strt: I ths pper the trler oorte ple s stue s geerle to the -esol ule spe -ler oorte sste s estlshe The oplr theore of -pots the ourret theore of -hpe

Transcription

1 -ler Coorte Sste Its ppltos Te Meer : Yu Xhg Teher : L Xhu The fflte Hgh Shool of South Ch Norl Uverst Pge - 3

2 strt: I ths pper the trler oorte ple s stue s geerle to the -esol ule spe -ler oorte sste s estlshe The oplr theore of -pots the ourret theore of -hperples re estlshe -ler oorte spe The uthor propose the oepts of opg spe horotl le trler ple horotl ple ts opg spe The uthor estlshe the prllel theore orete le theore trler ple et Kewors: trler oorte; -ler oorte; opg spe; orete le; hperple Pge - 33

3 Itrouto I ple geoetr we ofte eet the prole o le pssg through three pots or eple the prole of uler le: BC H (orthoeter G (etro O(rueter re ll oller I sovere tht the vlue of thr-orer etert s equl to ero osstg of stes HB HBC HC GB GBC GC OB OBC OC fro HGO to three seles BBCC respetvel Let B C e three gles of BC let R e rus of rurle of BCThe HB GB OB HBC GBC OBC HC GC OC R os os B Rs s B 3 R osc R os B osc Rs Bs C 3 R os R osc os RsC s 3 R os B I forwr put questo to three pots ple re oller f ol f tht the vlue of thr-orer etert s equl to ero osstg of stes fro the three pots to three seles of trgle ple? Theore (Coller theore o three pots: Three pots ple s oller f ol f tht the vlue of thr-orer etert s equl to ero osstg of stes fro the three pots to three seles of trgle BC C B BC Nel BC BC C C B B g Proofs show gure Let XOY e the esrtes oorte sste ple wth B s the org wth BC s the X-s Let e the esrtes oortes of Let S e the re of BC BC= BC= BC= S R s s Bs C If oe of s ot equl to ero the BC ot BC But Pge - 34

4 BC B so B B ot ot B BC ot BC ot B s s( B s B s B BC Slrl B BC ot BC ot B s B BC B BC ot BC ot B s B BC S BC B C S BC B C C S BC B Three pots re oller f ol f the re etere the three pots s equl to ero s oller B s B B s B B s B BC BC BC ot ot ot B B B BC BC BC s B B B B BC BC BC 4S s B B B B BC BC BC S BC S BC S BC B B B BC C B BC C B 4R s s B s C If re ll equl to ero the re ll equl to ero re ll sele BC If two of s equl to ero for eple the the vlue of BC C B thr-orer etert s equl to ero BC re ll sele BC If oe of s equl to ero for eple the the vlue of 3 Pge - 35

5 thr-orer etert s equl to ero re ll oller s BC re ot sele BC Q I put questo to further: I -esol ule spe + pots re ll se ple (hperple f ol f tht the vlue of (+th-orer etert s equl to ero osstg of stes fro eh of the + pots to + hperples of -esol sple respetvel? Ths pper strts fro the reserh to ths questo I two-esol ule spe I te three stes fro oe pot to three seles of the trgle s the pot s oorte I sovere ths of oorte self However I fou lter tht referee [] t s lle s trler oorte I stue eepl trler oorte of two-esol ule spe geerle t to geerl ule spe I estlshe the sste of -ler oorte --esol ule spe propose -ler oorte of -esol -esol-esol --esol hperple so o -esol hperple s pot -esol hperple s le But for oveet esrpto the re ll lle s hperple or ple I ths pper eotes -esol ule spe The oepts o the ler struture ler epeee ler epeee ste gle slr prout vetor prout e prout etert r et e fou orr tetoos of ve lger I retl te these oepts wthout eplto I ths pper s postve teger (-esol ule spe s ot susse ths pper stlshet of -ler Coorte Sste efto (The sple -esol ule spe []: Let P P P ( e pots of ler epeee -esol ule spe el the vetors The Crtes oorte of p P P P s gve s p p p re ler epeet The set of pots 4 Pge - 36

