Toán học và tuổi trẻ Số 122 (6/1981)

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Toán học và tuổi trẻ Số 122 (6/1981) trình bày về mở rộng khái niệm đường tròn Ơle và đường thẳng Ơle cho đa giác nội tiếp; đường thẳng đồng quy; công thức nội suy; khái niệm trọng tâm và ứng dụng trong hình học. Mời các bạn tham khảo.

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Nội dung Text: Toán học và tuổi trẻ Số 122 (6/1981)

l' +r 't tn$ H*r 'roAfi fItlC ylEr xru 8ACI RA ,1, rn,{Nc MOr xy Chri ni:ifni, nOUYtX cimm roln yho kg tdta loan HoANc cn0nc Trr,r r&: 7{) 'Irlrr IIung tlgo Hra N6i - [)5y rror : 52825 ,H# B$n;G 8.t-{Ar NIFM OUGNG TRON CLE YA SLTSNS THANC OLI CHO DA GIAC NOI rfiP l I t Cle ban thil mdn I ' f inh chflt .1 : C6c ilulng trdn co tArn lin lugt I Kh{i nignr du0ng trr)n C_Ile vA.itudng lhf,ng l*- tl t, llz, H3, vi bin ktnlr b{lg bin kinh I - dA etugc xAy drng d6i Ote cfia itudog trr\n ngo4i ti6p cirer tunr-giec..{rA.l,le i 'roi tarn giue. Droi sE gdc tlQ cfr* verto, & bAi ndy t6i xin gi&i thi6u clt rrhau lai m6t fii$m chung H chi.h IA m0t cdeh nr,l rdng cA,: khAi ni0m niy d6i v6i tr-gc tflm cria ta nr gilc AtAzAi. da gi{c nOi ti6p. Tru6c h6t ta hiy nhlu lri (Ctc tran hfly ttr chring mintr ba tiull ch{t trCn) lrudng hgp iam gi6c ir g6c ilQ ndy. D6i vdi tarn gi{c -lr^4e.{s, t.{c barr d& dang Biy gio ggi E tAdilm Bt0a cua OH, thd tbi " shung minh iluocrngay hai tinh ch{t sau tliyl 0- = (tl?) (Otr+ 0Az + OAil (t) 'linh ch?it 1; EiE:r ki€n c&n vd drl dB r', lf CAc bq.n de bi6t rludng trdn tim E bin kinh trong tAm eria tam giAc :ltdzCs li : t hfrng ltia bAn kinh cria clu&ng trdn ngoai ti6p +-.-4 rlitl = (l/3i(o.{r + oAz * ii-13) trm gi6c rl1.{rds di qua cAc trur,g di}m Mr, }Is, il} 'i113 cria cic eanh .,12.43, ,.11.131 At.4z ; cAc tr,rng di6m vdi S la {6m d*dng trdn sgoai ti6p tam gi&e (.r, Cz, /i3 crle cAc doao t'hing l/.tr, uaz) tt .lt; 1tt.{?As. c{c ehla ilrrdng vu6og g6c Br, Bz, BJ n+ tu I Tinh ehdt 2: $i6u"kien ciin v{ tlrl i[€ J{ tt ele dinb dr, .1t, ,{s xudng cic larih a6i A;en. trr:e lAm cria tam gi{c .{t;{rrlt l$ : During trdn ni.r. duoc gpi Id drbng trin g dilm, + .< lray cdn ggi li dubng trdn Ote Ifra tarn gi{c Olt*OAr*0Az*0.{g, (2) .{til2.l s. r&i 0 l*m tAm iludng trdn ngoli lilp eia tam Ngo&i ra, do (r), (Z); (B) n€n 4 (li0m o, M, H, E gi{c.41r13d3. In th&ng hlng vA ilrrdng th&ng ili gua 4 rtigrn -r\.oAi ra ndu gcri //r. ltt, Ita l6n lugt l& c{e nny ggt lA iludng thing Ole crla taur giitc AtAzAs. tli0m il6i xrlng cria t) qLra rlrAl, .ls.{.r, .4r,{3 thi Btroe ([Au ta thii m& rOng cjte khAr ni{nr tr6n ta s€ c6 tinh ehdt: cho m$t tu girc nQi ti{p.
