Vr o- .(-
{ {,* vtct- wq rl rr
L4
HOl 'rol,x rioe
Yrfrr xeu
[},
lr-
LNII
Tru
nhi(nt: lcotiY6x Cinu roAx
s&: 70 l'r'E.u l{ung .Dgo * Hr) NQi
BAo RA HAI THANG 1 MOT KY .t
ra6n.r Eo*lNc cqSNG
DAy rrtii : itZEZS
I'iu /rt' Irio
N6i chuyQn vrf i e6e bqn trt ycu to6n
KHIT CO Gi lffI ?
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Y&#'VE QUA
AN rrila 11[ trin"il hoc i11tr> I mA qudn duuc
tiii toltr ilon qiiri riltrng thir vi sau
ctir' :
Biii toAn l. r:hlrng ,rrrinir r.rilry, tt(u ,1! ki ntsl
rliPrn lriu 1i trdn ilu'rini7 trin nqoai tilp taDl (tiic
rl?u :llJO. tli trong btt tloan thin(t ilI 3,, MIl, it(,
co n.6t dtton cd r1f drii bi'tnq t''6ng dO ddi hai doan
thtirtg ri,n lt.ti.
' Dil tJinh r., il gii :ti rliirrn ^i/ nirn l16r,, r'urtg
irhil IlC. r'i chf n{ rrrinh ill A : IIB * i}|1.'. Mrrd;;r
vA-v, tr'on ,?I.-1 ra Il,r diit|i ,lJ sao (.ho lJl) :. ll il.
Khi, do trrnr giat' ,lfIJO r'1ilu. rrdn [..1R1)=Af;AM.
Suy la AD:ltL, vir dri tld ccl dpcnr.
Circh giii 1r'in lu-r' tlou giiin rrlninlvide ritr:
dinlr LIrtim cli&rr 1l lilr0ng phrii dtl drlng dt5i vCri
cla. sd r:hring ta. 'l'liri hili i:dn eitch oAo 11<rrr gi',in
vd'trr uhiAn hrrn lihfing'.' \'iii kidr.l. thfc irip ?,
cdu tri Icvi lA: khorrg.
LE QU6C HAN
M
. lli;th l
Nhrritp r0i, lerr Irili S, tir rlrror, grip riirrlr lri
P,filffnr,,\ \ 'l'tonll tngt tL? gtar.: n$i !ii; p, ri1,!1 a,tr',
dnbng chia bdnq tknq ci* tit,h cfiu e dc t:grth diii
difri u 'l'u hiirlg drrng uhri t(ri li.)i to6a tr€n
-!,
't: .
ii,
[..
"!'r:i. I r- -: I
tht vi nbung khdlg ph&i lA i.6 ich. l'l6u l;air c6
rl0t <t con m6t nhny cirrr tairtr hoc t !r:rrr sG th{v
ngay (1) vd (ll) ehi ta :
(rr f ,'r)2 -f-(,rt * ,r,e)? =1ri + lt:\,
hay ri + ",1 =. .,1 + .l
'l irrr r:r r'6i ! Hf th nc rr jrr. iilrA tlc1r. r-r't di
nhi0n e6 thb chung nrinh -lEng nd khOng phu
thr-rdc vio vi tri iliiinr -iy' uirrn i'r crrng n.io. I)o..
d6, ta di ,I6n: ,,,,
Bai tohn 2. Npu III iri rarit tii?ut lig rj ririni *l
trin dvirng 1ri)n ngorii-lli i, hirh t'uirg -1Br'; D. ltti \
ta ci M AD + M.{:e -= MR{ r- .V/l-.
S:iu pliirt sung srrt'rlg tliitt tiirt, hav billr tlirl
]aitIauhgi:rdiinglirrngiirir.iciiahiitolin2.
IJd thirc tlat duoc tlui 1i ttr.'p tirat. song leu 6arr
tinh j, bau sE thir-r' ng;ry riirg: cr,i ilil tiiri rluc,e
nr) blng cieh sti tlung tlilh lr. I)itago, r'i 1116i
vtr l:rrnA {[", trcnrl r16 ll iii irirn liinh rlu..:rg
trdn ngoqi tiip iiirih lLr0ng. liiri toin f tt.& uiin ,_+!;
qrrA rte I
A" ---*---
":./ n"
.3
:re ..1.