6 X X P s K-esol sple se o the vertes P P P It s epresse s otto p ( P P P If the s (K--esol sple s lle the ple (or hperple orrespog to the verte P The K-esol sple hs K+ vertes K+ ples C eges efto (the -ler oorte sste the -ler oorte spe: Bse o - esol sple se o stes fro the pot to ever hperple of the sple we ostrut the oorte sste We ll the s -ler oorte spe or -spe short We ll the oorte sste s -ler oorte sste or -oorte short efto 3 (The solute oorte of pot -spe: The solute oortes re ostrute stes fro oe pot K to hperples of --esol sple -spe K K K K g eotes the solute oortes If the pot K the orrespog verte re the se fl wth the orrespog hperple the s postve otherwse K s egtve K g 3 -esol sple s show g g 3 s trgle ple I g K = K =3 K = K(3 I g 3 K =- K =3 K = K (-3 Suh s K g 3 K re the fferet fl of sele BC so K s egtve efto 4 (The reue oorte -spe: There s pot K -spe If K : K : K : : the the reue oorte of K pot s 5 Pge - 37

7 If the rto of K K K s eterte the the loto of pot K s eterte Beuse Let -esol sple e eple show g g 3 : : : the K K K K K K : Suppose the res of BCBKCCKKB re S S S S 3 respetvel Let e the legths of the three seles The lulte re s orete If the heght s egtve the re vlue s lso egtve S = BC K = S = C K = S = 3 B K = S= S + S + S 3 S = Therefore K S SS S S S 3 K = S S I the epresso of K K K K S = S = K the uertor eotor re hoogeeous of Therefore the re ot ffete proportol rese or erese of To rw the prllel le of BC the ste of whh to BC s K (the ste s orete so t s ol to rw the prllel le of B the ste of whh to B s K the tersetg pot of two prllel les s K Therefore the uque pot K e lote the reue oorte ( If t s ssue tht K represets the pot ftel fr The two-esol sple orr ple s the trgle The three-esol sple orr spe s the tetrhero To -esol sple (+ vertes + hperples: K : K : : K : : : K ( The volue of sple s equl to the su of + volues osstg of K to + hperples 6 Pge - 38

8 V V ( V = VV V = V ( K = V K ( (Ths prgrph esres the oputtol etho of V ( Ths prgrph s sur of relte otet o the referee we lst the results ol V s volue of the sple s re of --esol hperple orrespog to verte P To the eterte sple V ( re ll ostt B Brts forul V! s s -esol gle of vetors P P ( wth P s the org s os( os( os( os( os( os( os( os( os( os( os( os( os( os( os( os( os( os( os( os( os( os( s os( os( os( os( ( j represets the two-eges gle of P P P Pj wth P s the org p p j os( j p P P ( We prove s ( p ( p j There s rusre hpersphere to ever -esol sple Its rus ( p R to -esol sple P P P stsfes 7 Pge - 39

9 R ( P P P j represets the ste etwee vertes ( P P P P P j ( P P P ( P P P The re of --esol hperple orrespog to the verte gle s ( the s (R (! I the epresso of (R s ( (The se theore (! K ( the uertor eotor re hoogeeous of ( Therefore lote K pot ol If we ssue tht K represets the pot ftel fr I ths pper f there s o spel eplto we opt the reue oorte s the -ler oorte I trler oorte of for : The -ler oorte e reue oo ftor or eple K=(48 e reue s K=(4 pots Bse o BC the followgs re the trler oortes of soe spel Vertes: BC B C Cetro: S S S G s s B s C H R os B oscr os oscr os os B os os B osc Orthoeter: Crueter: O R os R os B R osc os os B osc Ieter: I r r r 8 Pge - 3

10 M MC MB R s s C R s s B s B s C pot of BC: efto 5 (opg spe: the opg spe s Crtes oorte spe orrespog to -spe The pot of -ler ( oorte spe orrespos wth the pot of Crtes oorte ( spe The forer s -ler oorte the ltter s Crtes oorte The -spe s --esol ule spe ut the opg spe s -esol ule spe Through ths orrespoee we stu proles of --esol spe -esol spe Slrl we stu proles of -esol spe --esol spe efto 6 (The orrespog relto etwee -spe opg spe: I -spe the pots ( for ll re se pot I opg spe for ll ( represets le pssg through the org el orete vetor We eote t wth the otto L K Slrl ever orete vetor L K of the opg spe orrespos pot K of -spe We estlsh oe-oe p etwee ll pots -spe ll orete vetor opg spe K f : K L The proles re oveet for tretet whe pots of -spe re trsfore to orete vetors of the opg spe Oe the sple of - spe s eterte the the opg spe orrespog wth t s ol efto 7 (-esol prllelotope[]: Let p p p fro the org O e ler epeet vetors -esol ule spe The set K X X t p t prllelotope wth the org O s verte wth If the Crtes oorte of p s p re lle s -esol p p s eges for the the volue of -esol prllelotope e epresse etert 9 Pge - 3

