which are generalizations of Ceva s theorem on the triangle

Transcription

1 Theorems for the dimesioal simple which are geeralizatios of Ceva s theorem o the triagle Kazyi HATADA Departmet of Mathematics, Faclty of Edcatio, Gif Uiversity -, Yaagido, Gif City, GIFU 50-93, Japa Abstract We show other geeralizatios of Ceva s theorem o the triagle to the dimesioal simple tha the theorems give i [2] They are Theorems, 2, 2 ad 3 below Let be ay iteger>2 Let PPP deote a arbitrary dimesioal simple i Eclidea space R, whose vertices are {P 0 } Let deote the stadard Eclidea measre (=volme) o R Let m deote the stadard Eclidea measre o R Let T be ay poit i the iterior of PPP For each iteger with 0, write C {P 0, } For each iteger with 0, let H deote the dimesioal hyper-plae cotaiig all the poits i C For each iteger with 0, let P deote the poit of the itersectio of lie P T ad hyper-plae H For arbitrary fiite poits {V } i Eclidea space deote R, let {V } the miimm closed cove sbset, of R, cotaiig all the poits {V } Sectio Case of ay iteger 2 Recall that PPP deote a arbitrary dimesioal simple i Eclidea space R, whose vertices are {P 0 } Let be ay iteger with

2 0 Recall C {P 0, } Let r be ay positive iteger 2 Let be ay seqece of positive itegers with r { } r ad 2 for all the itegers r For each iteger with 0 2, write E C C Write E C C For each iteger with 0 2 2, write E C C Write E 2 C 2 C Let s be ay iteger with s r For each iteger with s s 2, write E C C Pt a s s Write Eas Ca s Ca s Here we pt a 0 So, we have { E 0 } For each 0 write { S S C, S, S E } The we have We obtai m( {P } ) Theorem 0 m( {P } E ) Proof of Theorem By the same way as i the proof of Theorem 2 i [2] we have m( {P } ) ( {T,P } ) m( {P } E ) ( {T,P } E ) We obtai ( ( {T,P } )) = ( {T,P } E ) 0 0 Theorem is prove Theorem i [2] is the case of r of the above Theorem Corollary of Theorem Let (0) () (2) ( ) be ay elemet of S Rewrite P for P ( ) with all 0 ad P for P ( ) with all 0 The apply Theorem The we have the eqatio for simple P (0) P () P (2) P ( ) 2

3 (We have simple PPP = P (0) P () P (2) P ( ) We ca cosider Theorem for ay (P,P,P,,P,P ) ) order (0) () (2) ( ) (0) We obtai also Theorem 2 Let be ay elemet of S that (0) () (2) ( ) satisfies ( ) for all the itegers 0 For ay 0, pt F, {P 0,, ( )},, {P } F,, ad { S S C, S, S F } The we have,,, 0, m( {P } ) m( ) Proof of Theorem 2 By the same way as i the proof of Theorem 2 i [2] we have m( {P } ) ( {T,P } ),, m(, ) ( {T,P } {P 0,, ( )} ) We obtai, ( ( {T,P } )) = ( {T,P } {P 0,, ( )} ) 0 0 Theorem 2 is prove (Remar We have easily, for ay 0 ) Ay elemet of S is epressed iqely as a prodct of cyclic sbstittios ay two of which have disoit letters So, if we rember {P } 0 sitably for each, Theorem 2 has the same epressio that Theorem has For each dimesioal simple PPP, the mber of the eqatios give i Theorem 2 is eqal to d #{ S ( ) for ay 0 } Namely the mber of the geeralizatios of Theorem type of Ceva s theorem o the triagle to simple 3

4 PPP is eqal to d It is well ow that d ( ) ( )!! 0 Eample Let 3 We have d 4 9 There are 9 eqatios of Theorem type for ay tetrahedro P0P P2P 3 The sbstittios to cosider are the followig As i Theorem 2, we have (023), (032), (023), (023), (032), (032), (0)(23), (02)(3), (03)(2) The cases of (023), (032), (023), (023), (032) ad (032) are treated i Theorem i [] The cases of (0)(23), (02)(3) ad (03)(2) are treated i Theorem 2 i [] Oe may say that there are oly 9 geeralizatios of Ceva s theorem of the triagle to the tetrahedro Sectio 2 Theorem of aother type for ay iteger 2 Let be ay iteger>2 Let PPP be ay dimesioal simple with vertices {P } 0 i R Let T be ay poit i the iterior of PPP Let b be ay iteger larger tha Let {U } b 0 be sch ay seqece of poits i R that U { P 0 } for all the itegers with 0 b, that U U for all the itegers with 0 b ad that U0 U b The let I(, ) deote the itersectio poit of the lie UU ad (the hyper-plae ({ T} {P 0, P U, P U }) ) for each iteger [0, b ] For arbitrary poits P ad Q i R we write d(p,q) PQ Notatios beig as above we obtai Theorem 2 d(i(, ),U ) b 0 d(u,i(, )) Proof of Theorem 2 First we treat the case of b 3 Let 0 be the miimm R -liear 4

