[Vietmaths.com] Tonghop Chuyen de Boi Duong Hsg Toan Cua Khoi 11

8/10/2019 [Vietmaths.com] Tonghop Chuyen de Boi Duong Hsg Toan Cua Khoi 11

1/116

Kon Tum, thng 05 nm 2013.
http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/


2/116


3/116

- 2 -

LI M U

Ton hc l mn th thao ca tr tu

Tht vy, vic hc ton gip rn luyn cho hc sinh mt t duy khe, l cng c

cn thit gip cc em hc tp cc mn hc khc, gip x l tt cc tnh hung trong cuc

sng v l hnh trang tt trn con ng hc tp v lm vic d l trong bt k lnh vc

no.

i vi hc sinh hc mn chuyn, vic t hc, t tm ti nghin cu l mt vic

quan trng bc nht, quyt nh thnh cng cho cc em, to nim say m i vi v p

ca ton hc. Tp san ny xin gii thiu nhng kt qu ca vic nghin cu tm ti, nhng

kinh nghim nh rt ra c trong qu trnh hc tp ca cc em hc sinh chuyn ton lp

11 t nm hc 2010-2011 n 2012-2013. y cng l ba nm hc m b mn ton ca

Trng Trung hc Chuyn Nguyn Tt Thnh, tnh Kon Tum t c mt s thnh

tch vt tri so vi cc nm hc trc; tiu biu c cc hc sinh nh: Phan Hng Hnh

Trinh, Cao Thanh H, Trn Th T Trinh, Nguyn Ngc Khnh, L B Lc, T Quang

Hi...

Nhng ti ny ch l nhng bc i chp chng ca cc em vo lu i nguy ngaca Ton hc, nhng chnh l nn tng cho nhng bc i vng chi v sau v c bit

l tin , l ti liu tt cho cc th h sau hc hi v tip ni. Chnh v th, ni dung cc

chc chn cn rt nhiu va vp v thiu st, chng ti rt mong s gp ca qu thy c

v cc bn a ch mail ca nhm ton lp 11: [email protected].

Sau cng l li cm n v s quan tm, to iu kin ca ban gim hiu Trng

Trung hc Chuyn Nguyn Tt Thnh ra i tp san s 1 ny.

Kon Tum, ngy 30 thng 04 nm 2013.Ban bin tp: T ton

Trng THPT Chuyn Nguyn Tt Thnh, Kon Tum.
http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/


4/116

- 3 -

HAI THC CC BT NG THC HNH HCTRONG TAM GICHc sinh thc hin:Trn Th T Trinh, Nguyn Th Tuyt Anh, NguynTh Hng, Nguyn Th Thanh Thy ( Nhm chuyn ton lp 11A1- Nmhc 2010-2011).

Gio vin hng dn:V Th Ngc nh.

Li dn:Ton hc, vi nhng v p huyn b v sc li cun mnh m ca n, vang lm cc nh ton hc trong nhiu th k qua. Ton hc ging nh mt kho bu v

gi ca nhn loi m loi ngi lun kht khao c khm ph, chinh phc. V, chmtay vo chic cha kha m kho bu y, loi ngi ph tri qua mt qu trnh tch lylu di. Chng ti xin c v bt ng thc hnh hc nh mt vin kim cng trong kho qu . Bt ng thc hnh hc l mt mng rt th v, mi bt ng thc li c mtmi quan h mt thit vi mt s bt ng thc khc. Bi vy, chng ta c th thu cnhiu bi ton thng qua vic khai thc mt bt ng thc n gin.

Vi mc ch hng ti vic s dng nhng bi ton tm c vo vic tm cc tr

ca mt biu thc hnh hc, chng ti s ch tp trung vo nhng bt ng thc c dubng xy ra v xy ra khi tam gic u. Hy vng cc bn c th vn dng chng mt cchhiu qu.