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'\:di c 'tim ra A = 462! = 213444. =,1, Vdi c rr2, tim ra ,{ = 962r = 925444. 2) d =. 10a * 2, Bing ,c{c phep bi6n t18i tuong L lu nhu tr€n, te s€ cd A-500cr38. 'la cd 105 (.42 = (S00c * 3s)z < 106 'Bli t/lf .+ loo/;l { iooc + 3s < to3 ,+ c ,,- 1. 9. HfrA tixt tdt c&. adc s6 ehinh Khi do dr - JitB' = 28g.lq l. phuong cd sdu chti sd md tdn eing eia nd ld ba Vfly cd 3 s6 th6a rlEn dE ra Ii: 2131.1{. 2891+{, cir? sd khtic khdng gtdng nhau. vn 92544+. Lbi gidt (c'le Ygug6n liug lftrii Srn, 808 NgrryEn LluF, llud|. NhQn cdl: [Jan, 8fin91 ltotng An \l0T Ngt, Si :Li€n, B{c Giang. Hi Bfre} clng c6 ldi giii dhng, Trrrdc h6t ta cd v:ii nhin xdt: tuy tihi8n trinh biy cdr rusm rir. t) Sd chinh phuong c6 ttn'eung li i,$ s0 cd chri sd hrlng chuc la s6 chAn. ThAt v$y. sd c:hinh N. ti . lt. irlruulg 5v si' cri dang: (10a -l- t )2 : t00a2,-! 20a -t I , (lkz ;[3)z =100a2 :i:6Oa +{} ; 6- 0, i,2,... Bni 2/If 9.'fim bdn sd Is nhiin cd tinh r,hlt '1'* hai d&ng thuc nA-r suv'ra dpcrn saa; Binh..phtrongl cfia mdi sd lrong chitng egng uoi ba sd . bb t)> d ({) (ttla:L 4)2 : l00al -}' 30a * td: a:0, 1,2.... ,{ = u2 * b*c +d, B:o* bz + c -l d, 'l'u dSr tld dAng syy ra rlpcnl. L -r.r* b.yc2 +d, D=:6* }i- c*d2 'l'r& Iai bli toln : Gi6 sfr sd ph&i rim, vj6t li c{c s6 chlnh phrrtrrrg, du6i dangrlhAp phfin la ,{z =;[};r o,. n,},..r .Tr! cliBu kign (*) ta c6 t ) tr, t. 2.....9 I vir rn r * {t. ,2 ( .r ( u2 +'ro ( (a t z)2. \ri Az li s6 chinh phuor;g vi do r =7- 0 nen r ehi cir th0 nhan cic gi5. .tri, I,4. b.6,0. Lai do v4y r{ :ot + t * c * d:(a * "t)z 3 .rrhAn -rdt tr6n rrdrr {:l chi c6 r - {. I)o vflr' ha.v d-(0*e+d-1112. (lt sr1 irhii lilir cc drrro. Tir (1) vi {'t) suy ra a (30/:, va .42 = rnnp444, tr { g < ia,/2+ bz + ztt : bz + ?blz 1. Tri (3r ta c6 c ( a (2c. r'rr (c * 1)? Suy ra d, = 60b * 12 : 500c * 38. *zc +-cz + l( () < 4c + c2 +c ( rc + t)2. yA]' Ta c6 t05(a2 = (500c - 38)2 ( to6 + t00VIo c = {c1 2)2,trlc ln2a } ca I d =(c 4 2)2 lrrry . {soOc - 3s < 103 "+ c = I ho+c c::,t.. 2a*d=4e*4' ("1 ) {
Gifii he(iB), (4) ta ctugc Vidc linh toarr sd tr& n6n nhe nhang vi ilon giin _ ' e:itdl"L - i, hon rrhi6u. n6u dE { rdng. [(r + rtz)Bl - |* * r/:)3J : Iu day la thfly ;=;:'; ' *'- { 3lr2 n6u {.- i (mod 4) I i=t .2,8 I r*' * r n6u r.: o (mod 4) "-+3> o = 5m - { } 1=t- d =.-!m. Suv mb 1. D : 2o * c -]- d?-=4mz +Jnt -- 11>> (2m'a' 2)! tdo Khi d6 ta c6 ngn-t -it m > rt, vL. 4mz * l3m - tt ( (2m + +.r2. Ve.v chi c6 tlrF: Ir s - X. (g*?rr.) + t/{ -F llti -+- rlt,J + 2g/tl i a) I]= 4*2 + IBm-- tt:(z'm +2)?(+m =8. r a-r-i 4 = 6=5m- 4 : II, c=Sm * S =6, rJ = 2m:6. 1t h) D - (2nr -t- 3)2 +* m = 20. a = D = 5m - 4:96, c = l}n - B = 17. rt - Zm E k+tl4+ =$ k:1 t/s+l/l?+2&/13 =,10. 'l'6m lni cAc sfi tu nhi6n tl.nr rlucrc lA : = :ao I9,1 312 rl=l)=c:3Jt1 +2k (t. I{ s6 tu,,tien), rl:1; 11 = g: i = 3*1 N.K.t. - 2A (& I* sd trl nhi6n), rl = I I a:b= II. c:d=6: Bet 4lll-9. .Chnng mtnh rdng u6i moi ctich (1 =b=96, c=51. ri=40. chialQp hop 1,2. ...9 (gbm g sd tV nhittr ddtt tient thgnh ra hai tQp hgp cirt lu6n lu6n c6 mgt tSp hyp coh chnq ba sd llp thanh mgt cdp s6 cgng. 3 BAi 3iIl9. fry hiQu ao tds6 nguq€n san li l.di gilti (cria .Vgd Duu Ninil. , nhdt. ildg tinh tbng Gii ugrrqc lqi : t6n tai m6t cAch ehia tip - chosr"r di -- tlst * tlaz + ... + t/arsar. S thanh hai t{p con L vA B sao cho trong Lb!