Geometric Transformations IV: Circular Transformations

294 Pages·2009·4.42 MB·English

by  I. M. Yaglom

Geometric Transformations IV
Circular Transformations
c 2009by
(cid:13)
TheMathematicalAssociationofAmerica(Incorporated)
LibraryofCongressCatalogCardNumber2009933072
PrinteditionISBN978-0-88385-648-2
ElectroniceditionISBN978-0-88385-958-2
PrintedintheUnitedStatesofAmerica
CurrentPrinting(lastdigit):
10987654321
Geometric Transformations IV
Circular Transformations
I. M. Yaglom
Translated by
A. Shenitzer
PublishedandDistributedby
TheMathematicalAssociationofAmerica
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Contents
1 Reflectioninacircle(inversion) 1
NotestoSection1 . . . . . . . . . . . . . . . . . . . . . . . . . 31
2 Applicationofinversionstothesolutionofconstructions 33
Problems.Constructionswithcompassalone. . . . . . . . . . . 33
Problemsinvolvingtheconstructionofcircles . . . . . . . . . . 35
NotestoSection2 . . . . . . . . . . . . . . . . . . . . . . . . . 42
3 Pencilsofcircles.Theradicalaxisoftwocircles 43
NotestoSection3 . . . . . . . . . . . . . . . . . . . . . . . . . 59
4 Inversion(concludingsection) 61
NotestoSection4 . . . . . . . . . . . . . . . . . . . . . . . . . 77
5 Axialcirculartransformations 81
A.Dilatation. . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
B.Axialinversion . . . . . . . . . . . . . . . . . . . . . . . . . 100
NotestoSection5 . . . . . . . . . . . . . . . . . . . . . . . . . 135
Supplement 143
Non-EuclideanGeometryofLobachevski˘ı-Bolyai,orHyperbolic
Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . 143
NotestoSupplement . . . . . . . . . . . . . . . . . . . . . . . 166
Solutions 171
Section1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
vii
viii Contents
Section2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
Section3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
Section4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
Section5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243
Supplement . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272
AbouttheAuthor 285
1
Reflection in a circle (inversion)
ToconstructtheimageA ofapointAbyreflectionina linel weusually
0
proceedasfollows.Wedrawtwocircleswithcentersonl passingthrough
A.TherequiredpointA isthesecondpointofintersectionofthetwocircles
0
(Figure1).WesayofA thatitissymmetrictoAwithrespecttol.
0
FIGURE 1
Here weare makinguseofthefactthatallcircleswithcentersonaline
l passing through a point A pass also through the point A symmetric to
0
A with respect to l (Figure 2). This fact can be used as a definition of a
reflectionina line:PointsAandA are saidtobesymmetric withrespect
0
to a line l if every circle withcenter on l passing through A passes also
throughA.ItisclearthatthisdefinitionisequivalenttotheoneinNML8,
0
p.41.
Inthissectionweconsiderareflectioninacircle.Thistransformationis
analogousinmanyrespectstoareflectioninalineandisoftenusefulinthe
solutionofgeometricproblems.
1