Ni dung chuyn

I.Mt s kin thc lin quan:1. Mt s k hiu dng ch cc yu t trong tam gic:+ a, b, c tng ng l di ba cnh BC, AC, AB ca tam gic ABC.+ ma, mb, mctng ng l di cc trung tuyn k t cc nh A, B, C.

+ ha, hb, hctng ng l di cc ng cao k t cc nh A, B, C.+ la, lb, lctng ng l di cc phn gic dng t ba nh A, B, C.+ R, r tng ng l bn knh ng trn ngoi tip v ni tip ca tam gic ABC.+ S l din tch tam gic ABC.2. Cc kin thc c bn:

ma2 = 1

2(b2+c2) - 1

4a2

la=2 cos

2A

bc

b c
http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/


5/116

- 4 -

1 1 1 1

a b ch h h r

S= 1 .2

aa h =1

.2

bb h =1

.2

cc h

= 1 sin2

ab C= 12bcsinA= 1

2acsinB

= pr

=4abc

R

= ( )( )( )p p a p b p c .

sinA+sinB+sinC 3 32

3 3cos cos cos2 2 2 2A B C

cosA+ cosB+ cosC 32

sin2A+sin2B+sin2C 94

II. Bt ng thc hnh hc:T nhng ng thc phn I, cng thm mt cht bin ha, chng ta s thu c

bt ng thc tng chng nh n gin nhng li v cng hu dng.1. Cc bt ng thc lin quan n ng trung tuyn:

Ta c: ma2 = 14 (2b2+2c2-a2) (1)

Ta c th vit li l: ma2=14

(b2+c2+ (b2+c2-a2))

mt cht ta thy: b2+c2a2 = 2bc.cosA

Do : ma2=14

(b2+c2+2bc.cosA) = 14

((b-c)2+4bc.cos2 2A

) bc.cos2 2A

Hay ma cos2A

bc

Tng t i vi mb, mc ta c: mb cos2

Bac , mc cos

2

Cab .

Tip theo ch cn nhn v theo v cc bt ng thc va tm c vi nhau ta lic bi ton mi:

Bi ton 1.1:Trong ABC ta c: ma.mb.mc abc.cos2A cos

2B cos

2C

Chng minh:

By gi hy vit li (1) theo mt cch khc: ma2=14

(2(b2+c2) a2)

Ta c: b2+c22( )

2

b c
http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/


6/116

- 5 -

Suy ra: ma214

((b+c)2 a2) = 14

(b+c-a)(b+c+a) = p(p-a)

Hay ma ( )p p a (*)

p dng (*) va tm c ta c th gii quyt bi ton sau mt cch nhanh gn.Bi ton 1.2: Trong ABC, chng minh:a) ma+mb+mc (p p a p b p c )

b) ma.mb.mc p.S

Bi ton 1.3: Chng minh rng:

ma+mb+mc 92R .

Chng minh:

Ta c: ma2

+ mb2

+ mc2

=

3

4 (a2

+b2

+c2

) = 3R2

(sin2

A + sin2

B + sin2

C)

Suy ra: ma2+ mb2+ mc2227

4R

Ta c: (ma2+ mb2+ mc2)2 3(ma2+ mb2+ mc2) 281

4R ma+mb+mc

92R .

Chng ta c th s dng bt ng thc ny chng minh mt s bi ton sau:Bi ton 1.4:Chng minh rng:

P= 21

am+ 2 2

1 1

b cm m 2

43R

Chng minh:Ta c:

P 3

21 1 1

3a b cm m m

3 (2

3

a b cm m m

P 3

239

2R

= 3

2

2

2 43 3R R

(pcm).

Bi ton 1.5: Chng minh rng:

P= 2 2 2b c a

a b c

m m m

m m m 2

R

Chng minh:p dng bt ng thc:

2 2 2a b c

b c a a+b+c

Ta c: P 1 1 1

a b cm m m 9

a b cm m m 9 2

92R R

(pcm)

Qua vic xy dng cc bt ng thc lin quan lin quan n ng trung tuyn trn, cc bn c t t cu hi rng: i vi ng phn gic v ng cao liu c thtng t ha hay khng? nu c th hy tip tc tm cu tr li cc phn tip theo.