-!idi_ (cria Ngd Dug Ninh { cfrng nhu trong B kh6ng c6 3 o6 ndc, Iep _ Qu-v Nhorr). - b? Trtn phri, thtrrh mQt c{p s6 c6ng. xct 3 c{p s6 .c6ng (1, s,trt; {3, +.5) va (3,5, 7), Do Vn li nrgt sd tr nhiCrr ho6c mOt s6 r,6 tj. 1'a th{-v cdp sd i3,b) klrd,ng thE nAm tror)g I n6n crieh x{e dinh oo c{ri bAi ra ll dur nhdt.l tap hqp, vi n6u ngugc lai thi cde s6 1,4,7 iap Ta nhAn x6t moin6 tu nhi&n n th6erndn,** thAnh rnOt c(p s6 c6ng vi crlng thuOc m6t, t$p n,)(r. ';,;;.'ilu hop, m0l lhu6n vdi gii,thi6t. tl.2),3.> - t!z)3 ae"u so.(k f t!z)" ve & _ rlzt" t.6"Jri-r. ci ta (t o,u, Trrrrng tu, .ret edc cdp s6 cgng (S. S, Z) ; (5,6,7) + i/zrg vA (5, 7.9), Q,4.6) I (4, S, 6) vA (4, 6,8) ta c6 : - (L' - ll2\' sd ttr nhi€n. r'a lai..cd (tS + l/il3 ) rlisi ) ,.. ) lgr4 > . tr!)ts:12 "c{p (q,7) cting nhu crp (+,ot ttrong th} nim trorrg mdt tAp hop. ( 13 - + l/:)3) t95:t, rrJn ,$rr ,+ lrt + r,zts] _ ltt .* trys .Do ctrp (3,5) ctng rrhu ctrp (5, ?) khOng th& t_. * L ttlul' ndrn trong mOt ttp hop ndn B,T crtng thu6e nr6t "_ i:I k:l A. tAp hgp vA 5 thudc tAp hop kia. Kh6ng miit t6rrg qu6t, gii s,i I,7 G i{ va b G R. + (rsrr * 195{-t- 1)it3 Xet hai trudng hcp: .-r ji,i + t, * - 3 + l2l2 | 2713 + 4gt4 t0st-6 + 14717 1)4t,.{. Vi 3, 4 G C fi6n 2 G A. Vi 4.7 C/ + lgriS + :43i9 + 300110 + 363irl + 43$/t2 n6n 1 C B. \ri 2,8GB n0n S €,4. Vi Z,S t ri + 28/13 r ndr. I G B. Nhung khi d6 1,5,9 (:E vd IripthAnh mgt cf,p s6 cdng mAu thudrr v0i gii rhi6t. IL 2) 4 c a, khi .td 6 c d. vi J;6 G rl r,ofi g L" =J --E arl: ,nr I * {ir{) (1+ 1 ! + tl3) + 28/tB E, Vi 5,0 € A,ir€n t c r4. \ri l,g q A rr€n t € B; vA vi 2,5 G B l$n I G .4. Nhrrdg khi d6 ti,7, 8 C A vA IAp th{nh rnQt c{p s6 eQng, mAu - 236-l!L thudn vcyi giA thtdt. 312 C[c mtu thu6u tr*n chAng t6 kdi lu{n crla --.S.ed" rcdt,:-Cteh giii tr€n cria ban Ng6 Duy bai toin Ii t*ing: Ninh g{p phli mOr sg rinh to{n }h{ c6ui tfntr, N.K.M.
=- { L Bdi 5/1L9. 'trong ddq tdc sd ch[nh,phaong cdn n6u chou u = i - q2, u'=2p,4" thi di de-n r?. :2. :12,...,rr2.... ' -' \**P, zpq-,r' hd.u ltiu ra tiit cit cdc h0 ba s6. sno eho rndi ttS ba tt = t" *u-z (:)- $ .sd tiu Idp thrintt ntQt cdp s?i c|nE. l2:1,ri2pq_ qz Lot gr&i. Br'ri to{rr trrorrg llrrrrng vcri vi€c tinr D& -;,' r{ng ctic bibrr thftg cria .r trong (,1) va t{t ci cAcnglii€rn rgrrvt\n dutrng (r: g; :, (-* ( U trong (5) chi hh6c rrhau b6i ddu, mA tn phii (:) cria lrhu'on;,. trirrh ) 0, thauh thri ta di ddn kEt lus.n : nghiSrn chr,rrr :u ngul'6n duong tbng quAl cria ptruong trirh (t), ,r? + ,2= 21Jz (1) th6a mrlri diFu ki0n (2), dugc cho b&i Di hni6n ta clri vi6c tim c{c nghiQm nguyCn drrorrg (:r: i/ i. :) th6a rrrirr th6.m didrr kiQn phrr : l.=l(q+q)2-zpzt r, y, : ngu-v0n t6 cilng nhau, 'tuc li I o= r'* q' (r, r7. ;; = i. (2) {.