EuclideanGeometry
Thc subjectrnatrer of thrs boot rs non-Eucldcr onem-usl know wna, Euc,idean Beonre,r), . 1i,,.;liJlil.lJ,llil;li::;i::,T::.: mofe.dimculrtask th.in it ,nirhr lpp.af rniria,)"i].;llil'Jll r,, r:,",.;, ."1 .*r,'",r,,"i"t";: ;;,;;;":;;;:,., producethe vaflou! cunEnlr] ," accepreddesc prions.sirtch areari basedon rhcrerarnerr ofDavid Hitberr recenrwort o86: 1e.'.r).Nee,ess ro s;). fu*. g_.,.",r,,,. ;;; ;;;,;;;:;;;1";'i::,,"", to bedeficient and op r for yer orherchar.acter.rzatrons jde.ar ,e.,r l| new oI rhesccons ions,the rcaders ura\ no,be \umri .ed , , nd,,r,.Ju,r.. n., .hJ,en,,,e\.,r_ . ::l',.."".,' ".,..,";'',_.:; rroridi,rg^l^ rheor s'i,h:r shofthisror).orrhesubjec, ,".,",0"ou",.i.ii;:i',",Jft ili:.ll''''t, -o - Ill Anlntroduction to EuclideanGeomerrv Geomctrfin rhesense ol mensurrrionof ti!ui de\eroped bI malr)cuhurer and datesback to mrr"""i,'"i ii"',i;",ii " T ip"lll"**'rl *,"''r,,"*l ;:;;;; ;i""iil:;:;"11.:;:nil j;J,i,iJT, Ii,i.J,li.: :i:I,l.rreason,was-creared by rheG.eeks. Hrsrrlrrlrns ::ii"Jll: asr.erhar rh€ orisn of gcomerr)."" t". ,.,"J i".* ,,, tlrehres of Thatesol Milerus Ncirher or.r,i. r*.;;;-;::;#;T,:i:::il'ii:i,lirhe.\ i':filiijlli:'J:Jjli,J,il: eclips€-*npr,,r,,""u,,of 585Bc. jr is believcdthar hc ln.eddunnr rhc;jrrh cenru,-;it ;,i,,ili;."il;, probablyt'om forrotrel in m.uv orher ".." 'Lrd counrries.rnttcenrurres 1, r, fr ,,,,rr. ,t,"i r," ,- ro a:uti.e |harborh ioundhis conr : burior\ useluta|d cared . p,",o.]. ,r,.",-i,,"". ,n" lutureby ilcorTrorarngthem ".,g1 rnroi(s cd!carionorsj.sren s;me rrrccreete orihi o"r *r-,'..^-e boththe denrocraLjctbrm of governmenr and$e idcaof r trratht r,I .r p."., ,, to..e\e.) ciLizenrhar he " slr(jutdbe abte ro argn.borl cogurr4 ura perurLsiverr.or,;"" "",;,,p*"r,. srxtr cenrury ,r," ,ii" ,. Bc,rtre rask of edu.aunsrhcif r.e'o{ uar.n". ,u rt ,ii',' s),slen ."qo,.;,t il,i ;;;; ;;t,t."l u'a\ assmed by rheSotirjsrs. t.hel $.er€irrneranl pedagogucs " rrarnng f,n.rn*,ry theirpupls mos t rnrhetorjc. Since rheJ i,,, r r,,nr, tl. ,1,,,,,.u,,n"lro -ra.,rc, of ; i:,;,.",;;;:,r,., Seometry,robeuseful rool\inlhis rairnng. tnrlc, L,erLrontofndanr.hcrnrh.ircLrni.utur,uno"";l was.eser red for psteriry js It let lillelt,rhat DemocrirLr\ (ca 4t0 p.om,rgar"an,to.i" tn"",, ;,;.,,;;,;;,;i E ;;il:,:,::1:,,illlli,ll':"Ji:fiil::[t:TJ; schoots.rr is n,orethan liketl. hou ever " thathrirlen,"'*,.";, *" ,,lii 1,;l;..,:;;,,"*' a.s,,uc,ure,ha,besins*',n*u,",,""**,""',i"uliii:,iLj'"xliTll",i'."";:1iilj".,:":l'i:l:,,,Il;Eudo.\us(408 355rBc) wa\ one of rheUrs "; nasnol surlived.and our intormarion abouthis acconrpxrtnnrntssecond hand Thereis no doubr. ho$ever,rhar he \rs a fiAFrxremathen'ocirn ^ u,ilhnn crcexcnr undcLsLandrn.r;i ,t,".;;;i";i;.-,,, h,, .'.; il"H::l':iiTH,::':l,1il:i i;ilii,H:tii:i:.J wnosccontent are ii,illtxi:::,Ti.i:iI;1,:::,iilTl Book' I I\ .,e. th( Jea-rerr,ur Jr. Jr J (r,. tf! Bnoh.v\|.ne||eur\ t fe.m.I rr "rut ar ur BooksVlt_IX rnettr,rr ol nunrherr Book X rhe rheor) of trario at lumbcrs Books XI XIII: solid geomerrl
ontetry. despilehrs fla\ s The nerl sedjon of rhrs '.u||.1 h: Jre\ ,r.
Bned)de.cnbe.he li\e, _iJ J.....r t,rJ,nnr,r. ., rtc In,,,.qrf j ( J.,. ,,.t r,,J a. rhrle\ b. Pl rnr3urri tl.€JJ. E;;;_i;" j d Ana\agoms 2. ::,1i:,t",., iilll::',,,,::,,,..,". ,,,a,..,,.tpri.n,.,1n,1,rh, r.{r,.q,. .r,. :r:ll,!e\.fbe'her,.ea. crchlnede. e..n,. rrLrr... h ADn unr r. c. pr,,t(l1 ' "., L t.,.,,' f,fp, " FromBook 0ne DEFINITIONS L Apri,/ is thar \rhich llas ro ta.r Srnceone cannotdeine \onrerlnr)Sb! snntl\,tisrnu a| rh. tropcnies ir doesnr{ have lhis cunnoL be takenas a genurnedefinrrion. tucljd In!sr ha\e rea|zed Lhxre!er) d.fillirnnr lnusrrctr on tre!roust] definedre,rns. and so it jmf,,ssibte is logjcaly lo dehnc all ot urc.s rerms.tr)ncad, ths:hould be vreu,edas an attempton Euctid s pai to tellhjs readersrl)rr his poi is sorncrhlrgtjke a doLone rnarks with a pcncil, but ar dre jr jJe, santctinre is an Jartrerthrn r |hlsical enLiri r).n;iriorr\ I and5 belos shoDldbe understoodin rhesaDre \LrI 2. A llr. is breadrl esslengrh Euclid s l,e is our .,/rt 3. The enr..r,iri.r lr1a line de porDh Sinceboth pojn ts and|ncs hale . aLead) beendc h n ed andrhe so d .rrr?,rn ! is subsequently used 10denoictheboDdary(fan)h8ure,!hats)c!er.itishardft)seeexacttywhxtrstcingdeJinedhere It.is possible thar Euclid fett rxt haljrg detinedborh tonrs ard lrncs ]re nou Deertcjto ctarirr rne telJljonshp berucPnrlfn Ir is rmplr!rtin rll|sJehnrU, rh,rre\ef\ rnehas crdpoinrs. erscquenlly. Eucljds lire rs xr ract a nnrre^ ffc, and hr\ rrr:lghr linr\, r,, bc dehDcLlnexr. are l]r ticr lne scgnrerts MDch has beenmade of the Greets iDsisrence(nr tbe finrrencs\of drc ohjecLs.f rh€ir in\esljgrrron\ Neverrhctcss,ELrctid ts qultepragmatic abou r ftis rssuc when e nec.rafis. rs rr rrocsin rhc statemcnt ot propositronl 2 below,Euclid is wiUiDgro bendhi\ o\!n rulesand meDrion r|iiniLe sLraighrtrnes 4. A sffaisht line i! at;ne \,,hichlies ere|ly s,ith rhcpoi s on itsell This too is a soDre\\,har obs.ur€ scnrencc lr rs possiblerhat rhe poi|rs () s.hrch Lhisdeiinrrolr refersne theexireniries ofthc pre\rousore Thus rhisdefinirrcn \lroul
