ch&
Trs
BAO RA HAI
TOAN
IId N+i
THANG MOT KY
'l'hn kg toa
f
soarr: HOANG CIIUI{G
D&y n6i: 52825
lE$ uqi h,uttrp
HOr 't'oAn Hec
vI$T NAil
nhiQm: NGUY6i{ CiNH
sdr : 70 'l'r'hu i{ung Dq.o -
"vOt car:yfr.\ \'t)'r cAr: nAN rqt \:ELt raA^,
fryc do
;
l,
. |'r .
.qs rg t & rr .. \rri{lT{y
vA t-,At- t'i-l ril' lt{fiF{ HiNffi
rt.trx 'rtruc TRiNH
fRoliG lrii ntii cLruy6n vcvi cic h4u tr€ 1,0u
ll tol, (('lhrrr 11uan i',.,og .ir" mQt c6ch nhin )r
(SO :l-:+ utrril l9?r(). g'r1o su Nguy€n C&nh
l'oin i1n chi ra thiEu hu6ng suy nghi gdu sec
lien quan.d€n vectry, sd phirc vi c6c.ph6p bi€n
hinh.
Chring.tOi nru6n,cu thb h6a them m6i li6n hd
gita .ba loei Lldi tu.Ing Iren (sd ph'rc, yecto .,.s
cric phep bi6n htnh) yA vin dung vio nr0t s6
biri lof n cu tir$ dB girip circ tran tilp drlqt thdm.
.,
1. 56 phuc z :, ir + bl ha-v .:. : r(cos{p f isiue)
trrrung ung v6i rngt di6m Z tlong nrf,t ph&ng
hay n:$t vertct 0Z (hiuh 1).
Tbng trai s6 phue tuollg irng vtri t6ng hai vecto.
Tich \ai ed piruc (zt.:2) ttong ringIvoi pheli
--
qlory OZ1 m$t g6c rp2, ti6p ttreo ph6p vi tu't&nr
O tf s6 rs:
z::1.7.2: r1rz x Ieos(rp1 * gr) + fsin(rpr + ez) j
Nhrr viy nhA.n s6 p.huc:.v*i iiiz,*t--0) tuolg
irng voi phdp quay OZ qu1nh 0 m(rt g6c gOe I
nhdn sd phuc z vdt-._n (oio + ct * I : 0) tuorrq
rlng voi phdp quay OZ quanh O nrOt g6c 120q
(ni a = -!3,+ it/e tz=eos(2ri13)*irjn(2nllt).
.lF..*
.!'ffii.
;,i.---- lrl'
L-, 2. Tron eo sri mdi li6n h6 vua
chring ta c6 thE tim thdY mSt s6 quY
sau tldy :
(a) Duong trdn bin kinh bing 1:
Eurlng trdn b6.n kinh bing r:
(b) ;;2:(a - b) {; -il.
T;2 : t;.
(c) Tam gi6c ABC li dbu:
(ndu tam gihc ABC eb
azatabar=0.
(n€u tanr gi{c -48C c6
dddz+at1=0.
Sau tIAy chrlng tOi xin giii thictr m$t vAi quy
t{c <dichn:
Quy tdc (cl.). TruOc t.i€n nh0 ring trong pham
vi sd thrfc, di$n ttch tam gi{c ABC mi cAc dlnh
A, B, C lhn luqt c6 toa dO (ra, gt\, (x2, y2),
(rs, y:) dugc tinh theo edng thitc :
xtgtT
rzgzl
:fi!3 1
Bty gid ilSt a=sl*igt, b=rz*iAz,
trong il6 cac ui, ul, d6u Ii s6 thqc.
c:r3 * iU3,
Th'g rhi;
a* ab4ozi_ o
huOng duong) hay
hu0ng 0.m), trong
C uinr tr€n iludng trdn
i (o*b)(/r-c)(c-a)
.abc
dqng v0i tam gi{c
trinh biry,
tic < dich >
:i * 1.