11 The e prout of p p p stsfes ( p p p p Therefore the volue e represet the e prout of vetors 3 The oplr theore -ler oorte of hperple Theore 3 (The oplr theore of + pots: I +-spe (-esol ule spe + pots re ll se hperple f ol f tht the vlue of (+th-orer etert osstg of stes etwee + pots ever hperple of -esol sple s equl to ero el tht the vlue of (+th-orer etert osstg of + +-oortes s equl to ero Proof: Let the -esol sple -esol ule spe e p ( P P P Let the re of the hperple orrespog to P e Let the volue of the sple e V Let the +-oorte of th pot + pots reltve to the sple e ( The j j j V V Ths s rell ler equto S V S V V V V If ll pots ( re se hperple the re ll ot equl to ero Pge - 3

12 et ( r( ( el + row vetors tr s ler epeet[3] + pots re ll se hperple [3] re ll ot equl to ero et ( represets the etert of tr r ( represets r of tr Q Theore 3 (The oplr theore of + pots: I +-spe (-esol ule spe + pots ol f tht the orrespog orete vetors L re ll se hperple f L opg L spe (+-esol ule spe re ll se hperple Nel the e prout = L L L L Spell f the ths theore s rell oller theore of three pots Three pots trler oorte ple s oller f ol f the orete vetors L L L opg spe orrespog to trler oorte ple re ll se ple Proof: org to theore 3 re ll se hperple l l l l l l l l l l l l l ( I +-esol opg spe the vlue of e prout o L L s the vlue of etert osstg of these vetors It represets L the volue of +-esol prllelotope L L L re ll se hperple f ol f the e prout o L Nel L s equl to ero L Pge - 33

13 l l l = l l l L L L L l l l If vetors + pots re ll se hperple the the e prout o + orete L L opg spe s equl to ero Ve vers f the L e prout o + orete vetors L equl to ero the + pots re ll se hperple Q re ople L opg spe s L L L L L The theore 3 s se s theore 3 rell The theore 3 els wth the proles fro -esol spe If = theore 3 els wth the proles fro two-esol ple The theore 3 els wth the proles fro +-esol opg spe If = theore 3 els wth the proles fro three-esol spe The theore 3 oets the pots of -esol spe wth the orete vetors of +-esol opg spe The theore 3 proues ore ssotos etwee two spes proves us wth wer eeshot The etert osstg of orete vetors represets the volue of -esol prllelotope osstg of les pssg through org Beuse the -ler oorte s flele the volue ot e eue -ler oorte Though we ot eue the volue the vlue of etert s equl to ero represets tht the volue s ero el ll pots re se hperple It represets the relto etwee pot le ple whe volue s equl to ero I g (trler oorte spe let trler oorte e ovg pot le If the trler oortes of re the the le equto pssg through trler oorte for s Pge - 34

14 fter reuto I -spe there re - ow pots I - pots the -ler oorte of th pot s X ( - pots re ll se --esol hperple I ths --esol hperple there s ovg pot the hperple equto pssg through - ow pots -ler oorte for s fter reuto s lger re suetert of Its epresso wth X X X s ( ( ( ( The seo susrpt of eleets etert lues to oule If ( j ( the j j If ( j ( the j s equl to reer of ( j ve efto 3 (The -ler oorte of hperple: I -spe (--esol ule spe there re - ow pots Ther -ler oortes s for We ll s -ler oorte of -esol hperple pssg through oe pot el the -ler oorte of pot ll re - The 3 Pge - 35

15 vetors L L L re orete vetors of orrespog opg spe We ll s -ler oorte of -esol hperple pssg through two pots el the -ler oorte of le j ( ( ( ( j( j( ll re C The orete vetor s orrespog to the hperple pssg through the org opg spe We ll s -ler oorte of -esl hperple pssg through three pots el -ler oorte of ple j ( ( 3 j( 3 ( 3 ( j( ( ( ( j( ( ll re 3 C The orete vetor s hperple pssg through the org opg spe orrespog to the We ll the vetor prout of ll - pots s -ler oorte of - esol hperple pssg through - pots ll s oe the vetor prout L L L ( s orete vetor orrespog to the hperple through the org opg spe ( ( ( ( If pot s lote the hperple pssg through the - pots the ( If there s ot eplto ths pper the -ler oorte of hperple s -ler oorte of - esol hperple pssg through - pots 4 Pge - 36