i\.both cqurlaLeral.nd fishr angled.rn,r/dr8 rh!( ther jr uhjch equitarfatbul not.ighrmgted:aDd r1 | 1,,.,n..Jn.l \ .qr.it..,Jr .eq. .r ,(rrr-. .l 23,.pa,,/L/,straighrrines a,es0!,ghr r,n.s qh,ch h::.T;i::iJ;$ .l'" . ,. hi,n: muely rn borhdirections. p,ocr!!eJi,,.,el. do nor nreerone anothcrrn errhcrdrccr,.f Poslulates Euclds bool is al ,(.mpr ro s\srem,lrz.rtre seon'uor the.rcms ccssor\.By a geonr.mca,;"-"",.;;;;;".,;.1'l.-\ P'odu'edhy his P'Ierrc r r|llid \t'rter crrrrcS r.ding rhe inrerplal o1' pornrs-sLflrght lrncs. angl",. .quot,q er"t,tnl,lrcrr i","rioiyr.",,,.""," aii,"t;;;-,"na jil;t^ii:i:il:,,i::l}i li"j:111,,:il:il#l['"1:,1"j"1.:; u''r. a s\sr€rr.ir rsmcessrr ro or(rer thc rheor.ms nnxesir-ill- usc"*1,",ir ot onty '".h in su.tra nranrerLhrl each freliousl)trs{cd rheorenrs (and J"ri"i,tu*. o,."n,,.",. rin,;-r,;;;il,, ol nrrch flustr il:,,," ron lo rheer l,crect uruhenrrt i( ,!, rs o urs. .er rhlt strchan ord.ang tLrrnsoul tu bc impossibleifonc wishesLo includc such inr!'r.snn! re$lr\ !( rlreTlrcorcnr ol p)rhr!o,eslrlrd thr rrrL rhatthe sum of lheangles of c\ crI ria rllc i\ (!\o ! LgtrrxDtlc\ lLturncltour thrLr \omc ttreorenrsnrust alwaysbe acceptedon lairh,$irhour lusrilicarior Tiresc ar..a ed /',!rld?.r Jusr$.hi.h fheorcms shouldbe listed as postulatesis.r quesrronlhat nNst be rcsol\ed on sLrbiccti\e !roun.ls t rs verf Ukcl) qas (hat Euclid's choice the cDlninaliolr ol hundred\ ct lcars ol Lreil dt\.u\\jo amongstCircel mathematiciaDsLater genemrionsnlodrhed hi\ rhor.cs dDdone \Lclt kno\n s\rsrcmis drscusscdrn somedetail in fie next sectlon Ilis impoflantto notedrat posrulares ltrc idenrr., toth€oren\r\trr rsrtrenlur.ofdre|re\pcu live contenlsarc concerned Ir is onl] r rhen.iulrrficarurs Ltrrtlhc) ditcr The lormcr xre lccepred wrthoutjuslificarnrn,whereas eachof rheht.r nrusLbe ac(onu rledbi x troof rhrrreties or |re\ xrus theorems.postulales. or deljnrLions This view of Tis ,lsr,.n/r as a wett grouniledrnd togiclll! co|srnent orderLng ol theorenrsrs k) beundentood as an ideal It rsa \{.ll r.knoutcdgcdtu.trh(rcr,enran\Uxw\inEn.hdsorgxnizaloll of geometry He nude rcfeated usc of hotb unJclinedlernrs rnd r *rlcd to: ta(es Sonrc of rhese effols will be pornredouL in the sequ.t Nelerthctcsi. becau:cot rrs usrm xnd hecauscol.irr bgcar strcngdr.Euclid's opus is jusriy regaded as onc of rhesuprunc r.hrelenrenrs0t LneekcjviliTNrioM panicularand ofthe hunrannrind in geneml \\r no\ g,) on ro ti\r rir.hn s .hoi.e ol posrulrrcs Let theJollo|ing b. postrldtcd 1. To .lra\ d sttaightlD1.lhrn ut\ toint to dtn t,n,n As dns posrularesrand\. _ it srmplysay( rhar eycr) fx ol disriDcLtoiILscrn h. iojled bl a srrrighr line Holvever in uer{ of lome of rhc rrgurnenrsgi\en i| thc froof ol profosirior .l r( \\,outdseem that Euclid undersroodrhir srarenrcnllo nr.hdc rhc rddrijold a\\urDt{ro. thaLrny tqo poinrscrn be Joinedby at mosl are sLraighrL e IfthegivenpolntsarenandBLhenEuctrddcnoresrhcslrxigtrltjnetornjn!rhenrbl-,tBaDotaoon thatruns coul]rer to moderncoDventro|s $hich denorethis ob ectb\,1, In prol)osrlroj].rheiaLne notationAj is also uscd to deoorcLhe i|finiLc srraShL llre toinins I at]d 1j tnctrd s nol ion (and anbigurties) wjll be rdofted r lhc subsequcDlchafrers assumin: rhal rhe conr.\r wrll inuirbl). clanl whrt is interdcd lhishas drcmalolpedrgo!r. adrarlageot\rrn|ir(ir\ dnd r h e dr s a rl r aD L a i e ol an rnsignificantlogical nraccur.i$i rheanrl$r beljcles rharLh. ii)rmerorLEcigh\ rhelarrcr 2. Totrarlute a finir. ntuirln t L.tottntuu\t\ itusndiqtrtnj. js This a clere, {ay oI sr.rrin!rhe a!\umprjol lhxLrhc ttaDe e\rend\ jnfirrirel\ iar in all directrons Wheneverpo\sibje. Geck nrathemlicixnsavoriled Lh. .)(|lrcrr r.nrionot mnnirl. Pobabt) becausc iuch [o.r, ,l . . , l(d ru r.,. n,d,. l. " , ont r..' L 3. Todes.rib. !.irc|e firh.n1 .ent! utkt.ti.rart: Thc statementof ProtosruonI helos makei it .tear rhatrhi\ f.\rulurc js r{rbe rnrerprercdIn r very nanoq selrseNrmely. grvcr a pornrI rnd ! linc \cgmefr;8. the.ccxrns a cxcle \1|lhcenrer 4 and AB Ii s as mfrion r\ sonclinr€srefhn\cd h\ nroderrn)rrhemrLicrans 'adrus h) llyjng rhar thcif GreekcounlerparLs used.,ll.q,rrt. conlpasscs.ir uhictr rh. angt.ben,ccD rtrc less $as tosr wheneverthe compdls Bas hfi.d ofl the pafcr 4. That all tisht dn\l.s dt! equal tt) otu d,orlkl The rlghr angleis Euclid s unrr lor mexsul|g rll rcctitin!:xlanglc\ rnd so tr( D.cd',ro knos rhar all right anglcsare indecdequrl Sinf. he hasrritcd ro pro\rJe ior t|c rjlid moLronsthxL q.out.l hale made it possible1or him ro prorc rhe .ong.ucn.. (all,] h.rcc rtro equllLryrof r ri:hr angles.somc suchpostulales a'c necessar)I or a tnrLlr.rd \cusso of cqnrtrD.the rcrderrs retcnrjdrc then,sr par!3raphoI Lhesectron Connnor) Norions helor Ihat,il.tsttuithrLin.ldtlnit.nt\oiraightliir\"rtkthtDi.rittanqt.rt,t _5. x,rore,i.cts! ttLut tr:o nqhj nngles lii lunl tht r\o \trdr!t!t tut.