He thilc luong trong tam gi6c 11 bdi... drrdng trdn ngoai ti6p tam gib"c ABC.. dudng trdn ngoai tiep o, R.. Cho tam gii"c ABC.Tim vi-tri c6c di6m P tren dudng trbn n6i ti€p sao cho t6ng
Trang 1\d,,* " / ,,-,)
DAr sO - Ldp o (DQT 2)
A Ddt 2
I N6i dytg
t] Oitrtr li thua,n vd ddu tam thitc bflc hai, dinh li Vi-6t (25 bbi).
2 Lrrgng gi6,c (ding thr3c Pitago, c6.u g6c c6 li€n quan dac biet) (30 bbi).
3 B5,t d8ng thfic Bu-nhi-a-c6p-xki (15 bhi)'
II Phan ngdn hdng cdu h6i
Dinh Ii f,huln dinh li' Vi'-dt ah' ttng dpng (cd'c bdi
Be,i 1 X6t d5,u cta cac tam thrlc bflc hai sau
tit 1-25)
c)-2r2*r-1.
Bei 2 T\m rn dc (hn7L)p'- (2m+L)x+* +3 2 0vdi moi x'
FEIg Tim m,Id (rfi-f 12 +2(m+ 1)r **:2 < 0vdi moi x.
Ae_4 Timrnddphrrongtrinh (rn2 *4m-S)g'-2(*- 1)r+2:0 c6hai nghi€m drrong.
Bij 5 Tin rz d6 ph,rnng tr\nh mz? -2(,m - 1)" *4m- 1 : 0 cd hai nghi€m tr6i
t1
dau.
Bei 6 Tim rn rld phrrongtrinh rnr2 -2,(m - 1)r *4nt'- 1 :0 c6 hainghiQm6,m
phd,n biQt.
bai Z Tim rn dd phrrong trinh (rn - l)rt - 4rnr * 3rn l- L0 : 0 c6 hai ngiri€m
phon biQt ldn hon 2.
bai e Tirn rn dti phrrong trinh (3 - *)rz *Zmx + n'L+ 2 : 0 c6 ha-i nghiQm phi;r
biQt nh6 hon 1.
th6amanrf i1I12.
Bei 10 Tim m ad'phriong trinh 13 + rnr *2m * 8 :0 c6 3 nghiQm phdn biet.
BAi 11 T\m rn a<j ptlrong tr:inh (m - l)rn -2(**2)t2 *2rn* 1 : 0 c6 4 lghiQrn
phsn biQt.
bal f Z Cho a, b, c ib cac s6 thuc th6a m6n 5a + 4b + 6c : 0 Chfing minh rling phuong trinh or2 * bx * c: 0 c6 it nh6t mQt nghiQm thUc'
bai fi Chrlng minh rh,ng vdi mqi Q,b,c phrrong trinh sau ludn c6 qghiQm (r
-a)(r - b) + (r - b)(" - c) + (r -^c)(t - a) :0'
minhrb,ngc<0
Bai 15 iim piruong trinh b6,c hai c6 c6c nghiQm lir, cic lfry thila bflc b:r crla c6c
nghiOm cria phrrong trinh 12 +5r -J : 0
gai f O T\rn m dii pS,rong tritrlt rnr2 - 2{* * 1)r * m- 4: 0 c6 hai nghiQrn ph6'n
biQt th6a rt6n 11 * 4s2 :3.
Trang 2Bei 17 T\m m d6 phrrong trinh 12 -2(*-2)r+m2 +2m-3:0 cd hai nghiern
phAn biQt th6a m6,n
i-a: b '
Bai 18 Gi6 sfi rr,rz lh, hai nghiQm cria phuong trinh x2 - 4x * I :0 Tinh gi6,
tri cria bi6u thrlc
I - I - -r 1 -i
l'* #l l'+ #rl
Bei 19 Tim m d6 phuong trinh 2r2 - 13r * 2m: 0 c6 1 nghiQm g6p doi mQt
nghiQm cria phrrong trinh 12 - 4r * m:0.