1
.\
.-l \
Phai
xem
ch(ro
flinh
th6 !
Itt
rbi, bAi to{n dy c6 c$i gt tuoug tU: N6u
MBAC ln tLt giirc n6i ti6p thi MA IA rlrrong
vA MB, -}r'il lri c$c can b. Ap d u ng d rroc
1i .Ft61OrnA clro hr\i tof n rly chI ng:? .lhrit
v$ AI):BC =It;i:+" bIA=MB+lIC. 5
I{$t ldi giAi qur'r ng[n gon cl.ro hii tojrn i, nhung
bqu iI[ bao gi<i nghi c16n cirua \rdn clil t'r& ltiri
tliti vi tfon, n6u tl nhri l:ri rlng do nr(inh riE
dfi.o cria -dlnh ii PrOI0rn6 ln rlring. n6n mdnh
rIE dio cia tjni tor'in chr.'rrip- t.a cInu riiing. Xirr
nllrrririH lrrn dr,c pirri I biFu hAi toan il,,r.
1'uy nhirln. hhOng rr0rr vi)i thila rniru vtri k,it
11u:i rtfi daI rlrrr.rc. \g:irrr ngh; lai : l_rti loar n cia
la phat biiiLr vii rnirt tinh chii t i:rirL tan"l gi6c
dFn ! lJdn rn& rdng rho da giirc ,rl0u. ci.r thu
ilnqe mQt hd thirc., trrolg trr tihOrg i, Li$u ttiuh
lj, Ptirl0m{: cri *irn gitip irh gt trong ligc phr.rt
hi6n r'ft rrhr.'ritE nrinh nhri,ng h6 thirc lrlcrng tH,16...
1'h*a irong ho,n, ta xrit tlrOrn rnr)t trurrng hgp
uita: srl carrh t:r'r,r ria gi:ic rliiu rlrl lrring bi,n.
Xel iu gi,i",lir, Altl|,lJA4n0i ticrp tlrrirng tron
!,. Db-_r]inli lv, hSy td-1. tlirini .il{ trdn crrnc ,11,14.
D[t Jf,4; = p-i (i = l, Z, B, .tr) vr) goi cani hintr
vu0rg ii a. l'a h,v von;1 thu t"irrqc h* thuc
et * ,zs =.a:.t *:r,4
hL4.llc = X,LB,A(; +],IC.Alt
fl inil 2
!,; /{
t
t,
. ilittlt ..J
Nlio Iriug' rr ir litr6ng lMaau sr-tn linh ttriin ti{rr
conq-c{ch nlr.r ng lroiig iihor hor,, cltir rrg lri illu
r:6 nin- lirnr:. J':r liho, I:ri,' hii toiiri i'ia ciro (b?ri
loSu I) r:6t v,,ri srli e:rrrh rr - B.'l'h0 iiri thir lem
v&i srJ carrh gip rjdi, h': thr'r,: cr'r < lirn thrrdng )r
li.hfirrg (r'tli nliiirrr rglrir i:ir:r tu. rril,). Ta h5v
gilt srL.lI nrirtr lt,1[i clrrq 11!.16,,ii,r tl rr,ing ll,on
ttgoai tidp Iue gitrc dctr _{r_l:-{3.{+_tsda. .\1-r clung
tlinh ti'PtOli.mi r.do cir: tr'r gi:ic niii ti6p
JI,dt.4:,:l:, u'{rA:.{a. ltf.{t-js,ts r,.ir .}-/:lrAl,{6, ta c6:
.. gp dt,ng ilinlr l-. Iltirltnr6 r:lro c;rrc iu giAc
tr!ALAZ.l.t, .t{lr,,trAa, ,,1.{AlJs_4a vi :,[ds.,tsCa, 1:i L]{i .