\, rt t)nrtk.d r\tthnrrtr, nk\1 .,t rht! tkte a, \ hich or. th? anglls L.!s thar tfu t) t tlhr dhltr\ Manyol Euclid !eadcrs ^\ s xnd iuccc!*r\ hehevedrhar Lhrs posrul0te sr\ unrrecclslryand fiar i$ validrrv could he denn)nstra(edorr lhe blsrs oi rhe oLherst.loqever Llre! rclrered aLtcnr;rsoler (wo nrillenrra10 subslantiaterhis bcrlel rn!a'rabr) tarlctr A br.frrsrorr or *'"." atr"n,par, p,o"ioeo,n chaprer 15 onenuvvervwcrconsidcrthcresto|rhr\rcxrasanciirLurLrrnroJrhcirracr'.rsL,cc*r For dral rerson. rhrsposrutarc $rll nor he dis.ussi:dIuficf hcre crcctL ro norerhaL yrne ot i., oetrel knos,n rartant! appearin thc trsl secri(lr oi rhi\ (t trer CommonNotions Thc .ommon nonons listed bctow arc atso|osrutdles bur din.e!Uonr rtx,rbove postuliresin rbal ther aremaJnlf concemed wirh thc.o cepLsol equrirr)and xrequrtrr) Linto(un.ret!, rtcre rs a rar anuunt of uncenahq'as ro \4r!r ELrctidundersbod b! tt)esctelrrs l,roruhis rruposii;on l5ofBook I (prf allelograrnswhi.h arc on the sNrne bascalld i; rt)esrnre prrr]tet" arc eqral ro one rnorjrer). ir rs clcxr tharat leasrsonrerimes Euclid was reterIrtt ro equatil) ot arcr \\,c proposchere rhat !_uc]id u,as acru alll consrsreDljn brs |se jl ot rhis refln aDdal$ays L,scd ro denotecq,d1ij ,r ji-. (fuilrlc) Jn oLhcl wo s, uhen Eucljd says thal lwo para|elograrls rre equ ne mcansrhaL rhe), hdve rhe sanrearcai whenJ'e saysfial rwo ljne segmenrsarc eqLal, he nrcaDsrtrl rhe! hrve the srure lengrh:and *hen he saysthrL two angles arc equat,he meanssomcthrng of rhe samelrarure Thu\ we \u-sgeslrhal Euclid hasan undedying!nd Lrnsrared rs{mpri(rr rtur rlt geomer.rcobjec(s hr\e an aste.Lof nurnericals]ze whose properticsafe scl forth in the Cojnnron Nor(lrs beio$. so nun bec;usc ttrey descfjberhc properLiesdrar arc sharedby lengrh arer. lotunre. and anguh nell.sureThis ertlanxrion accouursfor Euclid s failure ro provide any otherdcfini0oD of {hc nor;ns of a]eadDd volum., *u*;,rrr,r"at"g n,,, man)'proposllroD!abour these 'r'he vefy concetrs comn'on NoLrors,*e conrend.coDsrirurc lruc|d s smnlraneousdcfinitron of length area.loh nre,and rngotar nre.s u rc t.he.cader $ ho is f:mil ixL\r ux mooem measurcrheory $ rtt norjcethar ifthrs inreqr.ctario ofthe Conrn]onNorrons is \xld. thcn nen is a ver} rtnkng resemblcnceberseen u rnd rhedeiiniLiou oI r Harr nreasurc {n lnfo.nraldjscus,on of thls melsure is ro be found rn the tirsr sectur ofChap€r 7 Tbis interprerarron olEuctjd s Connon Noo,!r\ i! hereb-vcdopLed as a norauonalde\ rcdhrousn oul fie remaindefolrhi! .= boot The slnrbot ,s usco Lodel]orc cqualrl) rf srzcor nreasurcThus /AB( = IDEF
mflrns that the rwo srid anglcs halc rhe samemeasure O1 .our1c, rhe\ are atsocon! rerr. bur tnat r\ usL'allyjnelcvanr ro rheargumenr Similar\.
AB = CI)
me:nsthar Lhe srrrighr linesJornins1 ro, rnd a L. D hnveequx e grhs L.r u\ no( cxanrneEuctrd.s dclinrrionof equrln) 1. Thitlgt\i.hih oft c4udttattu n (thinBnt!dtt.rqt ruon rnnr)tt 2. euol s b( f ad.l..t k) eEu t t. tht \ h. t. I utr,. 4udl 3. IJequdlsbe nb lkredJtuneLttdts th. rnxtrh(k tt dtt t.ttul The modernfeadcr nla! bc . fuzzlld hy thc rced ft)r thc rhird posrutarc,sjnce rr sccmsLo he alrea(l) subsumedb) Lheprc!ious one Tho cfecks, ho\le\er. drd nor rccog|rzc (heexjsL.n.c ot nc!,rtrlc nunr bers.dnd so Euclid lbund ir lrecessa|}ro jnctud. borh CoD trD N;tnlrs I d -t m his tisL 4. Th itirs ritu h. oi tlcidt trtli on( tnot l k t o rc qudl n).nr anonr.r In view of lhe proof of Proposr(ron'l ihrs shouldbc unders rod as sr) rns Lha(rhnrg ! $ h1!lr. an be nade ro colncideqilh one arother a.e equal Loone anodrern neasure lr orherwoftls. congrl'en( ngureshave equal ueas or lengihs,as the crse urtl)'be Tlrrs can be vie"eil as Euclid s li.si rnenrlorl ofcongruencein $e senseof a grdll,nslornl.oon Euclid s amhr\alen.eaboul thc u\e ol suchrLa|s fbrrnationsis mrde obvlousby the conrradrcroDrtlrrLkle\ he drsphys rr rhe p.oofi 01 Proposrtrorsl and4. ln Lhenrsr of lhese.he alords sllupl] mo! g a lme scgmerrlrorn oneloc.uion lo u other al the cost of producingan elaboraLeprooflor au rnlui(i\ely ob{ious lacr O (hc {,lh.r ha .l. the prool of Proposition4 stans\rith ar app|.atron ol one t anglero anorhcrh other \\,ords one rrianglers Lited .rd placedon top oi the othe - - a clear-rui abandorflren( ol srandddso lhe p.rL 01 Euclid Ir rs gelr- enlly concededthat Eu.lid s trealmert ol (Jarslor|ralronand congrLrerce.or radrerrhe laci. lhereof constitutesone ofthc morc serio!5 flrqr rlr lrl E/.rkrti 5. The wholc is Bteatct tlLul thc pdn ObscNe that il we assumedLhat all geonrerr.al hgureshale Slze0- theu rhelirsr tou, coDnDon notionswould still hold relrti!e 1() this tr r!ral norion ot conrerrThlslas(conrrroD,ronoDex.ludesthe possibility of suchr degenerdLeuotron of srzesrnce it clearl) rnplie! thfl sonreobje.6 ha\e size It also turnsoul to be very co|venLenlrn Duny reducro ad ahnndunrargunren6 'rdrdo Propositions