Bai 20 T\rn rn dei plruong trirrh lr2 - (m-2)r+m2 +m*2:0 c6 hai nghiOm
tr,12 sao cho r! + rl ctat gi6 tri nh6 nh6t.
Bai 21 T)m nr, dii phuong+rinh ra - 2mn2 - m * 2 :0 c6 nghiQm k6p.
B,iij 22 Tlm m di5 phrrong trinh ua - (*" *4m)x2 *7m- I : 0 c6 4 nghiQm phdn
biQt vb tting binh phuong cac nghiQm d6 b!,ng 10.
Bai 23 Chrlng minh rB,ng vdi mgi s6 thUc r,g ta c6 12 +3g2 lZry *2r*69*3 > 0.
B,di 24 Cho x,y lb c6,c s6 thqc th6a md,n 5r2 + 5A' - 5x - Lly + 8 < 0 Tim gi6
tri ldn nh6t vd nh6 nh6t cfra bidu thrlc r * 33y + 1.
Bai 25 Cho r, y € R th6a mdn 12 +A2: 1 Tim gi6 tri ldn nh6t vd nh6 nh6t cira
o- r
' a+2'
Ludng gid,c (cd,c bdi til 26-55)
Bai 26.vTinh 2 (sin6z * cos6") - 3 (sinar * cosar) .
Bai z7.nTinh
(t - tan2 r)2 1
\,/
Bai 28.-Rrit gqn
n - sinz
_
^l_.V 1 + sinr Bai 29Y Chrlng minh rB,ng
tanz tan y : tanr * tan y
cotr*coty
Bei 30YChtlng minh rB,ng
1 + sinr
Trang 3Bai 31 Chrlng minh rhng
tanz x -tan2y sin2r - sinz3t
-;"7;;"7 a :
lit";;i'.Z;'
Bai g2.JCho sin tr : -i,,,
Bai g3.'Cho cos r: -1,0 r < zr Tinh
(sin r * tan r)(cos r * cot r) Bai 84.'Cho tan z * cot r : l Tfnh sina r + cosa r.
Bai gb Cho sina * cos
": | Tinh sin8 a * cos8 o.
1
Bai 36 Cho sino - cos
Bai 37 Cho sin a - cos a : \ Tinh sinee a + cosloo a'
Bai g8 Cho sin6 u + cos6 , : tn Tinh sinm z + coslo r.
Bai 39 Cho tan a : 2 Tinh gi6 tri cria bi6u thfic
S : sin2 a * 2sina coso - 3cos2 a'
./
Bai 40."Cho tanr : 2 Tinh gi6 tri cria biiSu thirc
P:sin3=r+2:otc'
cosss*sin3r' Bai 4liTfnh gi6 tri cira bidu thrlc S : sin2 10" + sin220o + "' + sinz80".
Fidi 4zlTinh gi6, tri cria bi6u thrlc P : tan 10o tan20" tan80"'
Bai 4g./Tinh "4 : cos
ft + "o, '# + o ff.
Bei 44.'Don gi6,n
B : sin2 (" * i)*sin2 (".'+)+sin2 (" T)*sin2 (" ?)
)
Bei 45' Don gi'n-bf.l[]",
i" (" - T) - tan (; ") "", (T -,)
Bei 46./cho A, B,C lds6 do ba g6c cria rn6t tanr gi6,c chrlng minh rh,ng
c) ta" (- z ):corV'
Bei 47' Tim gi5' tri L6n nh6t g : sin6 r * cos8 r'
Bei 48 Tim gi6, tri nh6 nh6t g ::,/ffi; + #cosr vdi t €
.)_
I 7T1
lo';l
Trang 4Elai 49 Tim gi5, tri ldn nh6t vb nh6 nh6t cria bi6u thrlc
^ _ lsinzl
^ - , +lcosfl
Bai 50 Tim gir6, tri ldn nh6t vd nh6 nh6t cria biiSu thrlc
1- sinar+1
cosd r * 2'
ElAi 5tri Tim gi6 tri l6n nhit vd nh6 nhdt cfia bitiu thrlc sinl2 r + cosr2.z.