-(p+s)2-2q2. rri (i) suv ra rlng r2 * z2 phii li mot sd voi p, g lA hai s6 drrong triy j', th6a rniri di€rr chin, nhu viy :r.: phiii d6nrr thoi ch.in hay 16. kiQir: ip, g) = I rd p) C. lihurrg nd,r r',: abng tloi clr[n, thi ,l + nz 'l rong sd cic l.rqn giri hAi giii. kh6rg bai: r:ao h6t cho 4, do dd theo (l), y ch&n, khi d6 "hio 2 ln gi&i t{ring bAi n}v. mQt udc chu ng crla c, g, z trai vf i (ll. l'hAnh thti vdi gi'i tbi{t (Z), ta co : r,: IA nhfrng s6 }-, K. \I, 16. Khi d6 ndrr p ld rn$t s6 ngur:6n 16, ur:c crjr c va : t.hi p 16, vA lheo (l). p la u0c cfia g'. BAi 6r'119. o, b. c, rt ki bin sd d,tonr; L:hac r"&v .p Ii nr6t urlc cira g., do d6 r,(i ld mQt udc chr rig cria r. v, :. 'f irirh thfr voi giA lhi€t (!), nhau, odl a * b = c * d. &dt phaong lrinh ta srry ra ring : r, z lir nh&ng sd 16, ngtry6rr l;;r + l;rbt: /;17 + 1'. -J l6 eLing nhau. Liti gidi. C6 thE.coi rdrrg a { c ( ri ( &. Ta Tu ket qui n*1' ta thd-r. r[ng r * z vl r - z hfly vidt phuong trinh tli eho dudi dqng It nlrring s6 chEn, do d6 cd thE vi€t (1) du0i *" + tl#-a = /rT7-, + v;+ai-.{ d+n H l hav [t* + ,llzl' + l(z *a;]lzJt =s', .t fi har' t q r-'-':o , :"1 Vr'= Ir' + -- at * a 1'l b rf*tn=!l' (3) *- vdi u: (z+r)12, o = \z r)r2 li nhtng sd nguJ€n - c -*+ --/;;7+, , i) tirrong. N6u a lA u0c chung l6n nhd.t cira tr vi u, Yo +jT+ a thi 2d ln m6t uoc chuttg cia .: * r va ; - .r. drr tt
'.lJ'Ll t)lJ ll,i ll, r,Jil0l] iri phurrng trinh'di cho, BAi 8/119. Cho m|t dudng ti'dn. Gei Sn ta co' i\ +R-{: + D (31 tti|n tich da 11iac tlbu, m canh n$i tilp tiudng ti,- do vS r. Atri =cD (4) (to Chnng minlL rdng ta ludn cit t u (.3) vri {4i su1' ra: ho{c A:.-', hoiic ;1 :1). 2Sz, } S,-t'1 Sartz(n2 4)' l Nhuns dri gi& ttridt a (c {.a. ta cd ,1 { c { if Liii (tiitt. I(hdng nr{t tbng quSt' ' ta gii' s& (vdi gi& ihi6t cac r-[u cri rrghia)' vi di tl6n rnau duong.irdn d[ cho co hin kinh b&ng- 1'.Khi '10 ihuhn. Thdnh ihr]' (2) kh6rig thB xiy ra, rriri cach aien lich da giSc clEu m canh nOi tidp ln kh{c phu'o'irg trinh cli cho co rghidrl duv r hiit s*: (ml2) 5in (t iltlm) * li r:0. a (bd.rrg t6ng rtiQn tich m tltn giArr cirn co c4nh Ban Ngd fiug h'inh (Qui Nhc-vrr) giii theo c6eh sau cIAv : cii t.hi c,-ri o(c(rj{b .+ (D-c) (r- d)>0 ir* iarg-r vir gor: & dinlr i:;'rrg llt/ml' + b? * hk + til + cd),0 + cd) b(c-i-rJ-D), t*c Xel- IrArn s6 cd) bo. f (d={r/Z} sin i2:rl r)' voi r "} $' Bir'lt pi111',,t;'1 i,r,i vd t:iiirrl). chi; r' 'i'a,'h.riirg minll a2,-2 to -- tc2,.t, f d l:etcd * abi di itdq phur:lg t t' (l) Irir,lr )f {2.ri } /r.r 'r- /i !' ed - ab+ l/fl+lx ) : VG+.4 (.'+;t Ta c6: (:t/r) cos i;rl:d 'l'idp tric }:inir pirurrng- hai r'6 vd udc luoc c6c / (.r'): (,r/2) si,, (2fi/el) = r sitt sd hqrrg *6rg d p.1:6 -s0 cli d€rr :2.r sin {:f,/2o-) r:os {3(/2r) *us i:(/'r)' {rr -- .1 : f ill[fl,:irFi /(2r) = 3c sin $t12il cos (3(/2r)' P)ltiir'::1 iri:;lr chi e6 irghi6rrr rr ( ab; binh f{4r) = 2;r sin (s(l2r)' \ piiucrnl Ir:.i ti 1ae(roc ltror'sd di d6n (1) tlrr t'irlrrg rninh tro thinh (a--blz.r':0 +)l':0' BAt dilrg thirc 4,r: sin (ttlZr) cos (3[12't) g65 (r{/r) SAi ? li}-S Cjro n sd drro'ng 3:t, t:2,..., :tL. } 2a'sin &l\d cos (fl/2'r) Chax1l minh bltt rLl.ng th{rc * 2r sin (XlZ*l (I f rr f ne,zi i l -! rz i nrr3)...( I i- so-tll:ro)( tr * r*"/nrr ) i+ 2 ) ct':s {fl12:t:} cc's (3(/r) * cos (rc/2r) 1 *=) 2 cos (stl2.rr ) cos (3t/2r) X ) rt t tlnt", Ldi gtiti (cri.r i;hi6u i;aii)' Ir - 2si;r2 (sti2r)] +1 - Cfic sd rt:i li\ a:r'i (i:1. 2..... a) {Iu la t6