Proposiiion1 Or ? girr, /tnilt strai!htIine 1o ton:n Lntdi cttutlttttll !tidrglt
LetAB be Lhegi!.n nniresrraighr liue lhus rt rsrequ;ed Lo coDsrfu.r an equilaLeraltriNnglc on the shaightline A3 (FiA 1 I ) with .ent'e A ard drs(aocelB ler the crcle B('D be described:a:xin w1|nceDtre B anddistan.e BA lei the crcle ACt be descnbedtaud tonr thel)orIL C rn *hr.h thc circlcs cut oneanoficr, b theNints A Bletrbestrarghl|nesCA.aBbeJorned Nos sjncc the poinl t js rhece,rrer ot' the crrcle aDn i( is equal to tB Again. since thc Dornt aisthecentreolrhccxcleatt,r( is equalto Bl But{,1 \as xlsotrov€d equll r(,.l1Jithereiore eachof drestraighL line\ a,1. {? is equdlro,1B A|d (hlLrgswhi.h rre equa I !) the srmc Lhing xrc rlso equal to onc rnothe,i thereliLeC4 is al$r eqLrrlLo Ct The eloie th. threestrarghr linc\ (11.',lli 1JC areequal rc ore arother Theftilorj lhe Lrimslc 4na is cquilatcral:and rr hasbeen co|\uucted on lhc grvensraight lire {B (Beingl rhlll ir $r\ rcqui.cd Lodo lhrs propositions proof delnon\rrares both sonr. ot Eu.hct \ snencLh( and some 01 hjs $eakncssesOn rheposirive side_ q,e ijnd hin) a c.returreasoncr and e\t)osiro, $.ho rs Ontritllngro accepras oblrous the ci isrenceof a lrirD:le. which rnosI feoplc rrke tbr gran re.i U ntirdunxlety.he js not carefule ough Spcci6calll,. he rmfticitli acccpLsrhar rhe lro aLr\itiaiycirctes drru.n nr rhisprool necessanl)mtersecr Nou'. asphvsi.ar objecLs.these ligL*s.leaJry m.sr irterscct.bLrl as ab,'rd enLi tics, vhose propcrliesnusr be rcducrblc ro Enctid,sdefinitrons posrurates _.-"" ,n,, clrim calls fbr I'erificarion This is ",,1 ";.i;.. nor a Dn.rn poinL Th. r'3cli\ ihat Euctd rairetrro pLorirrca t'rne work within which rhe jnrerio.s and errerlorsor contgurdhonscal] bc ds.ussed ard this is anoLber one offie najordefecrs oftis h:rcat cdilicc
Ptoposirion?Taplo..(tasierpon e\ dn e\trrtit^ ) a ti?itht tiL. .qudttuagiftnsttni|httiDt Let,.l be rhe gilen poilr and nC Lhegi\.en srrarghrIi re (FiB I 2 r t.hus ir . .r,J'.t rs rcqurredro Dlac. !r 'l"por j,-,,, .\,,".,,,r r,,..qu,rrorr, _,,,, _,,.,,,r, t".i,, t.lr.,,i.;.;", r,. rnc ,hc tornr 8 t(t nrd i, Lrr \Abe nrnedJ,dnnr.t.. n.e.tIt.ic Lr rr;teD,r/+o..,ir.rrr.reo Ler the straighl iines nI, BF be troduced nr a strar!:hrtine $i; ,A lr *;0, c.,r," L,a Jrs'n"e ,BCler thc cirde CGt be desc.jbed:and agrjD wjrtrccnlreDxnddjst,|eerClcrrhecjrcreG(Lbe describedThen sincerhe poinl 1] rs rhc cenLreof rhecircte CGtl, Ra is equal ro fr; ng,,i", .ln." dr. pojntl) is rhe centrcoI $e cntte cn'L. Dt is equrl to r(; Ard in lhcsedl is .q*t ,"iil, ,r,.*,r" thercnarnderAl- is equalLorhc re,nainderB(; BurBCwNalsofroledequatr,,iC:tterer,u"ead,,,r thesb aighL lines 4t BC is eq! al to BG And Lhnrgs$.hich lfe eqikt ro Ltresanrc th rnr xre alsoequ at ro onean other, lherefor e AZ is atsoequal /Ja Theref(ne a( rtrerr \ cn pornL A rhcstr ajgh; ti ne.11_ i s ;rnuco equal to rhegrven srraight line Ba. (Being) uhar jr s asrcqu ed to do Ar fi^r rr mighrsccn prur.sirxrr !lanre rhJr 2 bclaborsrhe obvious In rhis.urhor.shumbre oprnron LhoughIhLs proNsirinn anLl/\ prn,,i rre nothingslul of nnl\cious. As wxs Drenrroned above.fiis p.oposjtion indrcries rhar Eucliil hful co atsibi. corntxsses;, , n *fre,r r,",ror., r,x books The subllet) of this notidr nillstrxins lhc und.r.txnliIrS 01rnrrr\ oi toda) s stude|r( ltl(neo\ ef. Euclids $illrngnessto handicrt hnn\clf h\ rs\unrir! ui liLtlcrs he conldloslrblJ' ger .qay sith demoJrsLratesa faith thepo\.r ol humar rc!\orins LhrL$rs losr$rrh rlc c,tlxPseol the (n.ek civilizrtion and did noLrcsurfrcc unril rbc Retatss!n.. Etrchd could ha\e assurneda rigid comflss in Postulate4, an asnnnptionthar tould hr\e txler n,iLID rJ r rLes,, ds n, sfureand {ould hale obviatedthe needfbr the non Li!irl p(rf Protl,\rriol I The fu.1 rhft he .iose nor ro do so j'rdicaLes thathe enjoyediexirg his mcnrrl nnLs.lc\lusr Lbrtlc to) ol Ls ! tlren
Proposirion3Cfsn nU rn.q ol \trdrsh!lltr\ b Lrt 4i ltuni th( t:natt d trni\ht litt tqlalI. the
LclAB.Cbethet*ounequdlltrrighrljDe\.rndlcr.4SbcthcgrcLrLcrofdren)(Frgl3) Thusirls requiredto cut off'fron,18lhe gr.rL.r r sLrriehllinc .qual lo ( tllc less At thepojrLA let.4Dbc tlaccdcqualrr) rhc strright linc C xnd\\llh rhecenler.laDddsLance/11 letrhe circlerffbe describcdNow sinceLhc poinL .1 i\rhc.er)treollhecircierEF,AgrsequaL Lo ,4r. But C is alsoequal to AD lhereli)reexcholth.straightlincs,lt-arsequalto,lr:sodratAEis alsoequal to a Theretbre,giveD th. Luo sn,ighL lires.,1/J a irom AB rhe greaer AE hasbeen cuL off equal to C the less (Being) whdr iI wr! requircdro do
C
Fiqurel3
Pruposilion41/ nro r7.r,rlesha\e te tu. sirl.\ tqual 10 tro si.lt\ ntte.t^el\'. dnd ha\r th( onql(s cofitailed b\ th. eqnl straiqhl IiEr egudl. thrr |i11 dle hdtt th. bdt. cqudl k) tltc hase.th! trr a Kle vill bc eqwl n the ttid,glt, dtd ttu rcrninuts dnrl!: rill b..q|dl Io Ifu r.nduuns dn1lts respe. \'ch. nancl\ tliov rhidl th( equdi sm.\ att,lIt.1.