16 -spe I opg spe -oorte of hperple pssg through the org s orl vetor of ths hperple Let e hperple of -spe Let e hperple pssg through the org of opg spe orrespog to The We opt the rou rets for -oorte of pot the squre rets for -oorte of hperple I the -esol ple (the trler oorte ple let XY e two pots of le X Y X Y s the trler oorte of ths le oetg X Y I 3-esol opg spe the vetor prout s L X LY It presets orl vetor of ple etere two orete vetors Therefore X Y= I -spe (- esol ule spe let -oorte of - hperples e l - hperples s suh tht The the tersetg pot of el ( ( I two-esol ple let trler oortes of two les e l l respetvel 5 Pge - 37

17 The the tersetg pot of two les s the soluto of el efto 3 (The -oorte of tersetg pot of - hperples: I -spe(--eso let -oortes of - hperples (--eso e l hperples s The -oorte of tersetg pot of ( ( I trler oorte ple (two-esol ple the trler oorte of tersetg pot of les l s If l X l the X I opg spe t s rell trler oorte of tersetg le L X l Beuse L X so l X L L X L l L So L L L l X L l X l X I two-esol ple the vetor prout two trler oortes o two pots s trler oorte of le oetg two pots The vetor prout two trler oortes o two les s trler oorte of tersetg pot two les 4 The ottos reltoshp -ler oorte sste or oveet esrpto the upper letters suh s M represet pot The 6 Pge - 38

18 lower letters suh s represet le The gree letters wthout susrpt suh s represet the hperple Two upper letters suh s represet le oetg two pots Severl upper letters suh s B represet the hperple oetg severl pots The upper letters wth rrow suh s L represet orete vetor M opg spe The gree letters wth susrpt suh s represet the hperple pssg through the org opg spe The -oorte of pot orete vetor opg spe re eote rou rets The -oorte of hperple s eote squre rets efto 4 (The -oorte of hperple pssg through the org opg spe: The orete vetor of hperple opg spe s ts orl vetor The p oe oe etwee le pssg through the org hperple pssg through the org op spe e estlshe g : L M L s perpeulr to M = If L = M the represet -oorte of The equto epresses the hperple opg spe Let X e pot -spe Let e hperple -spe It s evet tht L X X s true f ol f the slr prout L L X L efto 4 (The opg relto the opg pot the opg hperple: Let the hperple -spe e orrespog to pssg through the org opg spe Let the orl vetor of e L M Let L M opg spe e orrespog to pot M -spe M= M = =[ ] M s lle s opg L = pot of s lle s opg hperple of M M eoe opg relto or eple two-esol ple the trler oorte of verte s =( the trler oorte of sele BC s =[] We ll s the opg pot of ll s the opg le of eoe 7 Pge - 39

19 opg relto I opg spe L s OX s s OYZ ple B=[] C=[] re gve Beuse terset so = =[] []=( Theore 4 (The ourret theore of hperples: I -spe (--esol ule spe hperples re ourret f ol f the e prout of -oortes of hperples s equl to ero I other wors the vlue of etert osstg of -oortes s equl to ero (I two-esol ple: The three les re ourret f ol f the vlue of etert osstg of trler oortes of three les s equl to ero Proof: I -spe tht pots X X X - re oplr s equvlet wth tht orete vetors opg spe L X L X L re oplr Let the X hperple etere the orete vetors opg spe e Its orl vetor s L K The hperple -spe orrespog to K L s hperple etere X X X - -spe Its opg pot s K L L X X L X L L X X L X L K L L K X LK I -spe pots X X X - re oplr the opg hperples of pots X X X - re ourret K I hperples of -spe the -oorte of th hperple s l l l l The tersetg pot s l l l l l l l l l the l l l l l l Ol f l l l the soluto s o-ero Q 8 Pge - 3