l ,'\( '':+)-11,,, )stt.,r-11Jj},p \ (') )-rr " .-. t ,-l Hei tam gidc vu6ne O.Up \.A {i.tIQ b{ng nha vi 0M=(),V, lp:Oe, \,ar i&=6X. .**\ T* dd : P0t1 = tr49y. (Xenr bii inOt ;rhuong phAp chuug ntinh vt . . Hai tanr giae cdn OM,l'ri t)pe t6 goe 6 diirh , sAng a . tao b:ii rlirs ttrrir,) -'ttH1rit s6 Linr lggl. blng nhaq n6n dbng clang. yav g..dtr li-1- q1 :.r, = l/2, d.du brit dfrng lhrrc p11g + /'\ vdi hin'r Idm trii chi6u vrii cif,u ring r"ii hnm lili). 0P{) = rMO Bdr- gio ta chrlug nrinlr vl rlo t16 bdn di8nr It. L p,(l rrinr tr6n n:0t rfudng trolr. ) Slr+a * ,!*o .Sn* Sn-l vi Sp:7"(m,r n€rr trr (2) suy ra (o*) Stry ra OI LPQ, vey 1 Ii. ditm giiia eira /,e. f # |, Sn * S,r-t )> Sr+r S, > Sn*r * .!o+r ) ... .T'e dn chring nrinh dltroe ilibnr giila 1 eria pe - ) n&nr tr6n &t.V. De ddng thdy .ing 7 ehi ninr tr6n >S?" - Szr--l =+ ,!-,r - Sn-l ) {,Sin ,- Sn),/n. (*) tt3trl -iis trdrr f/rV gicri hrrn b&i,1*i ilrrdng ch6o v* cri:r lr in lr t lr i rr g (lr . 2r. ' S4o+z* Sao (.\^cn - .S+,-z ( .}+n-g - Sro-t . Pl:-dl drio: Lri.r 1' tr6n IlS. Ndu I' ln diBni gita < "' So _i * .S*n+2. = }J.\ u B{i ding tirii,c n6u trong tl}u i:Ai duoc chtng *| -\\ QPO = pe} &+ A Ope cin *t Xp = ey. minlr. :1]U, r[ng trong trLrcrrrg hcrp rlrrdng rrung tJO .. Lrinh ,&f .Y c0a hinh thang khdng eit .XI."tirc le BAi 9lll9.CIto hinh lhanrs ed.n ABCD(,1tr= tr,rorrg hqp XyllMN thi trung- dr6n.r cia Be tidp trong rtrong trdn ldm A. IIAt jau Xf . nirn trln truc d6i xung IiI crlalinh thang. , ?-C\ "0: thla d'6i c&t dD tat p, BC tsi e, sao cio Xp QI'. l'ini qpS t{ch di?m gi{Iu cfio pe. = 'I'ir gii thidi Xp = ey . .Ldi gidi. su-v ra O rrim {:r'dn trrrng .lr:uc ci'ra },Q. Nhrr vay fl.eq lA tarn gi:ic c:in. \ ."D1rng-dudng trung binh Miy cria hinh thang, g.ii sri -&Ilf cflt X]. tai ^I. Tarn gi6c 0.]fiy efing . r.r i '1 d lii. ta rn giAc c{ rr. \l q,, \i I i/ ri, : iii: f Tirm lai quf richt'::'gi&a ciia pe ra hai iloan RS vh ItL r) hinh 2. f iF Ban trfgrigdrt Duu 'I'hdi Snn (Hrrd) e6 l6i gii i tttt. Bii l0/llg, t:ha mgr titi giat T ro diqn $ch S. ?'o chia m6i canh cr?n nd ,idnft .l phhn btng nlrau rdi ndi cdc cQp itilm. chia tuong *ng1 & cde eqnh d6i d.rcn. Ld.m nhr uQg, t* gtdc T dagc Hinh 1 clrfa {lronA I 6 htnh I* gidc. trong tl6 $ mdt 6
khdng kE uoi canh ndo cfia T md lq get td. 6 trung J tfrm. Ndi rinq diln tjr:h cria d trung tdm ,r b&ng Sl9 co dung khdag ? \ l Lil gidi. \:6i nh&rrg kt hif u nhu hinh.? ta ed SDAE = (l/3) .SDAB, .lBCc: U/g) !'BCD. .Tri rtd I I I c6 Sp6g + SSCC : * sEecr, (l/3){S11,1B SBCD, = SiB d t/ I I I i - 2513 t/ \l II I I M+t kh{e Sgpg : SEHG.'SCUF : SC*,U o ti g_ *.* _ SEr"-Glt = St.BGD12 has SBp611 = S.3. BA5r giCr ta chring rnintr rl ng pS vu ()/l 6.{1 ffintr 3 EII tai nh&ng clitm d. t chld doan EIi thnnh i phtn b*ng nhau. Thlt vi-v (xom hlnh 3) ta ccr l: t: PE = il813, f/S = (2r3) ]]B $, pE = HS/Z *+ fiE = KH12*EN=EHi3. Tunng t* H,!.:.EHi3. Tric le Ii, ,t e.hia doan E'Il thanh S phEn hrlng nhau. Tuong tg .lI,ry eltia iloan F6 thdoh :i phhn b*.ng nhau. Theo phlr trdn ta c6 SttLlt,r- = t ll3)S gpg11 =.S/9 .. Vriy tich diQn rieh d trung rim dring-b&qg tiS ai€u cria rti giic di cho. Htnh 4 t) rt{oz(...(&o, ii) vcri mgi /{= I, 2...., n, ta e6 a? + dl +...+ai { Chri,ng minh rhng: t) at * dZ + ... * an( h+bz* ... 1. bn f,&{ DE roA${ Yt{l stAt ToA}t t?s$ : Igg? r(i(i(,, l