:-_.2
1
$:- 2
i
'i
(d) Di€n tich tam gi6c ABCI
aa1
b !1
ccl
*t nt- igt I
.c2 f.2 * igL 1
13nB-iyS 1
rt*igtrt-igt7
xz+iAzrz-igz1
rsfiysrs-igg1
C-
J_
(Aesc)
D{c bidt n6u A, B'
bAn kinh b8ng 1 "thi
,s*
(Aasc) -
.l"ir
LI
-lntt
4tlccl
=41-8.
Trong A nh$n cdc phhn tfr cria cQt tbir nh{t
vdi (-1) rbi cgng viro c$t thu hai; trong B c0{g
circ phln t& c$t thu nhdt vdi cac phin tri cQt
thu hai ta sE c6:
+igt rt - igt 1
ig2 *2 - ig2 L
.tgt *z - iUs I
i37rrr1i lrrr/1 I
igtrzTl:-Zil e.2!t21
iy313 1l l13yrl
(d) Tam gi6c ABC il6ng
AtBrCt:
a-b
al-br
G\ AEIICD
trdn :i: 12 :
aal
bbt
cc1
i
4
b-c
:-:- l-
h-cl ln s6 phric). = -4iS. TrI tI6 S =
v6i A, B, C, D ni.m tr€n dudng
ab: cd.
Quy tic (i), E[t d = rt * ig1, D : 12+ iU2.
c=lr3 * igs,. thE thi
;_;
= .- bi€n thinh
b-c
a-b
(g),{BICD voi A, ts, C. D nxm tr€n tlu&ng
tron :l: 12:
ab*cd=0.
(h) S ld hiqh chi6u cta ,{/ tren c{t tuv-6n .4".B,
trong ato A va .8 nlm tr0n ilu]ng trOn :i : 12 :
g:(a * D - m- oU^liZ.
(i) .4, B, C cQng tuySn JID : *tl n*,
b-c b-c
'abc = abc.
*.' (r-aXI- 6+G-i>1a-ot
(k)cosUrg, l"l'n)=ffi:
b-c
ter-xil*ttyr-gr)
(rz*ril*i(gz-gs)
(rr*,;z) -iqlt-uz) . E{trt*iz=X,sz-
(rz-cs) -iqz*gs)
rs = X'. ,Jt * tJz = \', Az; C, = 1", ta sE c6:
, tuc'A, B, C cQng tuyGtr.
guy t6c cd'n i1i cAe bqn tg ttm l{y.
x+i}' x-ry x
= hay XY'- YX' tuc - -
x, + i1:' x'- iy' ' r
r
{,
\-
r'
C6c
\
h
B
lf
r
{
3. BA-v gid chirng ta hiy t{p v6.n dung c6c
qu] t*.c tr.en trong mgt sd bAi tgp cu th6.
Bii sd I. (5 tam gidc tlBu). Cho ba tam giAc
dEu .t1BiC; cung hudng (i: !,2.3) saci eho tam
gi6c At,A.zAt cfing la elEu vA ctng hu6ng. Chung
minh ring c6c tliilm giira p, e, li l[u luo,t efia
CtBz. C2Bz, t)sBt cfing IA tam gi6c dEu cting
hudng.
Bii s6 2. (6 tam giac rlEu). Cho hai tam
gi6c dEu A1,lzAs, BtBzBs ctng huong. Tr0n :liBi
dung nhirng tam gihc diiu. :liBtC'i cring hu6ng
Chung minh ring tam giAc CtCZCs cilng li tam
gi6c dBu vir crlng hudng.
BAi s6 3. TrOn cric canh Ilf.', C,.1, AB cria
tam gi.{c b5t k"\} dung nhirng tam giic dl6u cing
hudng DC.-lt, CABI, ABq. Chitng minh r.irrg:
a) C[c lAnt,46, Bo, Co cfra nhitng tam gidc il€u
vua dung duqc ll clinh c.ria nr6l. tam giic dbu
nguqc bu6ng. b) C:rc tliBm giira ,A', B', C' cfia
AAo, BB., CCr.la ctinh cira mgt tam giirc <1€u
vA - nguoc trudng v0i huong ba tam gitie dE
dgng.
Bdt sd 4. t3 tarr gi6c d6u). Cho tam gilc
ABC, Dang tam giec zloB6Co sao cho nhirng tam
giAc AoCoC, BoCoA, OoAoB ld dEu vi cL\ng
hudng.