20 5 The ppltos of trler oorte ple geoetr I ths setothe trgle eotes wth BC three gles eote wth BC The orrespog seles eote wth R s rus of rurle of BC r s rus of sre rle GOHI re etrorueterorthoeter eter respetvel ple I BC O s rueter G s etro H s orthoeter the trler oorte of le oetg OH Prove tht O G H re oller (uler le theore Proof: Costruts trler oorte sste se o BC The trler oortes of pots O G H re respetvel O os os B osc G H s s B s C os os B os C Let e trler oorte of le oetg OH the os B os C os C os os os B =O H= os B os C os C os os os B os B os os B osc C os C os os os osc os os os B B s os s( C B s B os B s( C s C os C s( B os os B os C os os B os C os os B os C s s( C B s B s( C s C s( B LO L G L H os os B os C os ( s s B s C s B os C os B s C os os B os C s C os B osc s B s( C B os ( os ( s B s C os B osc s Bs C os B osc s B s C ( ( s B s C os B osc s C osc s B os B Therefore G O H re oller Q ple Prove Meelus theore s show I g 4 re oller B C f ol f C B Proof: Costruts trler oorte sste se o BC The trler 9 Pge - 3

21 oortes of pots re respetvel ( B s B s ( C s C s ( CsC Bs B L L L Bs B CsC s CsC s Bs B g 4 = B C s s B s C B C s s B s C re oller B C s s B s C( B C B C Q C B ple 3 Prove Cev s theore s show I g 5 BC KPQ re respetvel pots o seles BCC B or o ther ete le KBP CQ re ourret f ol f BK CP Q SSS (gle for: SSS KC P QB Proof: Costruts trler oorte sste se o BC gle for K ( S S = g 5 K= K=( ( s s s s slrl BP s s s s CQ The e prout of K BP CQ s s s ( K BP CQ s s = s s s s s s s s s s s The K BP CQ re ourret ( K BP CQ s s s se for K KC s C BK s B Pge - 3

22 K K ( KC s C BK s B= BK s B KC s C Slrl BP Ps CPs C CQ Qs QBs B The e prout of KBPCQ s BK s B KC s C K BP CQ P s CP s C Q s QB s B = s Bs CQ BK CP P QB KC s Q BK CP K BP CQ re ourret Q P BQ CK ple 4 Prove esrgues theore s show I g 6there re three projetg les fro K B C Les BC re tersetg o O Les C re tersetg o N Les B re tersetg o M Prove tht pots MNO re oller Proof: Costruts trler oorte sste se o BC Suppose the trler oorte of K s g 6 Beuse s o K the Slrl B B re tersetg o M M B = = ( ( Slrl N ( ( O ( ( L M L N L O ( ( ( ( ( ( ( ( ( ( ( ( Pge - 33

23 Therefore M N O re oller Q ple 5 Costrut trler oorte sste se o BC K s pot trler oorte ple Suppose the trler oorte of K s the K s o rurle of BC f ol f Here re ot equl to ero (If K s oe of BC re equl to ero Proof: eesst s show g 7 KBC= BK Rs( BK s Rs( s K s Rs( B s g 7 BK s( B Rs( s( B K Rs( s Rs s( B Rs( Bs( s( B s( s The s s( B s Bs( s C s os( B os( B os( B os( B os( C os( C = Suffe The stes etwee K three seles: KBC= KBC KBs KB KB s( B S KBC KC The re of trgle s Beuse KB 4R s s BsC R KB(s s sc s( B Rs B S R s s BsC = the Rs RsC KBs KBs( B 4R 4R s B s s BsC R KB(s s sc s( B Pge - 34

24 3 fter reuto s( R KB Therefore K s o rurle of BC Q ple 6 Prove Psl s theore The three tersetg pots fro three-pr(opposte seles of hego sre rle re oller Proof: s show g 8 Costruts trler oorte sste se o BC B B B Slrl C C Suppose P s tersetg pot of B C the C B P Slrl Suppose Q s tersetg pot of C Q Suppose R s tersetg pot of B R R Q P L L L org to eple 5 euse re ll o rurle of BC so The re ll o ple of opg spe g 8 Pge - 35