2. Cc bt ng thc lin quan n ng phn gic:
http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/


7/116

- 6 -

Ta c: la=2 cos

2A

bc

b c os

2A

bcc (v b+c 2 bc )

Mt khc, ta li c:cosA =

2 2 2

2b c a

bc

cos2A =

2 21 cos ( ) ( )2 4

A b c a p p a

bc bc

Suy ra: la ( )p p a (**)Tng t cho lb, lc, ta thu c bi ton m sau:

Bi ton 2.1: Trong ABC, ta c:

a) la.lb.lc abc.cos cos cos2 2 2A B C (1)

b) la.lb.lc pS (2)c) la+lb+lc 3p (3)Bt ng thc (3) c chng minh nh sau:T (**) ta c: la+lb+lc ( )p a p b p c

3( ) 3p p a p b p c p (pcm)

Ta thy: la ( )p p a m ma ( )p p a suy ra: la maKt hp vi nh l ng xin v ng vung gc, ta c c mi quan h gia

ng trung tuyn, ng phn gic, ng cao trong mt tam gic nh sau: ha la ma(***)

T (***) chng ta s pht trin c thnh cc bi ton mi m chng ti strnh by cc phn tip theo.

3. Bt ng thc lin quan n ng cao:Ta c: ha laV th t phn 2 chng ta c th d dng suy ra c cc bt ng thc sau:

Bi ton 3.1: Trong ABC, ta c:

a) ha cos 2A

bc

b) ha ( )p p a

Bi ton 3.2: Trong ABC, chng minh:

a) ha.hb.hc abc cos cos cos2 2 2A B C

b) ha+hb + hc 3p (4)c) ha.hb.hc pS (5)

Ngoi ra, ta c: 1 1 1 1

a b ch h h r

1 1 1 1

( ) ( )( )a b c a b ca b c

h h h h h hr h h h

9.

a b ch h h 9rVy ta thu c bi ton sau:
http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/


8/116

- 7 -

Bi ton 3.3:Trong ABC , chng minh:a b ch h h 9r (6)

mt cht, ta thy (4) v (6) l 2 bt ng thc lin quan n tng a b ch h h

nhng ngc chiu. Vy th hy th ht hp 2 bt ng thc ny li, thm mt cht bini na xem no!(4), (6) 3p 9r

p 3 3 ry l 1 bt ng thc lin quan n p,rBt ng thc (5) cng em n cho chng ta s th v khng km

Ta c: (5) 2a b ch h h S

p p = r

2 1 1 1. .a b c a b c

p

h h h h h h

p2 hb.hc+ ha.hb + ha.hc

Th y, ch cn m ba la mt ci chng ta thu thm c bi ton mi.Bi ton 3.4: Trong ABC , chng minh:

p2 hb.hc+ ha.hb + ha.hc4. Mi quan h gia ng trung tuyn, ng phn gic, ng cao trong tam

gic:Nh chng ti cp cui phn 2, gia ng trung tuyn, ng phn gic,

ng cao trong tam gic c mi quan h sau:ha la ma (***)

Liu t mi quan h ny, chng ta c th xy dng nn cc bi ton mi c haykhng?

tr li cho cu hi ny, chng ta hy th i qua bi ton sau:

Bi ton 4.1: Trong ABC , chng minh

a) a b ch h h la+lb+lc ma + mb+ mc

b) 1 1 1 1 1 1 1 1 1

a b c a b c a b ch h h l l l m m m

c) 12

a b c a b c

b b c c a a b c a

m m m m m m

h l h l h l m m m

Bi ton 4.1 trn, bn c c th d dng chng minh da vo (***)Theo BT Nesbit ta c: 3

2a b c

b c a c a b

m m m

m m m m m m

Li s dng BT (***) ta c bi ton:Bi ton 4.2: Trong ABC , chng minh:

32

a b c

b c c a a b

m m m

h l h l h l

Xt bi ton sau:Bi ton 4.3:Trong ABC, chng minh rng:
http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/


9/116

- 8 -

2 3a b c

a b c

m m m

Chng minh:

Ta c: 4ma2=2(b2+c2)-a2=2(a2+b2+c2)-3a2 (2ma)2+(a 3 )2=2(a2+b2+c2)Li c: (2ma)

2+(a 3 )22(2maa 3 )

Suy ra: 2(2maa 3 )2(a2+b2+c2) ama2 2 2a b c

2 3

2 2 21 2 3

a b caam

2 2 2

2 3a b ca

a a

m

(7)

Tng t:

2 2 2 2 2 2

2 3 2 3;

a b c a b cb c

b b c c

m m

Do : 2 3a b c

a b cm m m

M halama, suy ra: 2 3a b c a b c a b c

a b c a b c a b c

h h h l l l m m m .

Ch :

T (7) ta cng suy ra:2

2 2 2

2 3a b c

a am m

a

Tng t:2 2

2 2 2 2 2 2

2 3 2 3;

a b c a b cb b c cm m m m

b c

Do : 2 2 22 3

a b ca b cm m m

a b c

(ma2+mb2+mc2)=

3 32

Vy, liu c tng t ha bt ng thc trn i vi ng phn gic v ngcao c hay khng?

Ta c:a b ch h h

a b c = 2 2 2 2 2 2 2 2 2

2 2 2 1 1 12 ( )

4 sin sin sinS S S S

a b c R A R B R C

= 18

(abcR 2 2 2 2 2 2

1 1 1)

sin sin sinR A R B R C

= sinA.sinB.sinC( 2 2 21 1 1

)sin sin sinA B C R rng, vi 90A cn 45B C th

a b ch h h

a b c = 5

2

3 32

Do bt ng thc trn khng ng i vi ng cao. Vy cn i vi ngphn gic th sao? Cu hi ny xin c dnh cho bn c suy ngh!

Bi ton 4.4:Trong ABC, chng minh rng:

3 3a b b c c al l l l l l

c a b

(8)
http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/


10/116

- 9 -

Chng minh:Ta c:

(8) 1 1 1 1 1 1( ) ( ) ( ) 3 3a b cl l l

b c a c a b

2 cos 2 cos 2 cos2 2 2. . . 3 3

A B Cbc ac abb c a c a b

b c bc a c ac a b ab

3 3

cos os os2 2 2 2A B C

c c

y l mt bt ng thc n gin trong tam gic.Vy, bt ng thc (8) c chng minh.M, trong tam gic ta lun c: hala, hblb, hclaV vy: t (8) ta c th suy ra c bt ng thc sau:

Bi ton 4.5:3 3a b b c c a

h h h h h h

c a b

T (8) chng ta li c th tip tc khai thc chuyn thnh mt bi ton mi

Ta c: ( )( ) 9a b b c c aa b a c a b

l l l l l l c b c

l l l l l l c a b

9

3a b b c c aa b a c a b

c b cl l l l l l l l l l l l

c a b

Hin nhin t y ta cng c th suy ra c cc bi ton sau:

Bi ton 4.6:Cho ABC, chng minh:a) 3

a b a c a b

c b c

l h l h l h

b) 3a b a c a b

c b c

h h h h h h

T tnga b a c a b

c b c

h h h h h h

, ta li c th khai thc mt bi ton mi

Bi ton 4.7:Tong ABC, chng minh rng:3

2a b a c c b

c b a R

h h h h h h r

Chng minh: Khng mt tnh tng qut, gi s: abc2 2 2

a b c

S S S

h h h hahbhc

hb+hc hc+ha ha+hb (1)

c c a a b

1 1 1h h h h h h

(2)

p dng bt ng thc Tr-b-sp vo hai dy (1), (2) ta c:

b c c a a b

1 1 1 1( )( )