-r Bei 5i 122. Ldy chin chii sd I, 1,..., I v i.it gdc'vdi lB" (ir.li (itt) lA tluong lrdrr ti6p xric voi t.hdnh bdn s6: mdt sd c6 ba chfi' s6 I'ri ba s6 circ d,ran CA, CD vi cung AC; (rrz) Li clu&ng eon lai e6 hai chir s6, sao cho chirng lip th Anh ttiori tidp -xrlc v6i c6c cloari C B, CD vi cung ('B' circ cip sd ci tilhtint rlhru__ C4c clrrdng trou (rrt), (uz) t16p xirc vdi AB Lai (*l Et, 82. Goi f ia tArn dtrdng tron ngoai ti6p tam abXctl:1gr.: X urr gil,e CEfiz. ()hririg nrinh Ci lu0n ludir di riuu Ch,-r bi€t e{ch vi6t irng r'6i gia iri cfr:t tir h tY mot r'!ii:rlr t:d dinh. rn6i vd cia (*) lti'n nh6t ln 1,6 Qude Hatt 98X 76: ir3l X 1{ (=-7l4rr. Hf,y tim tiit cir cic crlch vrdl rir'g v6i gia iIl 13di 9lLZZ. (,tro lanr giir dlu l8r-. ll-li m6t cfia tich nr6i r'6 cfra (*) cd chir s(i hiltrg rlirn v! rli0m bAt ki nam ngoirr rn[1. pirirrrg cira tarn Id 6. giae .1 Br . / t.hL?nfi minlr ling: Bei 61122. "I'r€n drrong tldn drror:g l
thi;ig ri6. giu srr clo Ii hai fluirng thing a va b. a. b (tuong [u c) n€n tl€u nirn trong P. \rfly n' Goi lJ lri giao cria a vrli b. l)r.rng nrirt phiug P dudng rhing ddng phhrg. chira a,6.\;i n tlrrring thiang dai,g \et khdilg Biy giti g-ii- srl a itudng th&ng't16i m6t c5t dbng qui nrin c6 it ntt{l n.rirt rlLrcrng thirng kh6ng nhau kh6i'rg ddng ph&ug. 'l'h6 thi n ducrng th&ng tli qua r1{. Gitr siL ri la mrir rluirng thirng khdrrg t16 ph8i cl6ng qui vi ndu khOtg thi theo chung cii qua ^U. Yi c cit cir u lrl D r€n c nim trorig mirh trd;r cht'rng.phii d6ng ph8.ng, trai vdi gii P. Cac rluirrg th&ng cr)r lai thi holc qua M va thi d r. cit c (tuong tu a, b) hoic lihdng qua M vn dEt DING y1f,5 (s6 Nsr) Ttm hibsl s*:r hryr: to6n hgc ph.& th0rg c$roc TH{rc NOr suY PHAN O['C CHINU { IiUNti plrd n };f thul'dl da i}r uc hoc & . ph& li ur0t. hiirg s6. nii h(:ei) = 0, do 'd6 ii.r) = Q, I thOrrg, r'6 nr6i vin dd khdng duoc dE lhirrlr thlr /(.r) = g("r ). cip 116n, nirurrg rtit cri hiq\u lirt: da giii'qu"vdt 'lri m{nh dE 2 ta rirt ra kdt IuAn : mAt da nr0t loal bii toin xurg qualh da thuc, 116 le thitc bdc n duqc hadn tadnrdc dinhbfit cec gid c6ng thti'c n6i su,s, h:ry cdng th*c La-grdng-gict. tri e&a nd tqt h+l diDm phd4 biQt. I-y tlo do;r giin cira vi6c kh6ng dE cAp d6n, Tu nhidn n6:y ra cflu h6i: bi6t a -p 1 gi{ iri lA c6ng thuc d1' r'in ihrrdng iluoc quan ni€rn /(ci)'(i = 1.2,...,.n + 1)"cfia da tli*c /(r) bac n IA phirc tap. \iru;:g r0t cho ctirg, su phuc tqp tai n4r diBrn ph[n tri€t ct. r2,...,x:a41, lAm th6 chi la vd rn{t hiirh th/tc, r:du bin ch6t cdng thuc ndo xf,c dinh /(r)'? Cong thuc nbi suy trl ldi thdt r u cing don lii.:. t:ho e0"u h6i d6. ${uc c!ich hdi nAv irLrric hdt IA gid}i thiQu v6i Vtvi m6ii = 1,2,,..rn;1, ta hIrr. r** rla thrire cAc ban c0ng thtrc riy, r,'A sau cl6, quan trg.ng buc n htyn, rninh hga ciich virr dung c6ng thirc qtta ph6p gi$i vrli bii toirn hry va l
