, 53 , 10 7 inscribed, 5 2

hyperbola, 2 Kiepert, 10 4 Apollonius, 11 4 equilateral, 2 , 9 9 Feuerbach, 10 3

inversion, 2 7 involution, 9 3 isogonal conjugate , 1 2 isogonal conjugation , 41 , 87 isotomic conjugation . 5 6

lemma Simson's, 2 2

line Aubert, 10 9 Gauss, 88 , 10 9 Simson's, 2 3

normal, 11 3

parabola, 2 , 1 9 pencil, 7 8

of circles , 5 8 elliptic, 5 8 hyperbolic, 5 8 parabolic, 5 8

perspector, 10 5 point

Apollonius, 5 3 Brocard, 4 8 Feuerbach, 2 9

133

134 INDEX

Gergonne, 5 5 isogonal conjugate , 4 1 Lemoine, 4 6 limit, o f a penci l o f circles , 5 9 Miquel, 10 9 Nagel, 5 5 Torricelli, 5 4

polar, 35 , 68 trilinear, 3 6

polar correspondence , 3 5 polar curve , 7 0 pole, 35 , 68

trilinear, 3 6 power o f a point , 5 7

radical center , 5 8

symmedian, 4 6

theorem Brianchon's, 33 , 63

converse, 6 6 Desargues', 3 2

Droz-Farny, 11 1 Emelyanov an d Emelyanova , 10 3 Feuerbach's, 2 8 four conies , 8 5

dual to , 8 7 Fregier, 76 , 96 Monge's, 9 0 Newton's, 3 6 on pencil s o f conies , 7 8 Pappus', 3 2 Pascal's, 33 , 45

converse, 6 5 Poncelet, 61 , 67, 93 , 115 , 12 4 Sondat's, 12 5 three conies , 8 6

dual to , 8 7 triangle

Ceva, 3 9 circumcevian, 3 9 pedal, 3 8 self-polar, 7 4

Titles i n Thi s Serie s

26 A . V . Akopya n an d A . A . Zaslavsky , Geometr y o f conies , 200 8

25 A n n e L . Young , Mathematica l Ciphers : Pro m Caesa r t o RSA , 200 6

24 Burkar d Polster , Th e shoelac e book : A mathematica l guid e t o th e bes t (an d worst ) way s

to lac e you r shoes , 200 6

23 Koj i Shig a an d Toshikaz u Sunada , A mathematica l gift , III : Th e interpla y betwee n

topology, functions , geometry , an d algebra , 200 5

22 Jonatha n K . Hodg e an d Richar d E . Kl ima , Th e mathematic s o f votin g an d elections :

A hands-o n approach , 200 5

21 Gille s Godefroy , Th e adventur e o f numbers , 200 4

20 Kenj i U e n o , Koj i Shiga , an d Shigeyuk i Morita , A mathematica l gift , II : Th e

interplay betwee n topology , functions , geometry , an d algebra , 200 4

19 Kenj i U e n o , Koj i Shiga , an d Shigeyuk i Morita , A mathematica l gift , I : Th e interpla y

between topology , functions , geometry , an d algebra , 200 3

18 T imoth y G . Feeman , Portrait s o f th e Earth : A mathematicia n look s a t maps , 200 1

17 Serg e Tabachnikov , Editor , Kvan t Selecta : Combinatorics , I , 200 1

16 V . V . Prasolov , Essay s o n numbe r an d figures, 200 0

15 Serg e Tabachnikov , Editor , Kvan t Selecta : Algebr a an d analysis . II , 199 9

14 Serg e Tabachnikov , Editor , Kvan t Selecta : Algebr a an d analysis . I , 199 9

13 Sau l Stahl , A gentl e introductio n t o gam e theory , 199 9

12 V . S . Varadarajan , Algebr a i n ancien t an d moder n times , 199 8

11 Kunihik o Kodaira , Editor , Basi c analysis : Japanes e grad e 11 , 199 6

10 Kunihik o Kodaira , Editor , Algebr a an d geometry : Japanes e grad e 11 , 199 6

9 Kunihik o Kodaira , Editor , Mathematic s 2 : Japanes e grad e 11 , 199 7

8 Kunihik o Kodaira , Editor , Mathematic s 1 : Japanes e grad e 10 , 199 6

7 D m i t r y Fomin , Serge y Genkin , an d Ili a Itenberg , Mathematica l circle s (Russia n

experience), 199 6

6 Davi d W . Farme r an d Theodor e B . Stanford , Knot s an d surfaces : A guid e t o

discovering mathematics , 199 6

5 Davi d W . Farmer , Group s an d symmetry : A guid e t o discoverin g mathematics , 199 6

4 V . V . Prasolov , Intuitiv e topology , 199 5

3 L . E . Sadovski i an d A . L . Sadovskii , Mathematic s an d sports , 199 3

2 Yu . A . Shashkin , Fixe d points , 199 1

1 V . M . Tikhomirov , Storie s abou t maxim a an d minima , 199 0