5 sbmaifold i R cotaiig T ad all the poits i {P 0, P U 0, P U, P U 2} Let 0 be the miimm R -liear sbmaifold i R cotaiig T, U 0, U ad U 2 Note that the dimesio of 0 is 2 ad that the dimesio of 0 is 3 We have 0 0 is a lie (amely R -liear sbmaifold of dimesio ) Let Q 0 be the itersectio poit of 0 ad the plae cotaiig U, 0 U ad U 2 The the lies UI(,2), 0 UI(2,3) ad UI(0,) 2 itersect at the poit Q 0 So, we ca apply Ceva s theorem to the triagle UUU We have Theorem 2 for b 3 Now assme Theorem 2 is tre for b =some iteger 3 The we cosider the case of b We have U U 0 If U U, this case resolves itself ito the case of b sice I (, ) I (, ) So, we assme U U Let I deote the itersectio poit of the lie U U ad (the hyper-plae ({T} {P 0, P U, P U }) ) By the case of b, we have 2 d(i,u (I(, ),U +) d ) Let be the miimm d(u,i) d(u,i(, )) 0 R -liear sbmaifold i R cotaiig T ad all the poits i {P 0, P U, P U, P U } Let be the miimm R -liear sbmaifold i R cotaiig T, U, U ad U Note that the dimesio of is 2 ad that the dimesio of is 3 We have is a lie (amely R -liear sbmaifold of dimesio ) Let Q be the itersectio poit of ad the plae cotaiig U, U ad U The the lies U I(, ), UI ad U I(, ) itersect at the poit Q Apply Ceva s theorem to the triagle U UU We have d(i,u +) d(i(, ),U )) d(i(, ),U )) Hece d(u,i) d(u,i(, )) d(u,i(, )) 2 d(i(, ),U )) (I(, ),U (I(, ),U )) ) d d Namely d(u,i(, )) d(u,i(, )) d(u,i(, )) 0 5

6 d(i(, ),U ) The case of b 0 d(u,i(, )) (The case of b 2 is trivial) is prove Theorem 2 is prove Sectio 3 Case of 4 The followig theorem is differet from Theorems ad 2 i this paper ad Theorem 2 i [2] We obtai Theorem 3 Let 4 Oe has m(p 0PP 2P 4) m(p 0PP 2P 3) m(p P2P0P 3) m(p P2P3P 4) m(p 0P2P3P 4) m(p 0PP 3P 4) m(p P0P3P 4) m(p P2P0P 4) m(p 2P3P 0P 4) m(p 2P3PP 4) m(p 3P4P0P ) m(p 3P4P0P 2) m(p 4PPP) m(pppp) m(p 2P0PP 4) m(p 2P3P 0P ) m(p 3P0PP 2) m(p 3P4PP 2) m(p 4PP 2P 3) m(p 4P0P2P 3) Proof of Theorem 3 We se If a: b: c: d t: : v: ww: : y: z, the a: b: c: d tw: : v y: wz Usig that ad comptig volmes of 4 dimesioal simplees we have m(p 0PP 3P 4) : m(p 0PP 2P 4) : m(p 0PP 2P 3) : m(p 0P2P3P 4) (P0P 0PP 3P 4) : (P0P 0PP 2P 4) : (P0P 0PP 2P 3): (P0P 0P2P3P 4) (TPPPP 0 3 4): (TPPPP 4): (TPPPP 3): (TPPPP ) = (TP P P P ) : (TP P P P ) : (TP P PP): (TPPPP) By the same method, m(p P2P0P 3) : m(p P2P0P 4) : m(p P2P3P 4) : m(p P0P3P 4) = (TP P P P ) : (TP P P P ) : (TP P P P ) : (TP P P P ), m(p 2P3P0P ) : m(p 2P3P0P 4) : m(p 2P3PP 4) : m(p 2P0PP 4) = (TP P P P ) : (TP P P P ) : (TP P P P ) : (TP P P P ), m(p 3P4P0P ) : m(p 3P4P0P 2) : m(p 3P4PP 2) : m(p 3P0PP 2) = (TP P P P ) : (TP P P P ) : (TP P P P ) : (TP P P P ), ad m(pppp): 4 m(pppp): m(pppp): m(pppp) = (TP PPP ): (TP P PP ): (TP PP P ): (TP PP P )

7 For simplicity, write A (TP P P P ), B(TP P P P ), C (TP P P P ), D(TP P P P ) ad E (TP P P P ) The the left side of the eqatio i Theorem 3 is eqal to CD DE AE BA CB A B C D E AB BC CD DE EA A B C D E Theorem 3 is prove Sice Theorem 3 holds tre for ay 4 dimesioal simple, we have Corollary of Theorem 3 Let 3 4 (0) () (2) (3) (4) be ay elemet of S 5 Oe ca replace P by P ( ) for ay 0 4 ad P by P ( ) for ay 0 4 i the eqatio of Theorem 3 Theorem 3 shows that Theorems ad 2 i Sectio do ot cover all possible geeralizatios of Ceva s theorem o the triagle to dimesioal simplees with 4 Refereces [] K Hatada, Geeralizatio of Ceva s theorem o the triagle to the tetrahedro, Sci Rep Fac Edc Gif Uiv (Nat Sci), 3, (2007), 7-9 [2] K Hatada, Geeralizatio of Ceva s theorem to the dimesioal simple, Sci Rep Fac Edc Gif Uiv (Nat Sci), 32, (2008), 9-2 7