. BAI TOAN l, Cho a, h, e fti ]n s6 liltdc nhou" do, o1,..., on*I, on *rt* da th{tc dg ld nhfrng s6 ' HfrA tinft Al(r) (&=G, 1, Z,B" 4), u
nguyen, ma f(ft- t), f&), f(k+t't ti +hiing nguy6n, vny /(r) ngu-vtn khi i nguyen. s6 , (, - i) I',,-as) (, . i) *":' ChLi g" l(h6og n6n k6t..ludn sai lhm rXng n6u ., t:3 /(z) ngu,vOn ktri e nguv€n, thi e) v6 phii cira (gt, c{c hd s6 oo, a1,..,, en ld" nhtrng sd nguy0n. 34 E3 kdt thric, ta hiy xern bAi to6n sau, phhn tr) ra rlng khi |rl{l Tdt bi6u thrle nA-y, ta rril' bii to:in nA1- dI'16 dE todn i.iri trong s6 J/t9g{l (nhd rlng ,1, B, (:, D co giA tri tuyQt rl6i bio 'l-llT I. hlrong v rloi qud l) BA^I 'l O,{N 4. t;ho dtt thirc : l*2.4 * tr-.r)* I (t-rt)x r bt.ic bal(.r.) ,r.r.3* ;/x(rti ( -'" tr- ) bxz * cr f Bi€t rans I f ttt i( rI. i tr,i i,i-(:, 34,3 h&u chftng minh rring L:cti rnpi fei { f: a) JdrJ * ccz 1or + nl -d 4, rr 1 -- li + -L(,-,r) (r r _l t + b) l,ca.t2*2br* c j
:-.. F - .;ong d6 I{ 1A mQt di6m tny f dr}oc goi It lrgng Voil chri y gia thi6r vti Or, o2, rheo (t) ta co drn cira h€ ch{t di6m nAy. H6 thirc (t) c6 lhb l vi6t gga liL : g.++I,*F+ L ui UtAi: u , . I b{.tzlti: 0 . r,0ir ta c6 : g z --) i:l .' j:l tta = at frIAivoi ai:aif (at*az *..,* a6i) (2) Il + i=l U + I ui OOr +X bi aor: g' m i:l j- t Vdi cAeh d*t Ehu v*v, ta cd.X ai=1 (Sl Yrlj,(1, Ot, Az rh?ng hr\:rg vi vE d0 rlni lrinh 1 l=1 Tt (1) ldy t[:0 ta cd nmi < < + hoc ta co Oot:A{)y = X a:/ : ot (apcm), ' OAt*...*a- Ol- = 0. at (4) "r' j: I i:I Ngugc lai vdi M tty f, tri (4) ta c6 l'Li (2) c6 4 < + + -,* at(aN * ,tf .4r) *... * a^(Ott * 1,, ,4*) : o
r didr: fai c6c cliEm A, B, C ia d[t cic kh6i \'a,r , ... lucrng tucrng ri'ng a, b. c, \/i -{41 Ia dLring phsn giac n6n Lltl, 6, a .,-_*l--*- lt'+ AfilA{: c'b "=} c.AtC : b..ttB a-l-'b*e .r*&*e '-.. €+ 4 ,+ + b.rttB+ c.,{1C : 0 (D.AlB * c.d1(; : 0 ). VAv lheo (4), Ar le trong tAm cia hQ hai i:hiit tiilrilr *i1-c B, C. 'Il.reo (6) vdi m:2,;iI:A r-n iheo il!rrlr nlhiu c1. o-2 ln c6: (Ltieo (?) r,ir (8), hay OI? * P,? - 2fir {trg thrte (Ote)' .{{?-*b-,tB3{-*' :c? -- ,{1t rr:n l') Lri tlrer, ({i) ta co : Gl? i- = a*b* c 6111 b*c b*c tb-l- c)2 - / : ,.' + cb' _ bcaz : ----- 'b ab b*c b*c; @+c>2 +- GIt"+ (r,*b*c)? u-1-b* c fo2 Il. - i,c I t -- ----; .. Qc *^ K | (b ;c)" I \ /1 _- ._-_;AL' --!L lelz \rtli chu j'A la lrong tAm cria h0 i ch{l- r'li}ni (o i-b*c)' {a*b ,4, n6.n thco tlnh ehdt (5) ta cd trgng tilm cira ( hQ ;1, B, C rrim tr6n itr.rring ph5.n giac Adt. T'uong .. .t -f l,-t , GBz --- tu trqng tim cria h6. A, 1i, C cring nim '.rdn o-l-D*c lf tong ph&n gif,c lJAt. V+y I chiuh la trong tirn cria t:g ,1 , B. (-. Do r1d theo (5) ta c(r: GCz - ?Rr IAitAt=(6*c)/o. {1 tIr+-( Nlrrin ril: N6rr t:ri A. B,t: cting dlt rrrOt khdi (theo (7) vi (8), Goi A: li trung di$m cia 8C, lucrng lrtr I tlri theo c{ch rAc clinh tlong tAm o ther.r c6ng ttruc rludrng lrtrng tuy6n la c6 vi dg 1 la dE tling thd-r: trong lAn-r crla hE 3 chdt rli3m A, B.C chinh lA trong tArn (gia.o g-l?* !- ,rr, 62- .. -- 11i3rn t ', ^ ^2 cfra 3 clrrdr:g tluog tuydn) cria A afC. ii {? = 'g"g2 ). Tuong tu I:l du 2. Goi H, I,(),G la truc teu), tim tf.utvrr.t ,, trdn nQi ii€p, tirm tlu