25 Therefore LP L Q L R P Q R re oller Q ple 7 s show g 9 the trgle osstg of three tget les o three pots BC of rurle of BC s Prove tht BC re ourret We ll ths pot s trpo eter W The trler oorte of W s W= W s = L = sb sc g 9 Proof: Beuse =B so the rto of stes fro to CCB s C : s CB = s B : s ll pots o le C e epresse s s s B Slrl ll pots o le e epresse s s B s C ll pots o le B e epresse s s s C B C = s B s C s C s B C C = s B s s s B = s C s B s B s W = s C s B s B s s B s C C s Slrl B s s B sc W So C B re ourret W Q efto 5 (The trpo eter: We ll W= W L = s s B s C= s the trpo eter the trler oorte ple efto 5 (The horotl le horotl ple: ll pots of opg le w of the trpo eter W re o ft fr Beuse w s 4 Pge - 36

26 perpeulr to L W opg spe the ple epresse w s ll pots ( of w re stsfe The ste fro K to BC s K s Whe K s ot sgft So ll orete vetors o w hve ot orrespog pots trler ple We ssue ever pot of w s o ft fr trler ple So the le epresse w s le ft fr we ll vsull w s horotl le ll s horotl ple Costruts trler oorte sste se o regulr BC The pot K=( s o ft fr w Let e les o trler ple Let e M M M N N N orrespog ples pssg through the org opg spe The tersetg le L of K pssg through the org s est ertl The pot K s the tersetg pot of trler ple orrespogl If re prllel the hve ot the tersetg pot trler ple But opg spe ll re pssg through the org the tersetg le pssg through the org s est ertl We ssue tht the tersetg pot s o ft fr whe re prllel el the tersetg pot s o w So f the tersetg le K L of s o w the re prllel Theore 5 (the theore of prllel les: re les trler ple re prllel f ol f the e prout of w s ero Proof: Suppose K If re prllel the the tersetg pot s o K w ft fr The e prout ( w= L L The LK w K w If the tersetg pot of s o ft fr the re prllel If re prllel el the opg pots MNW re oller the the e prout of w s ero el ( w= Q efto 53 (The orete le: If re prllel the gles etwee 5 Pge - 37

27 sele BC respetvel re se trler ple The tersetg les etwee horotl ple w re se opg spe If the le trler ple s orrespog wth the ple pssg through the org opg spe the tersetg le of w s le w we ll w s the orete le of w w w s le pssg through the org opg spe t s o w Therefore w hve ot orrespog pot trler ple (o ft fr the le reto of trler ple s epeet wth w Theore 5 (orete le theore: Gve w the the slope tht respet to sele BC of BC s s B os B (tget of gle Proof: s show g s le trler ple rws BK prllel to If K KB= the the slope of respet to sele BC s the slope of le BK respet to sele BC The trler oorte of BK s BK B K org to theore 5 ( w BK w w Nel g s( B s B ot os B s s B t Q os B org to setr the slopes of respet to sele C sele B of BC respetvel re 6 Pge - 38

28 s C t osc s t os w Theore 53 (Theore of the gle etwee le le: If the tget of gle etwee s ( s B ( os B Spell re prllel re perpeulr ( os B w ProofLet e gle etwee sele BC Let e gle etwee sele BC The gle etwee s So t = s B t = os B s B os B The t t t( t t ( s B ( os B Therefore re prllel re perpeulr ( os B Q org to setr we erve esl t( ( s C ( osc ( s ( os re prllel re perpeulr ( os C re prllel re perpeulr ( os ple 8 (Ru Olp test questo 5 Let R e rus of 7 Pge - 39

29 8 rurle of BC O s eter Let r e rus of sre rle of BC I s eter IO G s etro Prove: IGBC f ol f = or +=3 Proof: Costruts trler oorte sste se o BC etro G= eter I=( trpo eter W sele BC BC=[] G I IG Orete les: w BC w IG IGBC ( os B (reue fter susttuto os B os ( ( B ( (fter susttuto B os ( 3 ( ( or 3 Q Rer: The uthor oes eerses of geoetrl questo proess of stue trler oorte sste Beuse the ltto of spe there re o ts lsts Here re ol few fous questos thetl hstor Olp questo The oo etho (geoetrl for uler le questo s: Coets H rws O perpeulr to BC It s prove provg GH resele wth GO The oo etho for Meelus theore s: rws prllel le wth pssg through It s prove reselg rto The oo etho for esrgues theore s tht Meelus theore Cev theore re pple tes It s ver oplte t ls of esthet feelg These ethos re Pge - 33