Itt.,,, :r'3.{{:,3,,,1'2 " [.,.1+,g=.r.rli]* | t')
I e'r.ntr: I .r,4.o =,r'.3.rr o J .,., -.,r.a:.r.1 ! i (:)
Icr.a-F:ra.a1ii-:"'sj,, - 1.,.r-rr:,or,f (r)
f 11.rr *r;.tl:,13.o.,'2 l.,e+e'a=.rst/i- ({)
Tir (l) ta Ihiir,.:-nrurin c6 .r'1 l- ...3 .- ez .f ra. flri
phii c6 14 * (1.i't- * l)ee : nhung clilu nrl-r: lui
kh0ng lu6n 11i'rng t,ri,i irir.ri vi tri cfin diBnr M.
Do do, irl- . vi:trg ciia la khitug ilat rl rroc. Iilri i
ch5ng didu dir 151,1 chir1.{ ta - tIfit r.ong,1 IIdy
nhin lai bdl ti€ thLic dE rhu clurTc : errurr! khdng
t
I
!
t
.1'l 1- t'3 : ,i'21i 3
:ca -* 2.r:1 :3'61/il
'eI 'i .r'5 - -,',
2r:3 '- r'1 =' :r'4-ll i-l
trr tl6 :
;;,;.r)'t (, s -trr)2 .{ (p:rs-.r,r)i:;(.r! i "il-,.3i
*+ ri -l' ,3 + (.,,, -- 3:s)t: r.,2.r * 11: + rfi -
r*rrf +o-i"rx.A =,i+"Eii"i u
2
cc5
1--
D&y chinh'lil hc
dr (1en
Bii lodn 3. r;id
uin.g trin ngorlt iit P
thi ftr crl
lhitc mi ta efin tim, vA ta
N.l:i !.lIdB r'Mdg '= 1Ia2 + uAi + rr.16
Ndu rrhrr ai kh0ng 1,1E1 ,1rrS tlinh sul luAn
tr€rr dt dirn tiri hili Loan:1, lhi khl plrrii giii bii
toin 11i:, iit ching cd th'6.rern thndng nti (rhdc
gi t1tr tri6t tiring ili:rh lj' Pt0lt).ni0 d€ giiii chu
nherrh'.'!)
sfr: rl{ lri m$t t:li\m ttig $ trin
Lr.rc girit d[u CtAzAsA+lrds'
Ddn il0-y, hftn barr ciing nhrr t0i rilu rtghi rirng
bhi tuhn 2 r'ri hAi toArr li ct'r lhE iihiri quiit h6a
lhA'rh.
Bii to6n 4. irdu i{ &i nrdi di?m rriy * ttdn
clvbng trrtn nga1cti ti{ p tltt gtac il}rr 2m cqnh
-{tlz ,. Azn, thi ('r cti '
irlu,rri*...'i .HJln,--r=lrlB + ii-'tl r-' .'t lIAi*
ll:ii to{;r ,l cii 111!1g l,lirrt$. n'rrrorl I'ing r:irc han
s(i lil€tll Il'll n]r l
Cht l.rarr tri: viu tt;fin Ih&ti iirin ! iirio biil'" gio
t*c ban dI thi-r lItn qu:rn tt'cug r:it't viec dillg
- tihii ne tri tlr ti'c rniri iinh ltoi tluoe t[3 soi sAng
nitirng vrin tli ttotrg qrrrr lrirri rilii ohi I I
Tim hilu sdrr thtsm to6n h.oc, phti thdrrg
Err0r{G s0l cllc
HAI EUOHG THi}{G BONG
MoT BthM B$r vryx
Qfftr, psl vt]'E Btvtr'Nfi TR0N
Pr]{\'1 VAN HOAN
eua
I
TEn()NG tr\i 1i0u hinh hoc ldp I phb th0ng chuy0o
E torn cring nhir tt'ong bAo ( To6n hqc vA
tu$i tr6> s6 30 ra thing 3 nirm 196? c6 hai bni
tofrn dln r16n khni ni6m llrrorrg tl6i 'cyc cfia ur6t
dihm tl6i vrii hai iluong thing dbng qui, ctiii
vtli tlrrdng tron, I'a hr-r-v trfr lai circ Irii to{u ilri'
rii slu thtm ldo rn0t sd t{n d0 r'i m$t sd bfii
tuSn c6 lii n qu,,,'.