,3) 'l heo (o) co : tt € qu&: Trong iam gi{c ,{8C ta c6: 1) 9nz > ,rz4lrt1"tb ab*ac*bc ) tr8r (13) o -rfirr+ t--#al+ -L,rr= a*b*c s*b*c Clr*ng minh: T'ri (lt) *| ar + b? +.2. ( gRe.'tu (a * l)2 +-, o5 aB'* (a-e)?+tt,- c\z)> rt * e!*6i+c! ir ab f nc-thc' 4 .*-.9-* HCz -- (a*D*c)-; ar htc LAi lheo (i0) _ _og t 3br * Slrc -j* 3rle * at - lr? * uz b l',t' Itr. it.r - (a*D*c)'-.u,, I (u-Fb*clr Nhur.rg do ab + oc * hc - oz -- 1,2 - cb'-i< 0c- - ?.6i' .lr Tla clri x6c dfnh HA?, HBz,I/Cz. Drog A't'B'C' 2trb * 2ac * 2hc2 3ba + Jlrcf 3Qc- a'- ? It0 llr *+ ah * atc * bc}lEIir' I'6ni iai ta c6 {13) cci cii: cqnh sonE song v0i cae cgnh irrong ulrg v{ di qua e{c ilinh c{a ltrBC. 2, e\ 2.102- HI d| IO> oc Do eic trl gi6c ACBC', ABCB'lA cAc htnh triuh b)/t. rc>GI e)3.Io>0H hinh n€a ,rd': AIJ' : BC vil fijl -L ,B'C'. Yi vav de tlirrg thf-r- lI li tAm. iluurrg .trutt c) 2.IA> {;n t)IGHz - il E2 ngoai ti6p Ar{'A'(-". vui balr kin}r ll ' Cli' A "'l 'B C + 6/G -4IOl = 0. a*A,r^al +lr'ln - B'c'lB{ +.R':zIi. Gidi:'Iheo cie c6ng thuc tlnh GHt, Iil't, lG?, T'u d6 HAz: ft'2 *C',.12 - 4.Rt * a!. fol la c6 : 'l-truug tr.r .irri i'p'rur+ 1 rc2 + oJoz =.(44-*'tP + 0r)fi1 = {Rt * 1r* I ffCt H Bz ac 4/i.2-e:. - 2}lr (18 + 2T + to) -p {p * Ti3) (u, + ac * bc) _ {cz + br + cs) (rnlS + F + t/Sr. i l4) Y*5, theo ii) vS i8) ta e6 ai Cho cr =0, T =0, P = - l,eo =,l, tu (l'1) c6: III2=- q- Uns*a|\ *" ' a*bic (On?*tP) + tloz - tHz - o? + br -',-'c2 - dh * bc - o' ]> 0 a'rb*c (theo (13)). * 2fir = dli? * b)Cho a=F :0. = - l,o' =2, tu t'6: + **a (4Rz - c2) 2ftr Itoi T -(i{)b2 * if - in, * inr, + ,,r + bdls -F (ar,+ 4 a'l-&+t - ez)/g =[(taRz - 2ob - Zac ' 2Dr) l- (42+ br * c'- ,r+Ds+c3 n-l-b * c ab - oc - t,c)llc >0 (theo'.{13}i. 4, ttr-(l{) ta cS : tlr = c) Cho cr =- 1, 9=T=0, = 4ft2 - 2ftr - (ot * b2 * c2 * * Gls)(az +b2+cr))$ oD - ac-&c) -GHL =-8fir* 4rc2 (13)). 3ql,cl(a * Il + c)'(rheo (9)1. (ttreo , Y{y theo i8) ta e6 : d) vt e). 'Iheo rluong thf,ng t)'le co GH -2' lItz'4t\z*iRr*(;*46r* ee - ab*dc -be) ( l,) GO, GH = (3/3) OIl. Vay t* c) ta cb otb OG vi 4) DE x6c ilinh OG!, ta d{t ti ,{, l,, C er}ng m$t ua> oH. khdi ltrong l. theo nh0n xdt tr0n thl trong tArn f) Thay fl =3, P= -Z,ni =S: tlr= - I vio G eia AjEC chtah ii trqng tAm c&a h0 3 ch{t (l.t) ta ed ngay kdt luAn. di$rn A. B, C. 'l'rr d6 theo i6) ta c6 ' 96t-6.11/i: CAc b4n c6 thE suy di6n ti6p circ tinh chdt lf + oyzlS -t oc lz-- A82lg - aczfi*Bt-t;9. vav thii kh{c ttr cAe k€t qu& & trdn. f}6 gili vi du oGz = ft! * q,r2 + br + c\/g. 2. cic phuong ph{p kh{c iIXu g{p nhiiu klt6 khen hoac UinE ifre. Phooug phip clnng kh{i nifm Theo drrdng th&ng Ole. ta c6 o'Iy'=2OG: trong tAm e6 uu itiBrn trorrg uhi6u bii toia ttnh OII = 3OG. YA;- : toAB vdr chung nrinh, llu cllBur co bran cria phuolg l- GItz s4ftu * 4 (ot + trz -i- c?Ug ph{p nn1' li sitc xAc dinh c6c khdi luqng thich ,rf hqp s€ eho ltii giiii ngin gr.in vi m& rQng phr.rm ) rlllz =9fi2 * (a? + b2 + c2) (12) vi ting dgng linh hoat cho nhiBrr triri lo6n. TITOHO B,iO Bd,a lladn hot ud tubt tr| ttl sd 1 nam 1982 sE drgc bdrt thco gid ntdi td. 1 dbng m$t tir. lu ta i .\i rrglrigp Tn 75. 7i I{d*g Bb - I{A NSi. Chi s6: 12884 6{d i o,20d Sfl in 147/Sl, In rong v& gui luu ehiluth{ug I2-19E1.