Bai 52 Tlm gi5, tri *6rrr*#t vh, nh6 nh6t cria bidu thrlc sina r * cosa
" + ;f +
I
cos4 r'
Bai 53 Gi6i phrrong trinh sin3 z * coss *1 : 0.
Bei 54 GiAi phrrong trinh tanr *cotr {tan2 r *cot2 e * tan3z *cot3 x:6.
Bai 55 Chrlng minh rXng phrrong trinh / + tane + $ - cotr: 1 vO nghiQm.
B6t dd,ng thtc Bu-nhia-c5p-xki (cd,c bd,,i tit 56-70).
Elei 56.i Cho c2 * 2A2 *-322 : 1 Tim gi6 tri ldn nh6t vh, nh6 nh6t cria bidu thrlc
Bai 57 Cho r, y, z ld, c5,c s6 thgc th6a m5,n z *2A - 2z : I Tim gi6, tri nh6 nh6t
cfra bia5u thrlc
r'+y2 + 22.
Bai 58 Cho a, b,c ) 0 Tim gi6 tri nh6 nh6t cria bii5u thrlc
" - a-fzb+3c' b*2c'*3a' c*2q*3b' {9 <-,
{v"
L
\r_ *-'4 ,
eA,
Bai 59 Cho a, b, c 7 0 T\m gi6 tri l6n nh6,t vb nh6 nh6t cria bitiu thrlc
-a4b9c,
5:ffi+
"+o+ otrb' Lcs'
Bei 60 Cho a, b,c ) 0 Chrfng minh ring
-_ \
Bei 61 Cho a, b, c ) 0 Chfing minh rlng
*"'4_q s
(b+ c)2 (c * o)2 (a + b)' - 4(a+b+c)'
Trang 5Bai 62 Cho a, b,c) 0 Chrrng minh rl'ng
Bai 63 Cho o, b,c)0.Chtrng.minh rB'ng
o , b , : ix|*1*lt
t6+D * 4o *.1 t o("+ o) - -\o b ' c'
Bai 64 Cho a, b,c) 0 Chrlng minh rX'ng
W +WW + JW *GTTF + J+FTTATEP > 4(o+ b + c)'
Bai 65 Cho a, b,c) 0 Chr?ng minh r}'ng
a 2b 3c -6(o*b+c)
r+t*z+a*3+{>6+o+b+c' Bai 66 Cho a, b,c) 0.Chrrng minh rX'ng
"+%- b+2"- r+zos t '
Bai 67 cho a, b, c,d, > 0 th6a m6,n ab + bc + cd, + d,a: 3 T\m gi6, tri nh6 nhdt
Bei 68 Cho ab + bc+ ca:Sabc Chtrng minh rX'ng
111-9
Bai 69 Cho a, b,c) O,abc: 1 Chrlng minh rbng
F6.T + rcTA + A;+ b) 2 zo* %+ 2"'
70 cho a,b,c> 0 th6a m5,n a+6+c:3 Tim gi6 tri nh6 nh6t cira bidu
cria bi6u thrlc
Bai
thrlc
^-
Trang 6NGAN sAxc cAu uor
HINH HeC - Ldp 10 (DdT 2)
(Diing cho hoc,sinh Kh6i 10, Trttdng TH?T chuy€n KHTN)
I NQi dung
l Luong giac (5 bni).