1. fiAi toSrn tr, (Treng. tii li&l hinh hor: lrip
8 pt h tLdng irirq"rr€tt loin).
'Cho rnil riiilln P u& hni dtra-ng thrtng a, h ctit
nltatr it (). Qttd p arp N11r)t crit tttydn ,",&t a ud b tr
A ud Il . 'l r€n crit tu11?n do lin di?rn Q sao clto
(,1 OI'Q) iti lrrin.g di0m t?ii;u tfiu.'l'irn qur lich di?tt
() khi cdt futlljt qtritu .cltung qaanh P.
l.dl ql(tl.
'! lt,r r.i tt;
\r*r nrOt e*t tu-v6n h{t kj'P. /J. 'l'rEn r:it tu-vdn
tI6 lriy di6rtr Q sao clt(, ('1/JPQ) iir hArrg '1 iSnr
rli€u iida. lihu var' (O.rlBl']Q) iir clirlrl i!iilu ht)a'
Cirirrrr 116 c0 lt tia (..ta, {ib, OP c6 djnh, v{",- tia
di€u hda thri trr cria chtrm cl6 ciing c6 tlinh"l'a
th{y Q nlm tr0n mQt dttdng thirig cd dinh l!6rr
hgp v6i OP d6i '""0i 0o vA 00.
tliuh 1
-.,
ilda i
i-fiv rnlrt iliiirn f)' hril k'i lli'l t.luo'rr.u ilii' Ir0' cr5.t
a v,h ii'r,1'vi 1l'. 'i'rr tirril (A'ti"IrrJ') lrr rnirt hirng
cli0rn rliEu hi,r vi J',JJ', l', (l'li' gilo cliiirrr oira
nr0t cit [uvdrr r r?i circ i,irt ,'t'rll Iil0t chtirrr tJiiiu frrin'
YA-v tlui lich cria (i tii tlriirnq j.hirrg 1i0n hop
vtii OP cl6i r'0i a'rri b, *triitiq "iliitng rlir ilttrTe
gqi li r!rcitttt ,16i c*c c&a dilrn P ddi t'si ha'i
drtdrtg thhng Ot. Ub.
/! b'
ri
i'
li
'u
{_
I
Cho mQt dibni P
(O) cd diuh tdm 0,
sao cho drdng {.ron
dubag frrirr (0).
Ld'i qiili:
7'hudn:
cd dinh ud n$t d.ubn51 ttin
'f im qui.tich nhfrng tli?tn {)
dar|ng kinh FQ lrgc giats uoi
lr
!r.
i{l
2, Bdi tohn Z. (Bdi 166166 rrong. THvTf s6
30 lhtng 3-1967 sfra Iai d6i clril.). -l_. "Ot tinh clr6t l-i thh e iia drtorrg d6i cue ld.;
n N6u clrrtrng ridi cr1.c cia tli6m p rul.r di qu;r
diSrl Q thi rluong elrji cutr cira rliiiur (j tli que
di6nr. P >.
a) Tintt clril rli, lA hi,;rr nlritrrr lror:g lrrrutrg
lrcrp du'ring_ddi .,-:rtc cfni rrrdt diirnr d6i lrri hli
,ltr&r'g tlr,i rrg itBng qui rrrrrr liirrh lr.
b) (iiir sri p, 17 l:i drrirrrg dtji cuc ctia p, e ddi
','iri during trdu (0) vir e niiri tran p.
I'a cri :
l'a thdy q ph&i iliqua P. vi gii sri gc{t dLrcrns
rnot tlirdng tLg:" (fD+ Ffrd: rSrroi ta suy ra:
OQ. OI:OIt. OF'7.R!, tu. rJ(: p' phii rrung
r,6i P.
