Chapter 4 Topic 3 Module PDF

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Chapter 4: EUCLIDEAN GEOMETRY OF THE POLYGON AND CIRCLE Topic 3: The Nine-Point Circle and Early Nineteenth Century Synthetic Geometry _______________________________________________________________________________________________Learning OutcomesAt the end of the lesson, you should be able to: 1. d...

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Modern Geometry

Chapt er4:EUCLI DEAN GEOMETRYOFTHE POLYGON AND CI RCLE Topi c3:TheNi nePoi ntCi r cl eandEar l yNi net eent hCent ur ySynt het i cGeomet r y _ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ Lear ni ngOut comes Att heendoft hel esson,youshoul dbeabl et o: 1.defineani ne-poi ntci r c l e;and 2.di s cussandpr ov eTheor ems4. 17t o4. 20. _ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ Lear ni ngCont ent Renewedi nt er esti ndr amat i cext ensi onsoft hecl assi calgeomet r yoft het r i angl eandt he ci r cl ebegani nt heear l yni net eent hcent ury .Pr obabl yt hemostsi gni ficantoft headvancesatt hi s t i mewast heconceptoft heni nepoi ntci r c l e,somet i mescr edi t edt ot heGer manmat hemat i ci an Feuer bachi n1822. THEOREM 4. 16.Themi dpoi nt soft hesi de sofat r i angl e,t hepoi nt sofi nt er sec t i onoft he al t i t udes and t he si des,and t he mi dpoi nt s oft he segment sj oi ni ng t he or t hocent erand t he v er t i cesal lt i eonaci r cl ecal l edt heni nepoi ntci r c l e. A ni nepoi ntci r cl ei sshown i n Fi gur e4. 17.I n gener al ,t heci r cl ei nt er sec t seach si deof t het r i angl ei nt wodi s t i nctpoi nt s.

Thepr oofoft hef ol l owi ngt heor em f ol l owst hi spat t er n: a.Ther ei saci r c l et hr oughA’ ,B’ ,C’ ,t hemi dpoi nt soft hesi des. b.Show t hatpoi nt sD,E,Far eont hi ssameci r c l e . c. Show t hatpoi nt s G , H , I ar eont hi ssameci r cl e . Topr ovet hatD i sont hesameci r c l easA’ ,B’ ,C’ ,i ti spossi bl et oshow t hat DB ' C ' A ' i s ´ ´ ´ an i soscel e st r apez oi d and hence,i nscr i bed i nt he c i r cl e. A ' C ' ≅ DB ' because A ' C ' connect st hemi dpoi nt soft hesi desoft r i angl eABC and t her e f or ehasameasur eequalt ohal f ´ ' connect t hemeasur eoft hebaseand DB st hever t exandt hemi dpoi ntoft hehypot enuseof r i ghtt r i angl eADC whi c hi mpl i est hatDB i shal ft hehypot enuse . ´ Now consi dert heci r c l ewi t h IA ' asdi ame t er .Poi ntB'i sont hi sci r cl e,si nceangl eIB' A' i sar i ghtangl e;poi ntC'i sal soont heci r cl e.Ther ef or e , I l i esont hesameci r c l easA' B' C' . Ther ei s a weal t h of addi t i onal i nf or mat i on about t he ni ne- poi nt ci r cl e and i t s r el at i onshi pswi t h ot herse t sofpoi nt si n Euc l i dean geome t r y .Oneoft hei mpor t anti deasi st he

Modern Geometry

l ocat i on oft hecent eroft heni ne-poi ntci r c l ei nr e f er ence t o sev er alot herpoi nt s pr evi ousl y ment i oned.Thi si nf or mat i on depe ndson at heor em est abl i shed byt heGe r man mat hemat i ci an Eul eri n1765. THEOREM 4. 17.Thecent r oi d ofa t r i angl et r i sect st hesegmentj oi ni ngt heci r c umc ent erand t heor t hocent er . Thel i necont ai ni ngt het hr eepoi nt si scal l edt heEul erl i ne .“ Tr i sect "asusedher emeans t hatt hedi s t anceal ongt hel i nef r om t heci r c umcent ert ot hecent r oi di sonet hi r doft hedi s t ance al ongt hesamel i nef r om t heci r cumcent ert ot heor t hocent er . heci r cumcent er , G t hecent r oi d,and H t heor t hocent er . I nFi gur e4. 18,l et O bet ´ ´ Themeasur eof AH i st wi cet hatof OA ' .Thi si st r uebecauset r i angl es CBI∧OA ' ar e IB = AH si mi l arwi t h ar at i oofsi mi l ar i t yof2 t o1,and because AHBI i sapar al l el ogr am. e si mi l arwi t h ar at i o of1 t o 2;t her e f or e Now t r i angl es GO A' ∧GHA ar