Lel ABC, DEF be two lrla|gle( hu\ iLr! th€ rso sitlc18,,14 equ l to the rwo sides/lt-, D, rc\pcc tively namel). AB 1(rD, alld AC to DF, and tlre ansle BAa .qual kr thc angle EDF (Frg 1 .11 I \ay ihat the b,se AC it also equal to thc blsc I/r. drc Li.rrsl.lBa sill xlso hc e.tLrrlto rhe rriangleDta and drercmaining angles qill be equ!l to the rcmainiDgrnslcs respecti!elr.na.reLy rhose \hich equal sides subrend.thaL rs. the ansle ABC to the an:le llil rnd Lhcangle ,1CA to the aDgle,Dft For if the tianglc.,lrc be appliedto the rfraugle,tF, alrd iI drc foinl ,1 bc lla(ed ou the norn /t and urc straightlineAii oDDZ, rhcnrhc p ntr$illal\oconrcide$irht.trecau\crltiscqualtortASainAB coincidingwrrlr Dt. dresrraighL line Aa will allr conrcidcslth Da. be.an\erh. 1n31c3 a is equal to rbe aDglctDI- Bur I also corncrded\!ilh r-i her.e rhe bascr(l sill coucrd€ uith the ba\e Z?' lF(trrI wtren,9coincrdesrrirtrtLrnd(.$tth/.rhebrs.Aadocsr.l eorrr(idc\rifhr|ebrseA/..rwo .silrlc I h!'rcli)rrrtre br(e Zr(.ujlJcoincjdeu,irh,Fl 1/l( s ill .oinrrdc{rttr rbcwhote rianglc DZf and rt!r q.nh corncrdc (hc rcmfuningrnSles and !\i1l be lrd tlrcrnglc ACa ro ltr€angte DF . Thereiu,eerc.
readen,itt r€n)B zc p()tosil on 4 as !re ore _ .,,The urlr r nN.. .onrnronlyrri.c.r.e.t n) a\ the sjde a'gre srde(sAs ) congruencctheo*nr A\ {as tonrLcdouLearlicr: ns proofs,'ftirs from fie flau Lharrr nars ur orarorio,r or,r?1n'.rtu,, l,f u'hrcrrno dcrj,ririorsorarnrns arero r. r;;;.;.,;","..,, oi rheproposilion is atsoawk*ard. .. fl.\ rhnrrecun rn sn'e trrerpropoliriorsas u.etl Tlrjs.onclude\ rlre del.rjled . dircussn)nol rtre firsr fer paeesol Etr.trd.s hopes(hxr E1.,irn6 The:ruthor rerboityofsoneof,hes,.,.;i,",;t..;;:li,:1,:.;l:::ilili,:HiliiT,l,iJllllii,lli"f:despirethe extoscd fl.rws ir rh. to ,nsor rheaccraim hc has ,eceired orer rirecerro ies. ,,orr r.or wrrnL r,e lccorrfia,h", ;,,;;;;,",,, co oD ro sraLchofos iols 5 t8 01 Book I as well". cluded rhettr)ofs ot {heothers can be toun. rn me lu nonsrhc readc, \rith thescope o( huclids wort
r.if o\n .ighr.arc e\\errLia l, teDlnaslc,r prulDsr
Proposirion s 1, Nor.dt.r /tidnqte\ ( dnltcx nt bdk ar. equat )nn. unojhrt: and, i.l th. equot struisl1tlines be ptudtu.tl i. :ht futh.t. oite\ nd.t th. Ld\t! titt!h. qt(t rc on. dnt)ther Proposilion,S1l;, d tritol.qt. nio dngra.\h tqunl k).)h: on.thet trlt ri.r.\ rrtith \ubt.ntr the qunl dngle\ )rill nt50h. eqtuttr.) tat ("1.thtt:
Proposirion7 crj.dr n c r/rutqht lit.\ rah\tnt.t.d .r d vttntht !tt( tnDr it\ ern..nji!i(t) dtut nk?trn! 1, d potht. th(tr cornot b. .nnstt u.L..t ot the \tur) snailht titu lr.th i6 4trct,,t,t"rl ,,i,i.'r) ,i, , tit1,..\tl,ettD{ ,,,,r,, :.::.,_:.Lt:: :: ..::,", :,1h' in ano,h,,pia, ,,,i,"'r. ndn.i\', e.kh t. ihdt rtir.tt hd, th! ttuD. .\tk1nttr. ",d "q*r
Proposition8 f . o iungh\hnr.trr.\itt\equot tu tt.\)ctn[t\, uid htN. lt.sI)th. btse eqmltu ebasc.ttk\ti dtsohtttrhtangteteqnotrj|dtft...ntdtr.ttb\ttt,qtat^\.liLttr $jltifht turs ,i'clid s narro\ i.anre$ork. otsonre srand.rroclxr ,J ri. ., .r . D.rtclJ.cr t.. r, , ' -o,r.|| !r,el Ir ofPn)poiiLio ll rnclu.lesa fefercn.e ro an rnh c delhirronsthrt clcrl srrarshrlrnc is nccessaf|), Proposirion9Drlr..r, Eir?n rc.tiliredt nnstl
Ptopnritinnl0 To bise.r a giwn finik, ttaighj tnn.
PtopositioillTo draw a lr.dighthn.lttit:htdDqt.\k)oeivnrntrrtntttult.ntdlr.etpt)irjtontr
P.oposiriontz Ta a gi|en uflnt! stmqht litt, inr d !trln pitl I tlith t\ nDtut tj j. tt,n tr pttr(r diri ar straight litle
\!hen exanining Proposirron 13 Lhercaders sholld bcar rn nllld rlr,rrrc.oRtjng to Definlro| 8 the sidesof any angle cannorlie on ! single srrarlll lrDe.tn urd.e .*r.- U".r;at svsren doesnot anor fof angtesuhose meaiure rs errhe,0" or 18(,. o'e rnulr"rrt"ir"g" femenrber tba, zc,o was nor recognjzedas a boni fide lumbe ,ntrl nro,e rhr,r a rlusrnd yc.LLslrrei llowever. I $as arguedabove thar Euclid implicrrty assumed thaL.r| at]gteslas $.e11rs s..DrenLs6g!res. anrtsotrds) had some numericalsjze so u,harangulu size could hc Fssibh a,,ignin a p^t, ;i";;;jd*, ,,"" ixcLUdcrrre ..ro drgl{, once fi)s ao8lers excruo.o. inrch nreasnrcst8t)i) mus(also be exctudcdThe ofta|cc to rh.!e nj.erres.Thcl are issuesof sryle posroonrs ro tacilirlLe rhe conrpan$rr ot rhe sum :lor! Proposnio|ll is rr\ pr€d..essors con\ersc I As wc shallsee. it serlcsas a cnreiallerm,a ror
Pto.os-inonn loit.s.1 If,.sndLght up.)narftnlhr !ut. ni.k. dn Ptoposilion14 Ifwint an\ sttaightLin!. unl arap.njl arit \ttdilht titt\n.t t.r Lqonth? w. rhe o1|!ej eq\at ta ntu tisht j. ^t\) ::.:.!::ak: ltta.:ar dr!tts. t,,a ,txnqht ttn.\ r.j b( nt d \ttuisht tne Proposirionl5 II t\,o sr.aigh!IiD.s.Lt on. anorh.t lntrnkrk.rht\tfl!ttttn!t.\.qtutk)ottcuno*?r The followirg p|rposilrorr is o\ershrdo\cd rn nranr \rudenrs r..o e(Lionb) ttre strcngerone rhar arry exerior ansle of :_T:1"':"' a tnrngre r\ acruaLrlcquat ro rhc 5unt of rhc two rntclor ano opposrtcangres. The proor oI firs larrerler\ion .elics on EuirL|s PrxLurarc5 .,, .,,n'" r.gJ.",,l ,:',.p',:" \uch ptJ)turr, psrutarc. xs of on rhe tucr rhat tlre surr of orthelnanEleeqrir\rsorirhLaf!re\r\eese.rton:::1"-,j:1, rrrernrer.ror ar,gres r.4 of rhischarr.r I E,,"I,,i.r,",,e,,", ;akJ"."