Hinh S
4. i,li,i t.t,irr sau tli-y si eho ph6p suv lrr ticlr
\ F iluiyrlll ilrir crrc,:iia nrOt riibm e lii l;inu thrror kE.
a ) '[ a iril gili Ir:Ii tuun lien quirn (ld ir ctirorrg
ddi rtrr ci:l nrdt r:1i6rrr ddi v&i hai rlrrtrng th,irrg
CJir5 ,1ui. fI r
/a
Gt: rt'.
Ilinh 2
Gifl si'r Q 1A nrgt cli6m sao cho ilrr&rrg i lCr rr
dudng kirrh FQ trur: gii,ro r.iii duirng irdn (0).
gndirg tiring P0 c5.t dtrring trdn (pe) & ..tI ri
iltrdng trtin (0) & A vi .B, Vi hai tltrtrng trdn
(PQ) r'I (0) trrrc giao rr0n (AI)PH) lA mOr hing
di6m diiiu lrul. tril tlo. srrl rn:
Gfi '=TtZ -'Gt'77i'
Yi ff rrliur.irdn cltrdng trt)n clut)ng k{nir lie ri$r: :
PllO :()i)0.
T:r th{1'diirrn Q ninr tr0tr rrrdI cl rrr.rng thing
e6 dintr ti tlrri)'ng thing p vudng g6c v(ri tluring
th&ng Pri trri cti'6in If sao cyoT{ -{ez lu1r.
D&.o:
Li[y rnr)[ rliilrrr" Q' Lf,l 'hi' trdir tlrrirng p. Ia
t}{y-d,ri','rg trt)n tluor,g kinh /'aJ' di qua 11. vi
pHO' : {it)o. J a c6: OiZ = e8z = OXL OF, \,i vl-r-
(ABPU) ld trOt hing di0m diBu hda, t.rk it6 sul
"ra dLrirng trr)n (O) rluring kinlr rt8 lrLrc giao i 0i
drrrrirg tlou 6*6on kinlr 1'Q'.
\,+,v r;ui tich cira Q lA tlriorig lltgng*p vu6lg
girc viri P(l iai r1i6m fI sao chr; OP. Aii = R3, lt
l. irAn liinh ciia rlulrrg trdn (0). Otrd,.ng thing
d6 goi ia ddang rtdi cqc c&a elii-m P ddi udi dvdng
gy5n (()).
Ctti i;. Irdu rlu0ng thirr,g \iiii P voi mOt r1i6m
O niro 116 r;ria p rrii cit ilrr0rrg trdn (O) t6i iiai
di3rn C, f, thi {U/)PQ) le mgt trirrg (tiirrr di€u hoa.
4
f,
Ilinh 4
n
-*40
44"k i.\
l"#
it
\{\o ,
U
I
,. 1
a
t
n
t
VE hai c{l 1u._r.6n hf,t ki pAB, FCI) AD vd BC
,c51 nhau ri .E Chrrng minlr rArrg Oli lii irrro,rg
diii cuc c[a rlii rrr P t'l6i v6i a. D.
List qidi. Gii srr OE cht lt,lB it i_). i a ha l
clriing rninh rriug (ABFQ) ll lr;rng di&nr riiEu hoa.
Ap duug tlintr l;i lld-n4-la-{rt li rlinh lv Xd-
va ri\rr t;rtrr {ii,. -ll}(t. 1r, c6 l;irr lun! :
It,4 , ltB t tt (J.t tit i,
.-''-'-:"ivlt-' ''.-----!
PI] no r.-A AG ri, ,fi
ili6rn -B cIa'' hai ilrrring th$ng C0, r.] .4/1, Op s6
Ii drrong phii tim.
b) Oach r-E clrrdng ildi cu.c cria mi)t di6rn (&
lrong hay 6 ngoai -urrlt duong tr.on rli cho) d6i
v(yi rlucrng lrdn dci eirng tuong l.t,r {xenr hluh 5)r
VF hai cAt tuydn FALS,pcfl brit ki. Cric iltung
thring AD, BC cit nhau & e, AC, -bD cft rhau
,", Ei"
Dd drirrg th{y ring drro.lg rl6i, cuc cua p cldi
ro'i tlrrdrrg trr)n (O) la dudng thling p di qua
E vA Q vd tluong ddi cuc cira e adi-vOi dtrolog
tldn O Li dLrcrng lhing q di qua p vd .8.