2 Tfch v0 hudng (11 bdi).
3 He thilc luong trong tam gi6c (11 bdi).
4 He thrlc luong trong drldng trdn (3 bd,i).
1 Luong gi6c
Cdu 1 Cho o.!1,
* +: *+ Chrlrng minh rbng
coslou , sinlor 1
Cau 2.t Cho 6sinar - 2cosa r I, tinh gi6 tri c6c bi6u thrlc
a) 6sinar - cos6:r.
b) sin6r+6cosar
=-f-* ]
theo a, b.
Cau al Cho r : sin g, U : cos g sin g, z : cos p cos d, tinh rz + y2 + zz
cdu 5/chfing minh rbng bi6u thrlc GC%T4.;F; +
'/"osn;+ 4s=[z; khong
phu thuoc vb.o r
2.Tich vO hrrdng
6"-qCL 1,qggi6c ABC vd P 1) rliOrn b6t kj, Clnnrg rnirrtr rturg
+
PA.BC +
ceu 71 cho hai vector AB,fu A'Al rb, hinh chi€u vuong g6c cria A, B l€ndirdng tlring C.D Chirrrg rninh angZB.CB :TB.eD.
Trang 7Cau 8 Cho tam gid,c ABC b6t kj,, c6c dudng cao AAr, BBr,CCr Az, Bz,C2 ld
t'urrg rli6m BC, cA, AB chtrng minh reng lfrrm +Ttr'ri;ffi";; : o
Cd.u 9 Ch, tam gi6c ABC trung tuy6n AA,i.
-t/a)
chrrng minh ri,ng TB Td - AB2 + AC2 - BC2 .
'2
b) Chring minh rhng IB.ft = ALI| - P9:4
c) Dung hinh binh hdnh ABDC Chrrng riinh hng
=t +
2T6.BV : AD2 + BC2 - 4AB2.z4Bcl: AD2 + BCz _ 4AC2.
CAu 10 Cho tam gi6c ABC trong tdm G, p lh di6m bAt k,,
a) Chrltng minh rdng
3
,: b) Gqi (o,R) lb drrdng trdn ngoai ti6p tam gib"c ABC K la trong t5.m tam
gi6c OBC Chfing minh rBng 3AK2 : AB2 + AC2 + Rt - Bq'
C6u 11 Cho tam gid,c ABC,^-L ld tAm rlrrdng t.On nOi tiel,
a) Chfing minh rbng AB.AC BC.BA :{: - :y:- - CA.CB=t9:= :;
b) chrlng minh rbng AB.AC + BC.B.4+c,q.cn > eA+ IB + IC)r.
cdu 12 cho tam gi6c ABC truc ram H dudng trdn ngoai tiep (o, R) chrlrng
rninh rXng
Cdu 13 Cho tam gii"c ABC.Tim vi-tri c6c di6m P tren dudng trbn n6i ti€p sao cho t6ng binh phuong khoing cdc trl p dcn c6c dinh tam gi6c lZn nh6t.
c6u 14 cho ? yt tacdc so rhuc a, b Tim vector ? 1ui6u dion ? theo ?, ?
vb.a.b)saocho {Ai:o
I b.t:u
'ecror A,t ? rinr di€' kien Ae f
".71? : -d1l 21.
cdu 16.,.cho P ld di€lm b6t kj'tr6n d rdng trbn ngoai ti€p tam gi6c d6u ABC a) Chfng minh rd,ng
"orFTA*cos pOB*cospOC:0.
b) chirng minh ra,ng\ os' FTA* cos2 FoE + cos2 Fdd khong doi.