., Pans ro avoid usilg this posrulate 10, as ton! as possibte.on.r so t,. y,u"s rhi. ;;ri;.". Panial as it is. it ptuvide l*"i;i doe" rhe touDdarjonin r''p'overarer (P'oposirion:l that par'rtrer rin"sdo inde"d exi.i-F;;;t;; =;J;, #;;il:::ssa'rer () rerre\lrflre srude ls. rncmory.tuctid.s proof sumnarizedbetou is Ptoposirion16Inan\tritlrgt.,iIoi.Dfrhr,sid.\btptotrtftt tht.\t(tetonqh,isRt"okt rrr.trtl qt the nltet iot dnll optd ite ui qtt \. PR00I:LetrlBCbcatn r:lc andtcrone\ideoirrB(.berrodrccdoD(FLg l5).irulll beshu$! that tlc exr0rior/DC,1 is greaterrtran rhe inrerio..tnd opposrtetrla"and -.,1Aa LerACbe bisecredar t and tel BE bcjojnedand produced nl ,r sbd'ght iine to F; ler tF be made equalto BI. and1er FC beioined ThenAAEA = ACEFsince ,4t = Cll, and RE = FL. afi IAEB = ICEF. Conseqo€ntlyIBAC = IFCA < IDCA A srmilarfigumenr using a bisecrionoftlre sjdeACrcs!]ts in theconclusion rhat lD(:A < IABC CE.D, The sumof rheangles ola lriaDgleis rfietheme thar unjtes mosr oI rhechapters of rhtsbook, and in amoregeneralized form. jr oneofthe central issuesofmodcrn geometuy rlJr"l."l"g o-p"*-" constirutesrhe fifsr variaLion on this rheme Propositionl7 In an! triangtetuvo dn7let taken ta|eth?t in a11.rnnnjLet alc tessthan nrotight ah!!!es P80OF|Let ABa bea riangle (Fjg. I 6); n wi be shownthat IC.AA + IACB is lessthan r$o lgbr angles.Ler BC be produced ro D. Therrsince /DCA rs an exrerjorargte of AnrC ;, g."u,", .u, theinterior and opposire ICBA ConsequenLlt + IACB < lDiA = " -CBA +:a(a ,*" igii rog,", 0.E.D. ( D FrOUrct 6 Moslpeople. when asked ro characGrizea stmjght line, u,illrcspond by sayrnethal it js the snon_ est.dislanceberween two pomrs Thrs is quire reasonrbte,and we shalt larerm;euscof the same underlving inh'ritionwher) dennins sraishr lines in other geomcl es Eucl',r. howe'cr has; J;i;."n. q) dennilionfbf straightLnes and r( bclnxnes hnn ro trove ri,l iIs sr,.,sht tinc do.! jn ia.r possess rhe sho'1esldrstrn.e fropertv di.rared b) Ihat rn css.nce,i! rtresubiccl lnarrcroj Proposilions18 22 Proposition18 1, dtr) triarSlt ih( qkdttt \i(tt ttbtu trts tht gt.dttt dn(tr Proposiriont9/,rdn!r/trrgl.th.Snrt.t dnflr i\ rtlrottd h\ t!.lktjit r.tL Proposition20l atL\ttilill. j.)!!Lthel LAaed.\ttk., it1@\ rrtn(t nh. lt.th.r tkuttt rerrMrL! PtoposirionzlIlanoneol thrsitl.\oldjti.)nsk,fj1,ntt\e\h?n tr\ ttn,R.b...nstrtuted stnryht lines ^to vithin the tt i.trgle, tht sh11i!httitj.s r) tornntt(t \ iu bt I.ss jhuD thr )rhuinng si.te\ol th. tridngle,but rill tontuin d gnato dn!1. ^ro Proposirion 22 Out of thter s uightliie!rh hta" etttul tothtct tjret tuilht tn.\, k) canshu.to trianglc; thu\ itts n..(sy,\ tt ol thenraigh! tues uklt Dl iD ndme;. \ho Lf ^. bt:ttedret thL,l The pmposrrron readerwrll recall lhaLrhe prcoi ol .t (SAS) entajtedan apthcatioDof o|e rrj angle onto anodrer.a stef of quesrionrbtclalidiry wrrhin l_u.lii \ lianrelrort io"t,l nu* ,.r" ,", Io proYc two more congruencctheorenrs. ASA (anglesrdc rngtet and SAr\ (side anglo angte).rn Proposition 26 ProposiLion2:l lltoq,shinr roa!oid ttee\ltirir useol.lpptrcrlions lrirt"piooror thrs proposidon propsilioD,t is bascdon (SAS) shose prooi. thc rcader$il recall. docs retv on applrcadons It ii very reorprinSro spcculare|hlll l:u.trd s crtjrulah)n and usc of apphcationsr| ihc prool of Pfoposition 4 cameonly allcr mar) lain (rcnuous and s efor ts to ptuduce ar unrainted 1)rooI The purpose ol Proposi0oD s 2,1and 25 rs L,nclearWhite rhe) be[ I iormat resernUarce ro Pfup" srtron.1.they havelery few appticarrons Pruposiiion23 OtagivosffaiShttin.dndajdp.jrt.ritk).oru/ urtdh\.jitinut ahgtaeqLdt loa g;l,enrertilin?dl angt. Pioposirion 24 lttut)tnatlglt\hd.erhtrrott.l(tryuat k) trr) s;t.\ ftst.tthch,. huj htn( the.)h.,.1 the angles contdinetth\ th. lqLdl \trdi\hr tine\ gnu..t ttn rtn othttt tx\ \\ dtnj hne t1t ba\. 8rcatet thin the ba!. Proposilion 25 lf^\o ttidtl|t.s hdt. the \jtes etlud b.\a nrlr,!?../ir.ht htr httcrtrc ^\a bdsr: Steaterthun ttu b4!!. the\ rrll .tl\o hdt. rtt. t"te ot' th. o?t.s..ntdulttt b\ tht qttt lnryht Ltht! Ste(ter than thc ath.l Proposition26lftr.bitlt!!!tthttj.ttt ta one ^\.'or,!t.t.rludtnr\oflnt:k\h,:t\ttr.tr.ln(l.tu\irJt,qud! srde.txtneh, eithlt rlt nrte adjoninr rh. aqMt nnlttt ot fut ,htu t.tinr o,e.f je utual 4netes,th. r\11dl!o he\. rtu rcnttihnr! sxtr! u.ltdt t. th. r.rktu!11! sittts utt the te,Lriniry dnqk n Hl\ing trovedall thesnndud congrue|cerheorenls Euctid lror Lurr\ro rhcro|ic ol prrx etLsnl In- thenej(l two proposLnons h. rDdicaLcsseleralmerltuds lin .o|sLructnS farrltelli;e, Ptoposiiion 27 It d stttisht titk lulliry o lrr flntiSht tn.x nntt?\ tlk ah(nnlt unltts equdl toon. anothet;th. sttuight line\ ri b. p(lrd et rc .nt. dnnth.l Proposiiion2SIfd taigh!hnrtillinqontn)eruightltn.snttAtthrcv!ritrdnpteequdtlothritt?rial antl oppositednillt oit rh. yir? sid. ot th. rnEit)t dn!.\ t,1 th. sttft ,iLl. quat b dro tilht mstes thr strdightlin?s \ill ht potullel kt atu antthLl None of the pro|osrtnrnslisred so tar mlke usc of I'osruhLc5. and jr has hecn arguedrhar mrs rs evidenceof a consciousrelucLxD.e on rhc prrt ol Euctid to ltl) on iL lle tharas ir nrcl_rn !ies, 01 l.tel developmentsil has becorncconvene|L r() g^e a di{Iere|r nam. Lo thar parLof ljucldean gcomeuy which does nor depend oD this postllaLc Jrnos Bolyai (1801 1860j. or)e of ttre fou ders of ro!