Chi s:
r*6u -F & lgudi dunng trou (0) vA u6u vd hai
ti€p tuy6n PA, PB r,6i drrcrng tron thi t10 ddng
Ihflyliing ducrng thSug JB le duung d6i cr,rc
i:r1a (li6nr p drji vtri r1 rrong trOn (O).:
C'ht't u:
Gid sr? pti&i r-e duong rl6i t:uc cria di6jn E
ildi vc_ri s, b (hinh {). Trong tr_ur}ng hqrp nav la
v€ hai r.f t lu,l,dn- AED, CEB.bet k-r;r6i [{v ujao
PI
Lu d{r su 'Fa
H inh
oa
-- j-'
11D
t:
I
!":,
ft
i,
ir
t'
Ndu 0 & trong duong trdn (O) vA n€iu 1,c cer
tu_r'6n dQIJ, r'iri v€ cdr t.i6p luy6n t} .{, B vsi
ilrr0ng tron cirt nhau tq.i p thi dti dring thf,v ling
dtdrg thirng l,I vuirng goe voi ilrrting iiring {,rg lt
iiuon,J tl(ii cuc crla tiiFnr eddi vrii atrong t,lOn {O).
Ch,.: ban t:6 1i1ir vr.n tlrJng ciic diiiu hi6u tri6{
'ndu trdn . r.6 rlrrirng dcji ci.r.e d6 girii c6c lrii toin
sarr rtftv (crte hAi to:in niy cfic Lan ciing co rhE
clri ri rilg t:au.kiEn Ilrrlc ilroc Iioc d lfrp g"chrri,dn
1.r.,irr rli rlifri).
lldt t. 'l'rin cdc eanh ;.lC vd dE cfia ta.ru gi6r:
.4lC iS1. hai ili&nr bdt.,kv D va E, goi 0 Ia giao
diiirrr t.rj.ir t,rlc,lrrdrrg th;rrrg BO ve t.ti.,f , r.,,i arllns
ll.l,,s j:] If,v rnot .rti6m b{t ri-v r. Cac duong
lhring /0 r;i 1.C,.ri tr,;ic rlu.ong thiing CE vd, BD
theo th* trr & citc ttiFrrr ff "a C. (lir.rng minh
:i:,g l:, 1.I.,:,rng rhing DE, r/G va Brt df,rrg qui
lai rrt6l rliFirr.
[!itrh 6
Bdi'2. Cho rn0t tam gi5c ndi tirjgr trrinc rn6t
dui;ng tlr5n. {-ihring nrinh rdng circ giru di&nr cria
lidp tuvdn & cric dinh tam giAc vo,i clnh tldi ld
ba di&rn cung ninr tr0u mdt iludng- iiri.irg.
Edt 3. Cho rlrrln6 lrri;r t0) vrl iliErl P 6 r:goAi
rlttong trr)Lr, ti6p tuyeiti 7,-1, PB vt'rj rirr;r:g tidn,
'Tu'"8 ta r,6 1lD vu6ug go,: vtri drlirirr i<irrjr "4fJC.
Chrlng nrinh ling diro ng thing l,C di ,luir irlrng
ali8m /i cria iior,rn thing 8D.
lJdi 4. Cho tlu<rng tron tAnr O, drrrrrig thing
rI v.{ clibm / tr6n ri. lu diiirn ll{ Irir. tiiii. kv tr.6rl
d vE hai tidp tuydn -tlz{., MB vt'ri drrring tron,
vA iri di&in.I vE clc duong thirrg sorg soug vdi
cic tidp tuytin dd, cit cac- ti€p lu;'€rr n,,y 6 f,
vA Q. Cht'rng minh ring khi,l{ chav rr.r)rr d, pe
di qua mdt di6m c6 d!uh.
Btii i. Cho gac rUA vi iliErrr p tlil canh Or,
vri duiing tr.On hi6n thi0n ti6p xric voi c{r: canh