= +(o.,q2 + oB2) + 4oC2 = 4(a2 + b2 + ,.2).
Suy ra 2@B? + 8,4 + OMI + Cttt2)
= 8(a2 + b2 + c21 (1)
Lai vOn dpng vdo c6c hinh binh henh AA1B1B,
ACB1M, CAIMB tac6
2(CM2+482+hA2)
:2(a2 +b2 +c2 +*2 +n2 + p\ (2)LAy (1) tru (2) theo v6 ta c6 LAy (1) tru (2) theo v6 ta c6
oM? =3(a2 +bz +cz)-(*'+n2 + p21.
suy ra
oG2 =!O' *b2 + c21-] r*' + n2 + p21.
PHIJONG THIJC 3. Khai thdc, s{r &1mg cdchinh trung gian ldm phuong tiQn cho hoqt hinh trung gian ldm phuong tiQn cho hoqt
ctQng chuydn hoa bdi todn kh6ng gian vi bdt
,'J
toan pkang.
Thlrc hiQn phuong thtc ndy ching nhirng girip
hqc sinh huy tlQng t6t hon chc tt'rthric ctra hinh
hgc phing d6 nghiCn cr?u hinh hgc kh6ng gian
,,.
md con girip khac phuc nhirlg h6 ngdn c6ch md hgc sinh thuong gap khi chuy6n tu viQc nghiCn ciru hinh hgc phlng sang hinh hgc
kh6ng gian do sg ph6t tri€n tri tudng tuqng kh6ng gian cdn han ch6.
TORN HOC
Do oG = Io*,
*Thi dq 5, Cho fih dien ABCD c6 cdc canh BC : CD : DB : AC -- AD : a vd AB : b. Hfu
t{nh bdn kfnh mdt cdu ngoqi tidp cila ta di€n
theo a vd b.
Khi gini bdi to6n ndy n6u hgc sinh sri dgng kitin thric vC truc duong trdn ngo4i tii5p tam
gi6c BCD vd m{t phing trung tr.uc cira tlo3n
th5ng dC x6c dinh t6m m{t cAu, titip d6 tinh b6n kinh thi kh6 phric t4p. Tuy nhi6n c6 th6 chuy6n viQc tinh b6n kinh m{t cAu ngopi ti6p ci.r-tfi diQn v€ tinh b6n kinh ducrng trdn ngoai titip tam giirc ABK. D6 ld duong trdn 16n cira
m{t cdu ngopi ti6p tu diQn, v6i K ld giao tli6m
ci-r- BM v6i dudmg trdn ngopi ti6p tam gi6c
BCD vit M ld tnmg <li6m canh CD (h.5).
ra c6 BK =+, AM =+, aru =!ax
n6n cos FER ti*ducv. c theo c6c cpnh cua tam
grdc ABM. Tt d6 AK tinh dugc theo a; b vir
"o"frR gi6c ABK. nOi ap dlmg Dinh nhd str dung Einh li c6sin cho tamli sin cho tam
gi6c ABK ta tinh dugc brin kinh dudmg trdn ngopi ti6p.
CHinh 5 Hinh 5
PHIIO|{G TIIIIC 4. Khdo sdt, phdt hi€n hinh trung gian ld dnh cila hinh cd:n tim qua mQt
phep bidn hinhf ndo d6.
Tfurc hiQn phucrng thric ndy lim s6ng to:
Oe tim t6p hqp (It) cfuc die:m M c6 tinh chAt
(a) ndo d6 ta chuy6n sang tim t$p hqp Gl)
c6c di0m M1c6 tinh ch6t (a1), trong d6 Mllit tao 6nh ci:r- M quaph6p bitin hinh I
__a-.,
*Thf dV 6. Cho hai dwdng trdn (O) vd (O2)
cdt nhau tqi hai di€m A, B phdn biet. Dwdng
thdng di dpng lu6n qua A cdt (O) vd (O) tdn
luqt tqi hai di€m M; N khdc A. Tim tQp hqp
trung di€m I ctia doqn thdng MN.
Ggi E, F 6n luqt ld c6c trung di6m cria c6c dAy cung AM, AN; D ld trung di6m cria do4n O1O2 vir 4 ld hinh chi6u cria D 16n MN.I<hi d6 ttr gi6c EOQ2F h hinh thang vu6ng. Tt
giirc AKD\ nQi ti6p trong mdt ducmg trdn,
cludmg trdn niy chinh ld tao 6nh cria dudmg
trdn qu! tich cAn tim qua ph6p vi W V} nhd
chimg minh dusc 7i = zVi. Do I thuQcduong trdn tluong krtlh AD n6n l thuQc ducrng duong trdn tluong krtlh AD n6n l thuQc ducrng
tron duong kinh ADr 6nh ctra AD qua ph6p vi
w v3 $.6).