Bii s6 5. Trong tam giic .lBC cho truocr
h{y tim,qui tich nhting di6m p sao cho hinh
chi€u cria M l0n ba canh cria tanr gi6c d6 lA
dlnh c&a nrSt tam gi{c c6 diQn tich .So kh6ng
col.
Bii ed 6. Cho tam giee b{r ky. Tr€n cAc
eqnh va vE phia ngodi cria tam giac tld dung
c{c tam gi6c tIEu. Chrtng minh tem cic tam gidc
vtla dung tao thinh mSt tem g[i6c d]u ?,1. BAy
gio dung tr6n c{c cquh vi. vE phia trong c6.c
tem gi6.e tIEu, goi tam gifc c6 dinh li tdm c6c
tam gi5.c vua durrg ln 72. Chring minh ring hiOu
clrc cliQn tich cira 7't yit Tz blng di6n tich crla
trm gidc tlfr cho.
Bii sd 7. (eai thi toan'qudc t6 lfin thu 19
nim 1977 do IIi Lan db ,igf,!). Cho hinh
vudng ,lB{:D. Dung l'6 phia trong hinh vu6ng
d6. tarn. g'i{c dbu co canh 'blng canh hiuh
vuOng: '1BI(, BCL, LDlt, D:l,\'. Chung minh
rlng di6nr,gita cria b6n do4o tlt8,ng t{L. Lil,
,11,\r, -\IIi cirng tt cli6m giila c{c itoan AIi. BK,
EL, CL, C)lt DII , Dj vd -t.V Iil Cilh cria mgt
hinh I2 ca rrh (lEu.
BAi s6 8. Cho tam giic ABC vh, trong tdnr
O. Dqng tam gi6c d6u O,4/) yh OAe.. Gpl pr vi
Q1 la dilm girla cfrr, CP vh Ce; (1) ln duo.ng
phtn gi6c. cia goc Ptoer. Chrtng rninh ring
ndn Pz, Qz lhn luot IA di8m gitip cria Bp vh Be
3{ ctng Ie duong phaLo gi6c cfia g6c
PzoQz.
Bii s6 9. nf0t da giirc 2n canh AtAz... Azt
nQi ti6p tro,ng mgt rludng trdn. Tir di&m 81 b6t
kj. tr.0n ilurrng trdn do A.i: Bfizll (hay l) vOi
41.12 ; sau iI6 k6 ,B2Asl/ (tr"y I) v6,i AzAs vA
ti6p tge nhu vey mi.i. Chrlng minh ring llzo9til
(tray l) v6i zlznAl.
Bdi sd 1O.
X (z - fjj):0,
l6i thi z ui.m
r31
Chrlng minh ring ndu c6 I 1,, )(
j:t
trong dd c1 lir fiinh cria a giAc
trong cla gi6c do.
*.
f
i
I
I
,l
m"]ruf; rrm r$Iuc vrerry
RONG nhi€u b&i to{n hinh hoc ph&ng cfing
nhrr kh6ng gian, phuong pirAp vect<r ti
ra rf,t c6 hiQu ltrc. BB girlp cic ban c6
thB vfi.n dBng duqc phuong ph6p dd, bai ndy
sE ndu l6n cAch x{c ttiuh b&ng vecto mgt s6
dibm il[c bi€t nhu trong tAm, t8m cf,c duong
lrdn (hay mtrt chu) ngoai ti6p, nQi ti6p, birng
ti6p,.. cira laru gi{c (hav tu di6n).
I. frudc h6t chtlng ta bit diu blrig nhirng
bAi toin trong mIt phfi.ng.
1. X6t bei toA.n sau diy: Clio m$t dibnt M ndm
trontt mdt tam gid.c A1A2A1. Hdg rdc ttinh cac sd
thrc kt, kz ua kz khang dbng tltdibitrg 0 sao cho,
I 1riffi: o
tkf hi0u X fi Aay vi sudt trong mgc I ituryc
Cl
hi8u la tiing X I.
i=ll
Trudc irdt ta thdy r&ng n6u (kr, kz, J<g) tt
m$t bQ s6 thr3c th6a mln h€ thtc d6 cho thi
mpi bi s6 ltruc (lft1. lk2, '1113) trong t'I6 i, Ie
m$t sd thrrc khic 0, dEu th6a m6"n h€ thuc dl
cho. Vi vAy nghidm cria bai toan duoc xde dinh
sai khfc nrdt thrla sd khfc 0.