30 the le setr of BC The vtge of trle oorte sste ple s tht we ot ee ulr les erl to solve geoetrl prole I for the epressos of B C re le setrl eutful The trler oorte sste proves us ew vewpot to el wth geoetrl proles fro trler oorte opg spe Puts plr proles to spe Solves plr proles spe The trler oorte sste proves us ew vewpot to vew the reltoshp etwee pot Le ple If we re eperee pple ths of etho proles e solve for ver ref oveee 6 The ppltos of -ler oorte sste ult-vrles regresso lss I sttsts the ult-regresso lss s ofte e wth the help of oputer or eple The sthet gs of stuets le shool re relte wth r of orl euto phsl euto ll ourses Its regresso epresso s or p p p ( s r of sthet gs re r of orl euto phsl euto ll ourses I ult-regresso lss the lest squre etho s oo etho Here tes etho of -ler oorte sste There re sple stuets the lss There re eg ees to ever stuet The vlue of full r s > I ult-regresso lss stuets re osere s pots -spe ever r s osere s oe of -ler oortes t frst stuets re ve to soe groups There re stuets ever group org to otorsll re C groups ver stuet s C groups Let C 9 Pge - 33

31 To jth group j j j j or j p p p j ( ( ( ( j ; ( re -ler oortes of --esol hperple j j j j The e vlues j j or p p p Let p the or p p Here If s ver gret the oputtol lo of j s ver gret for C j ; So the oputtol etho ust e ofe If s ve etl the let If s ot ve etl the let The lst group hve ot stuets (suppose q stuets lst group q s equl to reer of ve q< ll re groups ver stuet s ol oe group To frotl - groups -ler oorte of - esol hperple s opute To lst group -ler oorte of q--esol hperple s opute I groupg groups e rrge t ro e lso rrge org to level prple 7 The further proles to stu ojeture To the sste of -ler oorte the stu strts justl There re uh wor to o 3 Pge - 33

32 The followgs re soe proles the sste of -ler oorte tht s osere s further stues the uthor Oe pples -ler oorte sste to oputtol geoetr There re reltve proles o pot le ple suh s how to sert two le segets re tersetghow to sert pot s le? how to sert le s polgo? so o Cojeture: fter pplg -ler oorte sste the rthet etho oputtol geoetr e reue Two pples -ler oorte sste to ler progrg The restrt o of soe ler progrg proles s sple fter pplg -ler oorte sste the rthet effe of ler progrg e prove? Cojeture: fter pplg -ler oorte sste the rthet effe of ler progrg e prove Three I fgure solute trler oorte the legth of stsfes os s B s C ( os B s s C ( os C s B s ( os s B s C I reue trler oorte the legth of stsfes s ( s B ( s C ( s ( S ( ( s s Bs C Whether pplg solute or reue trler oorte the epressos of legth re ll oplte here s trler se I geerl -spe the epressos s ver ore oplte efto 7 etr etr spe[4]: (X s etr spe X s set s etr o X s futo to efte X X stsf followg four os to ll X 3 Pge - 333

33 s rel uer fte o-egtve (= If ol f = (=( ((+( org to ths efto stues of etr uer whh the epressos of legth -spe s sple setrlesl to opute our I -spe how to opute the gle of two hperples the esos of whh re fferet? ve The orete le theore trler ple e geerle to -spe? efto 7 Ier prout er prout spe[4]: The rel vetor spe X whh er prout s efe s lle s er prout spe The er prout s p fro X X to slr o K of X To ll pr X slr s orrespog wth t eote Stsfes = org to ths efto stues of er prout uer whh the epressos of slope -spe s sple setrl esl to opute The legth gle re two fouto stoes the ule spe The sple setrl epressos oputtol ethos o legth gle e -ler oorte sste to pl powerful role geoetr 3 Pge - 334

34 Referees: []Hug L Lu Hogwe ltl Geoetr Shgh utol Pulshg House p7 []Sheg Weu Itrouto To Sple---the stu o the ult-esol geerlto of trgle Hu Norl Uverst Press p49-5 p6-89 [3] lger Group Of lger Geoetr Stff I Mthets eprtet Of Bejg Uverst ve lger Hgh utol Pulshg House 987 [4] rw Kresg Itroutor utol lss Wth ppltos Bejg vto ollege Press 987 p-3 p8 [5]Sh Zu Tehque Of ltl Geoetr Ch Teholog Uverst Press 9 33 Pge - 335