BAP {mg dugc; ddy ld hinh trung gian rlugc
ph6t hiQn, h6 tq cho vi6c giei bdi toan (h.7). Tri
d6 c6 thd thgc hiQn c6c bu&c dpg nhu sau:
A
Av C:
o DunB tam siitc ,n::n:"ba c4nh da biot.
o DUng duong thtngBM song song vuA1C.
o DUnB di6m A tlOi ximg vdi t1i6m A1 qua tryc BM.
Tam gi6c ABcli:tamgilc cAn dpg.
BAI TAP
1. Cho hinh ch6p S.ABC. Ggi G ld trgng tdm
cria tam giSc ABC. MQt m{t ghing (P) Dn luqt
cbt cec canh SA, SB, ,SC vd cEt
^iG t4i c6c di6m
Ab Bb Clvir Gy : ,14 SB SC 3^lG untmg mmn rang .t4, * sB, * scr = sc, 2. Cho ba dudmg thing a, b, c d}i mOt ch6o
nhau kh6ng thu6c ba m[t ph[ng song song. Qua
a,b dtmgcflp m{tphlng song song (P), (Pr).
Qua b, c dgng cflp m{t phing song song
(Q), (Q).
Qua c, a dgng cflp m{t phing song song
(R), (Rr).
C6c c{p mpt phdng song song ndy d6i mQt cit
nhau tao thAnh mQt hinh hQp. Hdy tim tfp hqp tdm hinh hQp ndy khi hai tludng thing a, b
cO Ainn vd ducrng thing c di dQng c6 phuong
kh6ng AOi va thuQc mQt mdt phing ( a )
cho tru6c.
3. Cho duong trdn (Q, ddy cung ,nA cO dinn.
Ggi M ld di€m di dQng thu6c cung AxB cua ducrng fdn (Q. Tim tpp hqp c5c hinh chi6u cira tr.uc tdm H cta tam gi6c ABM 16n duong
phdn giSc g6c AMB cira tam gi6c d6.
M E {IA1 I
l-t \
:/
Ilinh 6
PHUONG THUC 5. Hinh trung gian daqc
khai thdc vQn dung voi muc ti€u ldm phtrong
tiQn trqc quan, sdng td trinh tf ting bu.6c
phdn tich hoqc lQp luQn trong ti€n trinh gidi
m6t bdi todn.
*Thi dr17. Dqrng tant gidc ABC bi€t AB : c;
BC: a; trung tuy€n BM chia g6c ABC thdnh
hai phin sao cho 78il = 26fri.
ViQc dpg tam gi6c ABC quy vC 4mg di)mA.
(h6 kh[" d6i voi.hec sinh ld de dqng di6m A
cdn phii c6.hai dieu ki6n, trong khi d6 chi m6i
bi6t mgt di€u kiQn ld,4 thuQc ttuong trdn t6m B,bfunkinh c.
Oe Uric phuc kh6 khdn d6 c6 th6 hu6ng ddn hgc sinh vE th€m dii)m Al tlOi xtmg vor A qua
lrqc BM. Khi d6 MBI/Ap n€n BAp h tamgi6c ctn c6 BA1: AtC: c. Nhu vdy,tam gi6c gi6c ctn c6 BA1: AtC: c. Nhu vdy,tam gi6c
TORN HQC..
i r't.nl., L\ ..LU:L H ,rltrr ,ai rl \rr r ro r LIt- _+- rv
NHIIU (ACH GIAI
EAI TOAN 9 NCG. Ilai fifiitg tt"in tam o t'd
tam Q ('dt nhaLt ttti t'tic diim -4, B srto cho
OAA - 90". Tren doan tltdng AB liy clilm T sao cho 7'khar: A t'r) TA < TB. Durmg thcing
OT ciit drong trdn tdm Q tqi diim E vo'i E ndm ngotii ltinh rrdrt lam O. Dtrt)ttg thung QT'
cdt chrdttg tron tdm O tqi diOm F t'ri'i F nittt
ngctdi hinlt trdn tant Q" Chrng ntinh rdng
Gi : azi
Ki hiQu H ld grao di6m cira hai cludng thing
AB vit OQ th\ AH L OQ. Gqi C ld giao di6m
cira do4n thtng OT v6i iluong trdn t6m Q vd D
ld giao di6m cira do4n thing QT vli ducrngtrdn tim O (h.1). trdn tim O (h.1).
DINH HUONG 1. Phdn chia hai g6c dang x6t
thdnh t6ng cdc g6cnho tucrng img bing nhau.
Cdch giiii 7. o Tri LOCA a A,OAE (g6c o dinh
O chung vir dk : 6il)suy m OC.OE: oA2