Eb gili bili tc6nr niy, ta hfiy gqi Bi lA giao
0i0m clra ilrrdng ndi ]14 vi ilinh ,4; vtri canh ddi
di€n (t: l, 2, 3); st, s2,. ss theo thu tn ]ir <ii6n tich
c{c tam gi6c .Urlg"t:,,11r1s;tri vd Jfil1.,{2 (hinh 1).
il .
I
,43
Nriu Nz vt\ N3 lEn luqt la chAn c6c dudng
vuOlre gdc ha tu .42 vd J3 xu6ng ,4IBl tht
* .-
BrAzl B].At : * AzNzl A31-s : - Ol2)'AzXz.
Miti ltlD rsl{s. M,4r = _ sg/sz,
. N6u P in mQr diEm btu tri trong mdt phang
thi hQ thrtc
{
8tA2l 617r: - ,s/sz.
cd th6 yi6t rhanh <iJ2- frl1rf,'-R!]= -,sr,,
tu d6 rrit ra (sz * r:) pEi - ,Ji, + ,3ffi trt
Ta h[y chon P !n'di8m chia doan lrBr theo
I i s 6 -- (sz + ss )i sr, r itc I iA I /p;i: - (sp l- sa) /s r b a y
+ .+
srlr.ll 1- (,sz*ss) pgr : 0
t* thuc n-Ay .cirng vdi (1) cho ta ,rE +
ssP.{2 -f- rrF.a3= g nay XrsiF?r - o. (2)
. Nhu v{y, cti3m p ric dtinh b&i hC thrlc (2)
nirnr trdn dudng th&ng lrBl
Db f ring trong h€ thirc (2). c6c chi s6 l, Z, B
c6 vai trd nhu nhau, ndn bing c6ch tuong. tu
ta ciin3 chung minh dugc ali6m p xac dinh b6,i
hQ thuc (2) cflng n:1m tr6n cac tlu&ng thfr.ng
,4zBz v$ .,la8s. N6i cich kh{c dii}m p *a* *int
niru vfry chinh li di6nr -11 cli cho.
Nhrr vAy IA ta di cht'tng rninh
ti le l'di s; (i : l, 2, y, vi uoi
nilint tron11 tam gidr. AtAzAg ta c6
4
Js;.1/,{1 : Q
{4fl
T["r
1,q
$)fl xAc
SIAC VA
m4e IinffiT Tffi$NG. TeM
plNi{
TIICI I{G
HIE M
rurEN
QUY
tlb Chi Minht
NGUYA\ (]ONG
G)ai hpc Sa pham 7,.p.
_l
I
1
I
i.:
i:;
t
I
L
I
i
-I
lr
I
I
L
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t-
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i.
i
l
t
\i.
t
I
,
I
I
iluoc rXng /ri
mei di7m hI
hc thic {
i
(r)
\Cu /l Ih nrdt.ditlrn tiry j tlong mdt ph&ng
thi lrd rhric rre,n c? rhir yi6t thenhXsi(5f, ,-6at -=o
tt d6 nir ra df,:XuG/Es1:XsioTi/s vdi s
lir di€n tich tam gi[c /rd2;t3. Db cho gon ta sE
quy 11} ki-Tg:theo *vecto b6n ktuh rirc ln
vi6t (1,11= :lt. oAi: -4i. Nhrt vay tr€ thuc tr0n
co thE vigr thtnh
+
.!4: )si,li,/,\ , (I )
l^-,.
\:
Hinh I
2. Trudng hqp di6m ,U nirn ,n61odi tarn gi6c
.{rAzAs. bing tlrrong l6i tuo.ng t\r tr6n ta tl8
dirng eti tdi k6t qu6 sau dey. Ndu ta d6nli s6
c6c mi6n cria mdt ph&ng nhu & hinh 2 thi trong
truirirg hop. tI thu0c miBn sl hoilc cr'l thi he
thrlc (I) ilrrdc thav rh6 b&i.