- geomery. Eucldern retercd Lorhis subscLofEuutidean leomc0).NVbrorr/c a.oftlrr Nowaod)s, il is also conrmon to rcflr ro it a! ,.rru] g.oD.r1 J,tuclid\ pi)nsi on\ I l8 ar€ all rhco.enlsof absoluE geometfy Theorenr I I 4 l]1 thrs chaprer.rs well rs lhc conrenlsof Chatrer 10. conshrures severalmore cxamplesof proposiron s of rt'solute gcoDre{r}\!e go on to lk I \onre more proposirrons ofEucld thal do rcly on the PDrllel PostulaLeand ro which we r\.il hale occlsi(m ro retertarcr Proposition29Asta\hr litl. litlhry.n y0ottetstRtish tntr klok?sth? dtkl1ktt( dnElt! equotbone dnother,hec]tertarahgl.equoltotht inktiat d o 4sne drglr. o th. intanot dngt.rt)Dthe rane lde equal n t||o tiehr ahllc: Pn00F.Lct the strarghtline tF fall on the parallcl srrarghtlmel, tR and C1l (Frg 17): ir wil be .AGH = ICHD. lt:GB = IGHD ZBGfl+ 1c,./, = lwo righr anglcs lf IACH and ICHD arcDnequal,lhen onc of rLrernrs grcarerLer .a,,ldH be rh. grcater Hencc. IGHD + IBCH < IAGH + UBG,g = rso rigtr angter Hencc. b) Posrulare5, rhe sraighr lines AB rnd (_D. jf producedr|delinitety. $rlt mceri bD( they do nor meet be.auserhey !re by h)porh.sji para et 'theretare.tACH = lcrl, The ptuposjlon.sorhcr assenionsno$ follos easily.and Lhcdetarls:\rc omirted O,E.D The following rwo proposrnonsre iuctuded herr be.au\c of their inrnrensexnporLxn.e The) consriure the ro mosLilluofranl sLdenrelrrsot Euctideangcomerry. if not ol a nattrcnarics Wc shall try to substantiatethis clainr in rhesuhsequcnL chaplers oi Lhisbool lltt !\tttio' Itt!l' i\ eqnl tu th' nNa Proposition32l, anr ,idn 31r' il aE o.f thL litl.s b. UrhkaL (.!dlIo n\|) tighi dhglL s i tetior (u1dopPosil dngfu5,ahd IhL thkt irk|ittr dnzlt!.l tht liotqtu or. eqml rr) Proposition47 hl riSht a r g l. L1t t i.u 1! 1.s tl1? vlttD. otr t1( tillt elltttdill! tL tryl angl' it thc squerc| on lhe sidet containhg th,: iqht uulr FromBook lll{Circlesl Euclideancircles play a tundamen la! tulc tu rlns book s delelopnrent oi non-Euclrd'rn gcomerv For it is convenienrlo resr.tc hcrc sol|e ot (hesranda.d theorcnN abo!l thctr rangent!' this reason 'lltles. the Proposiiiont8 f a\ttuirhtnewklla.inl( antldlttdigtt!litttb!iaindh? ttu te,tt1't' P"inr of conta,:t,|hertraisht lirt so futt..l rill tu Pery?nltulttl to thc Pt)ilt .J .ontd.l Prcposilion22 Th( opt).)ti!eunsles o.lqtdd, tlnotlt itt (ih lesdn (LILdlto o rilhi angtes ^i Ptotosirionz1ln equdtcuctt\ skltalingon equal.:i r.1d4ft knlrr at" equaltn otl. dnathet ||he rrth^ standat the cenprs ot at the (ircuhtfercn.es Proposition31 1n, .irtl, /re an\le it th. setuicikle t\ nghl Proposirion32IIastrci|h.Iitt'tunhaInle.andlurthaPoiNol.ontattth?!?bedtrt\nd.t'ss tn the circle, a stroigh! lin. cuttinB!h. (ircle. fi. dnqll's)\ hkh i,ntkas \ nh *e ta\.nt till be (qonl to the anglesin the alt?rn e segn?nLtofrh..itLl? FromBook Vl (GeometricalProportion) Any theoryof proportionsnNsr conlrorrrsomc fomidrhle Llilhcultiess4rcn rhe issk ofincommensu rability (inational ratios) is encoulrered trr ordct rc surmouL rhese Buclid chosea rrther surprisrng approach He fnt dealt \ ith inco mensurabilirytn tbe conle,(rof areasaDd proved Proposition I below Thrs enabledlrim Lo transferquestio|\ rcg{rding the rrtios of cedljn lengths ro thc ranos ol correspondingareas This cleler techniquehas been ncglec led bl l1loder n expositionsi n favor of nofe direcl approachesard is essenrial\'forgotlen Srnceth. deLarlsuould takeu! too 1.r alield onl) some of the beneFknownpoposiLions rbout similui(y are ,estatedherc withour an) atlemPlto sketchoul EtLcLids prooti Proposition1 lridr8l.r aitulpdrolleltrgtums ,ticlt at" undu th. wne hcight are to on( anothet a\ Proposition2.ftdrtnisnrlitlebe.lnrnparlLlhln,at.)flltr\i.]?rafthehiorgl. l\illcut th' \itles afthe trianlk ptaportiorallr:tntl iflhrlide\afth.t,xo18l.hetrt Pt1ryu1iondll),Ih't'lesjotntnF the poin$ of the seLtion\till hc patalltl o the knktinill! stuLof the Dio1!l( Proposition41" !4ai./n8!1arttknglil theel€5Ibatn theLqtnlanglt! ltt praPotnoinl lndthos'4t? canllpa dng sLdes\rhich trbrcnd nt eqrcl ansl(t thr tti.n31^ rllll'. and \itl Proposiiion5//tui.,r';dnSltthutcthtit'titltsltotDttioittl 'qutuirultt how thov anglescqudl t|hi.h tltc co'retPon(littSsitlct trbrcnd Exercises1.2 l. SLrnrnrarizeyour highschort'\ econeLr\ couLsc rn r li\r |!re do.uDrcnL Pnofs lbt tht pbrositi.,t\.md tut.\ \ttih n.\ irttr J u) rht\ ttldput npreo u1 "t)t 2 Clflricizefie sLaren)enlsrnd/(rr pr!)l\ oi Eotr)si ons j 9 R.qrrre rtrcnr$irh rrorc.mderi rer nnllology rnd roratrolr 3. Crjllcjze LhcsLaternents xnd/or p(!ni\ ot Ploln)lrtlolrsl0 l5 Re$!rt. theDr\\ilh ft)r€ nrodern ternxnologtrrnd Dolaljon 4. Cdnclze thc ttatenrenrs d/or proofs 01 protr,\Luor\ t6 l9 Reqnrc rhclnwrrlr more modem reffnnology and nonLion 5. Cdiicize{hc strtenenrsrnd/or proofs of EoposrrLo\ t0. tl drd ll R.qnre rhcmsirh more n)dern ternrinologr and notrnor) 6. Citjcize tlre sraLcDrenrsanl/o prooli of proposirionst6 tE Rrwrk thcmurth otur. nrodem temrinology rnd norarion tI i! tt..nrntn.l.tl thdr th( tedderscithet tnrr E\itat 7 thetr.l?t\.!.t lo.dk tndr proofs in satnc hiqh lha.l gtortet^ t.rr Iht,ptool\ \h.rtrt nt\ anh on th\. ptotbtirion, oJ Er(lid that ure qrcr(l in tltir.hat,!t): Motar.I th. t.utL r\ shoutltt.t kt. \rhi<.tt.t t!