1 OG= GH ,and 2

´ . G trisects OH I tmaybesur pr i si ngt ofindt hatt heEul erl i neal socont ai nst hecent eroft heni nepoi nt ci r cl e,asi ndi cat edi nt hef ol l owi ngt heor em.

THEOREM 4. 18.The c ent eroft he ni ne-poi ntci r c l ei st he mi dpoi ntoft he segmentwhose endpoi nt sar et heor t hocent erandt heci r cumcent eroft het r i angl e.( SeeFi gur e4. 19)

´ ≅ O´A ' , andt Fr om Theor em 4. 17, EH het wosegment sar eal sopar al l e l .I f0'i st hecent erof sapar al l el ogr am t heni ne-poi ntci r c l e,i ti st hemi dpoi ntoft hedi amet er A ´' E .But0AHE'i and t hedi agonal sofa par al l el ogr am,bi secteach ot her .Thi smeanst hat0'i st hemi dpoi ntof ´ OH .

Modern Geometry

Al lt het heor emsaboutconcur r enceconsi der ed sof arr e l at et ot heconcur r enceoft hr eel i nes. Mi quel ’ st heor em,pr ovedi n 1838,i ssi gni ficanti n par tbecausei tconsi der st heconcur r enceof se t soft hr eeci r c l esassoci at edwi t hanyt r i angl e . THEOREM 4. 19.I ft hr eepoi nt sar ec hosen,oneoneachsi deofat r i angl e,t hent het hr eeci r c l es de t er mi nebyaver t exandt het wopoi nt sont headj acentsi desmeetatapoi ntcal l edt heMi que l Poi nt . hear bi t r arypoi nt son t hesi desof Usi ng t henot at i on ofFi gur e4. 20,l et D , E , F bet t r i angl e ABC . Supposet heci r c l eswi t hcent er s

nt er sec tatpoi nt G . I n ci r c l e J, esuppl ement ar y ,and i n I ∧J i ∠ EGD∧∠ ECD ar esuppl ement ary .I nt her es toft hepr oof ,t henot at i on m means ci r cl e I , ∠ FGD ∧∠ DBA ar t hemeasur eoft heangl e.Si nce

m ∠ EGD +m ∠GDF +m ∠ EGF =360, 180−m∠C + 180−m ∠ B+ m∠ EGF =360, m ∠ EGF=m ∠C +m ∠ B=180−m ∠ A . Thi smeanst hat m ∠ A and ∠ EGF ar esuppl ement ar y ,so A , F , G , E ar eon a ci r cl eand al lt hr eeoft heci r c l esar econcur r ent .I ti spossi bl et hatt heMi quelpoi ntcoul d be out si det het r i angl e,i nwhi chcaset hepr oofmustbemodi fiedsl i ght l y .

LEARNI NG ACTI VI TI ES 1.At r i angl ehashow many: a.Ni ne-poi ntci r c l es ? b.Eul erl i nes ?

c.Mi quelpoi nt s

2.Pr ovet hatt hesegmentconnec t i ngt hev er t exoft her i ghtangl eofar i ghtt r i angl ewi t ht he mi dpoi ntoft hehypot enusehasameasur ehal ft hatoft hehypot enuse . 3.Pr ovet hatt her adi usoft heni ne-poi ntci r c l ei shal ft hatoft heci r cumci r cl e .

Modern Geometry

4.Pr ovet hatt heMi quelpoi nti sapoi ntont heci r cumci r c l ei ft het hr eepoi nt sont hesi des oft het r i angl ear ecol l i near . 5.Pr ovet hatt heMi quelpoi nti sont heci r cumci r c l e,t hent het hr eepoi nt sont hesi desof t het r i angl ear ecol l i near .

REFERENCES JAMESR.SMART Cal i f or ni aSt at eUni ver si t y,SanJose

Pr epar edby:PAULJOHN M.AGCAOI LI ,LPT I ns t r uct or...