$,: i' ,is
+1, r.,. tl,,'
xr;lfr=s
<
Zsin ,li IAi = 0
Trr rlri Buy ra c6c h{ thtc
*
I =I.ui Ar,2p
(tlong do p lri n&n chu vi cira tam giac)
r :. th2ius,4i * nstu T +_tt*z;ili
hzhs*hsht+hthzl
== X *in,4"i i?iE otn,t,
: E sin.4i f,?
I +fi casetill)
3
(v6i r( :.rr. lA kj, hi€u tich).
i_t
c) Ndu diEm ,lf trung v0i t{m ft crla dur}ng
tjgl bens ti6p tam gi{c Arl,:As nlm trong g6c
AiAtAs thi ta thu drroc cr{c h6 thirc
'-b+
* aL ltAr + az ItAz * as .{1d 3 = 0,
_: -t r+ --D
- h1 ' ltAl* h[' ItA2 i- h3-I I13, = 9,
-- ,in,rrT?r * siir,{z ffd, * sin-,13 ffi - 0,
4t#&iDdE
. 11 : (- u:At* az Az* as "4s\12q, - at\,
dP4*5
Ii : ( hzltt .lt * hsht Az + hthz ^qi/
lG hzhe * hsftr * ttthil,
FF d* *rF .+
It : (*" sindl .41 * sin;tz "{s * sind3 rls)i
l? sinAt * sin"4,e * sin.4s)
: ( sin,4t .,4r * sin,{2,42 * sin. e,{g)/
/4c os(lt/2)s i n( Azl 2 t x sin(,4s/2).
d) N6u tli6m 11 trr)ng vdi tAm O cria dudng
trdn ilgoai ti6p tam gi6c rtrCe/s (hinh B) thi
,- )''
- Hirth 3
/'\ -/ .\
-4-
st'sinlzO,4g = s2isinA3O-,lr = rs/rinarO,q]
lray
sr/sin2dt = szl sin[
Az = sg/sin2..{B
\. f{; I
n'rlrf, I
#
/1s
#
t
Htnh 2
,lM; + sz;rI + ssnffr: o, cirn (I') rhi
dunc thay the bfri cic he rhtc
{{-
M = (-sr.lt + s2j2 + ss,l3)/S n6u ill € qr,
hoxc ru*: rrr,il-rrJi-rsaI)ls n6u,41 €. e.1
Trudng hqp ,?l1 nim tlong cAc miEn khr{c
ctng ctuqc gili quy6t tuong tu. Con n6u M nirl
trdn mQt ho[c hai trong nh&ng du]ng th&ng
AzAs, AsAr, a:,"1.z thl cric h€ thri; .(I) vi{I') ven
dring. Lric d6 sE cci m6t holc hai irong nhri,ng
sd si bing 0.
- :. ryf gio ta hflv r-An dgng c6c k6t qufi da thu
duqc tlb xAe dinh c6c di6m d[c bi6t tlong tam gi6c.
a) Ndu rli6m .[f trung r'6i trong tflm G crja
tam gi6.c lt.4z.4s thi 16 ring s1rs2:sg va ta
thu duqc cdc k6t qu6. quen thudc
+
tr GAi: 0 vd G:X,4i/:.
' b) Ndu di6m ,4f trring voi tem 1 eria rlucrng
trdn ndi ti6p tam gi6c .4rJz-ls thi stl a1:s2f 62J
$lag=rlZ trong d6 al =lZJg, as-AgA1,at=AtAz
vi r le. btn kinh ilucrng rrdn n6i tiep. n4
thric (I) tr& thlinh
Ioilz{i: 0 (B)
DA y d6n cAe h€ thr?e atht = azh;= nan3 tf,t l-e
ilu.dng cao ha .tft Ai) vir allsin At: Azl*iaAz_
dglsin 43, tu h6 thrlc (3) ta suv ra
I
I
'i., l
Y1,
\,i \\\\
ii
I\-
_,,ts-,
;:\