\e e\cnBes dtt i'dlLl in dbsolutu,r.ome^ antt\hi.h .dt\ n L'.tit..1D {a.tr.t\ 7. Pro!c tharthe locusof all poinrsLtrat arc equrdrsranr r,ronr hro disrirct pornrst d B js thc sbarghr lme$atisperyendicularrothelinesegnrentA,4andbiserctthct/,1).ndi.r/dthi.re&rofAB) 8. Proye thal the perperrdiculr bjseclo.\ of rtrethrcc lidcs oi rr) Lrianltc arc .o cLrrenr(i e they all intersectin a conxno| poin g. Provethar thelocu!ofall rhepornrsirr rhemLcrid.lan angtcrhat tuc eqordisranrf()m jrs two sidcr is that anglc ! bisecLd 10. Prole that thc threeanglc bise.roLs of ant r.iaDglcrt cor)cLr.n L ll. Apoly:ouallofuhosediagoDat\Iallinitsmrcri(rissaLdrobc.rtrr.r prorerhaLrhcNnrotthe rnLeior anglesoI an] conlcx ,r srdedtollgon is (, - t)u 12. Prove thaLrhc opposrteangles ofany parellelognm areequal 13. Prole rbal if both prirs of oftosjLr' rn:les ol a quld latcral ! e equat to each orhcr, Ltre|rhe quadnlateirl is a paxllelo:raur 14. Prove ftat the opposiresides ol an) pafajlctoglrnra.e {udl l5 Proverhat if borhpair.s of opposircsides of a quadlrtarcr!l rre equatto ea(trorhcr rher rh. q!a.lfr- lareralis a parallelogram 16. Pro\c thal il onc pair ol opF)siresrdes ol a qurdllarcr!t a,c bor| equal rnil par! et t() cachorher. then tlrequrdihreral rs a parrltelo$rm 17. Pm\c that thc diaeonalsoIrDy para clo:mm bi\ed ea.h oLtr.r 18. Proveih.rl the locLrsof dll rhepornls Lbar a.e.quidrsranr lronr a gr\cn]llrf .onsi\rsof rwotarallej lS. Pfo\ethrfthehrejornrngLhenrLdpoin(soit\{jsid.\ot .rr.irnSle rs taraliet lo rhcrtrird srdc aDd eq|ul |o hall its lenldr 20. Plove thrt any r\ro meJi,rn' cut ciLch(rrher rnro se:nrenrs$h.se tcn:rhs havc rrLio 1 l 21. Lcl P. 0. .mdlR bc threefornr\ on thestrxighr l/Lre\ ,B(-. CA. andA,4 v\,hcr..1,B.anda forlu a trirngle Prove thrt rhepoinrs p ? r|d /? !|e cotljnc[ iiand ont] rj AR RP RBft #=-, (Here r\e folloq the con!€ntion thal iI th. thrcc toinrs i' \ . u arc .ollnexr. then thc rnLio# is positile or negatile a.cording rs I is nrsiJt or ourld. tl]t l|re se-Qnentarli lhb c\etcrle rs known as the TheorenrolNlenelaus ) 22. Let P, 0, and R bc rhreepoinrs on rhe srrarglrllnres B(. al1. and t/J $hcrc I. a a d a torln a tdangleP,ovc that the Iines AP nO ardaRarecon.urrenLi{andonl\ AR BP CO RB PC U-' (This is lhe TheorcmofCc\, ) 23, Provethat thethrce mediansof an! rrianglcdc concurcnL 24. Provethai the threc dlLitudcsof rny tri8glc arc coDcurent 25. Prole lhat in congruentcircles cqual ccntralxngles de(emine equalarcs and equll chotds 26. Provethar in congruenLci.clcs. cqual arcshare equrl chord\ ind nrbtendequNl ccnrrrl r gle.. 27. Proverhlll in congrre.r circlcs.equal cho.ds cur ofl equ,rlarci ind s'rbt.nd equalccntrl argles 28. Prole thrt thc.ugle rl Lhccircunricrencc ol a crrcleis equa]rc onehxlfihe ccnLrrlxngle srbGnded 29. Prole that lhe locus ofallpoinLs Irom \hich a given Lr)esegmenl !ubtend\ th. sanrcgiven angle consistsoltwo drs ofcqual .irclcs 30. Provethat fie sum ot rhc opposirernsles ()1a qrLadrilarexl thxt is inscribcdm a circle rs / 31. Let 7be apoinron thecircnrrfercnce ol r crcle cenLeredal ( Pro\cLhrlrlirerrconlxrnLnslrs langenrro the gileD circle if andonly rt il rs peflendi.ular to thr rrdiu\ aI lp Hilberts Axiomatization {Opti0nat) Wtrile rl1 the subsequenrgenerarions praiscd Euclid lor hrs greal a..onrflidrnrcnG md regardedhis work as cpitonrizingture rcrsoD.many nrathc atrcirns$ere lull!_ auare that thi! \,ork was incom plete In his proofs,Euclid repcaLcdlyuscll delinirir,nsnnd fostrhles thal had noLbccn InadeexDlicit previousl) Auemph to corecLthcsc d.liciencies se.e conLrnuedo\cr th. ccntuics andgalhercd g.eaL momentum in the secondhrll o{ th€ ninctccnLhccnlur}. o(l} be.ruse oI thc dr\colerl ot Don Euclideangeornetry during its fiht hdli Wc p.cscntlrere the ariorllrrilati(m ofTcrcdb\ Da\rd Hilben in18s9inhisboo(7h.Fourul idts ofG.on tt As nrenuonedearlier, thj\ syslc is colsrdered definiuve.bul il is al lhe sametinre also jgnored b) nrostpcdagogues because of its.oDrplexil) and subtlery What follos's is a sunmar) oi Chaprer I oi Hilbcrr's book For a deLailedde\elopmcrt of Euclidern geomelryfrom aDrriom sysrc r lhal rs lery closelo Hilhcrt s (llLhoughb) no mexnsi'ldr dcal to itj. the readeris .ei.md ro E E Moise s book ln .onoall to llu.lid s devekrtment Hilbei begmsbJ L(tirlg r coll.cti(nr of l.nns lor \ hr.h no dennirionis olTe|edThis apfroachhas thc rdvaDtagc01 bolh lreaier lenerxlit) md logrcalcofrechess M,thematlcianshale found rt usefulupon o..asion LoinLeT,r{ th. absrra.ridea of2arrr rs sortrethirtg olher tbaD.m inlinrrely snLll dor Thc mosrcommo| ol theserltcmate interpreLalro|sis tlrutol r poir I as a set:sometimes a ler olother points.and v)mcLirncsa setof|ne\ Thc abscnccol a dehnr|lor thlr tics do\\ n tlrenreamns ol the\\'ofdpoi/rr to aD! spccific\isral rmageshould hc .on,Lnrtd as a freedonr to aptll rhisabsracr sel of axiom\to otherLogi.al slsrenr\ b.sidcs EuclLds geomen) Inanlcase. Euclids firstsevcn dcnnitbns mrkc iLclcar drar rl rs irntossibltto dcfin. derr_thmgOn. musL(.rl with someundehned terms 0.Undefi ned 0uantilies. A classof undeiinedelemcn t\ callcdtdi,xr.dero(ed b! Lxtin cafiLalsl,/J a. A classof undetiredelement\ called /nr.r. dcnotedb\ Jrall Lari| lerrcrsd. , .