Trang 8cdu 1r Cha rarn gi6c ABC ddt.t : eA.ndFd+fcA.cAFB+tTB IAsd
clhLls nrinh ring tam gi6c ABC vuong khi vh chi khi ? co'phrrorrg uuo'g go"
v'<1i BC,
3 He thrlc luong trong tam gi6c
cAu 18 cho tam gid"c LABC vuong tai va tam gidc aA,B,c, ri0rrg darrg tar'
gi6c AABC AH,A'H'ldn ltrot lh dudng cao cria tam gi6c ABC ua turr,'gia"
A'B'C' Chrlng minh rhng
i11
ABA!B' + AgA'C:7gNJg,
cAu 19 cho tam gi(rc ABC vu6ng Lai A Dudng cao AH Ar, Azlan luot lb hinh chieu vuo.ng goc cria H len AB, AC Chrrng minil rbng
C)
=; .- -:- -{- r '- \/ - - "" )
-' HAr.HA2 ,AC' AB.HBtrxB-T7.HC)
cAu 2o cho tam g i6c ABC vu6ng tai Ar p rh dicm b6t ky tr€n drrdng rhx.g
Bc, E,.F la hinh chi€u cria p le' rlrrdrrg thdng AB,AC chrlng minh rh,r€
ME.MC : riA.EE +FA.F1.
C6u 21 Cho tam gi6c ABC
a) Chrlng minh rhng ,4: 60" khi vd chi khi a2 : bz + cz - bc
b) Chfing minh rdng A: 120" khi vA, chi khi a2 : b2 + c2 + bc
c) Chirng minh ri,ng neu ,4 > 60" thi bz * c2 .-2a2, b * c .-26,
Cdra 22 Cho tam gi6,c ABC Clrlng minh rX,ng
al (o^- c)coL- 1-( a)coti *@ -ir)cot i:,,
b) (b' - c2)cotA+ (c2 - a2)cotB + (a2 - bz)-cotC:0
Cdu 23 Cho tam gifLc ABC cdc phdn gi6c tlong c6 d6 dhi lo,16,l,.
a) chfing nrinh rhng # * AS cos $ 1 I I
La t6 - t" :;t b*;
b) Nou tamgi6c ABC nhon Chfngrninh rbng i.l lo 16 1,, * i ,/2t! ,a b *l * r:'tt
Trang 9CAu 24 Cha lam giS,c ABC b6n kinh duong trdn bing t,i€p g6c / liL ro, b6n kinh
rtrdng trdn n6i ti€p r bd,n kfnh drrdng trdn nqoai ti6p R.
rr) Clrrtrrg rninh rilg cosl - cosB + cosC: f
= '
Ro
c) Chfng minh ranf bct ca t ab : p2 + 4Rr - r''
R
d) Chrrng rninh rdng bc - ab - ca : (p - o)' - 4Rr^ + rf;
Cdu 25 Cho tam gi5.c ABC c6 /-A:2/.8 Chung mirrh rbng n2:b2 +bc
Cdu 26 Cho tam gi6c ABC tAm cludng trdn nOi ti6p 1, tam dudng trdn ngoai
ti€p O, tntc tdnr 11 Gie si't 01 : I H Chrrng minh rd,ng tam gi6c ABC c6 rndt
g6c bbrrg 60".
CAu 27 Clto tam gi6c ABC,
a) Cho {.,a,2lb c6c s6 thuc b6t kj' Chrlng minh rd.ng
azcosA* zrcosB * rycos 6 t : ! ') J z
b) Tam giac A'B'C' ld tam giSc b6t kj, Chtrng -inn".arrg
cotA * cotB * cotC )Sg + t:tf'
* cosC'
4 I{.e thr?c luong trong dudng trbn
CAu 28 Cho b6n didm hai dudng th&ng AB,CD cdt nhau tai L{ Chr'rng niinh
r},ng A, B , C , D cirng nhm tr€n drrdng trbn khi vd chi khi -trA.MB : Me MD .
AD Dtrdng trdn ngoai ti6p tarn.giilc ADM cdt AB,AC tai,E,F Chfng minh
rdngBE':CF.
Cdu 30 Cho tam gi6c ABC, c5.c: duilng c,ao AA', BB',CC'd6ng quy tai truc tArn
H
") Chrrng minh rbng
a) HA.HT : HA.HT : HB.HF : He .Ee.
t:) 2(x7.TH +EF.na +e7trE): AB2 + BC2 +CA2
"-,.':j'tr=!': :.- -l