3 h h h h h ha b a c c b

c b aa b c

h h h h h h

(3)
http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/


11/116

- 10 -

Ta c: a+b+c=2R(sinA+sinB+sinC) 3 3R (4)

Mt khc:b c c a a b

1 1 1 1 1 1 1 1( )

h h h h h h 2 2a b ch h h r

(5)

T (3, (4), (5) suy ra:3

2a b a c c b

c b a R

h h h h h h r

T bi ton ny, ta li thu c bi ton sau:

Bi ton 4.8: Trong ABC, chng minh:

a) 32a b b c c a

c a b R

h l h l h l r

b) 3

2a b b c c a

c a b R

l l l l l l r

c) 32

a b b b c c c a a

c a b R

rh l h h l h h l h

Cn rt nhiu cc bt ng thc khc c th khai thc, nhng chng ti xin c ktthc phn ny ti y i qua cc phn khc cng hp dn khng km.

5. Bt ng thc lin quan n p, r, R:Bi ton 5.1:Trong ABC, chng minh:

a2+b2+c2 9R2 (1)Chng minh:

Ta c: (1)4R2(sin2A+sin2B+sin2C)9R2 sin2A+sin2B+sin2C 94 y l bt ng thc c bn trong tam gic nn ta c c ccm.Chng ta hy khai thc bi ton trn tm li gii cho mt bi ton mi.

Ta c: (ab+bc+ac)( 1 1 1 ) 9ab bc ca

2 2 2 2 2

1 1 1 9 9 9 1ab bc ac a b c 9ab bc ca R R

Vy ta c bi ton sau:Bi ton 5.2:Trong ABC, chng minh:

21 1 1 1ab bc ca R Bi ton trn c rt nhiu cch gii, song chng ti ch gii thiu cho cc bn mt

cch gii. Vi nim say m ton hc hi vng cc bn s tm c cch gii ring chomnh.

Sau y, chng ti xin trnh by mt hng khai thc khc i vi bi ton 5.1:Ta c: ab+bc+ac 2 2 2a b c 29R M, ha+hb+hc9r

Suy ra:2 2 2

2 2

( ) (9 ) 99

a b bh h h r r

ab bc ac R R

2 2 2 2 2

2

( ) 9a b c a b bh h h h h h rbc ac ab ab bc ac R

ng thc xy ra khi v ch khi ABC u.
http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/


12/116

- 11 -

Bi ton 5.3: Trong ABC, chng minh; R2rTht vy, theo cng thc Euler, ta c:

OI2=R2-2Rr (vi O,I ln lt l tm ng trn ngoi tip, ni tip ABC)

R

2

-2Rr 0

R2rTrong qu rnh khia thc mi lin h ca ba ng trung tuyn, phn gic, ngcao phn 3, chng ti a ra cc bt ng thc m cc bn c th kt hp li an kt qu bi ton 5.3.

Bi ton 5.4: Chng minh rng:2 2 2a b c r2+p2+4Rr

Chng minh:Ta c: p2r2=S2=p(p-a)(p-b)(p-c)p2r2=p3-(a+b+c)p2+(ab+bc+ac)p-abc

=-p3+(ab+bc+ac)p-4pRrr2=-p2+ab+bc+ac -4Rr ab+bc+ca=r2+p2+4Rr

M

2 2 2

a b c

ab+bc+car

2

+p

2

+4Rr (dccm)Thc cht trong li gii ca bi ton trn chng ta i thit lp cng thc :2 2 2a b c =r2+p2+4Rr

2 2 2a b c =(a+b+c)2-2(ab+bc+ca) =2p2-8Rr-2r2T bi ton 5.4, suy ra:2p2-8Rr-2r2r2+p2+4Rrp23r2+12Rr3r(r+4R)p2

3.4sin 2 2 2 2sin sin .4 (1 sin sin sin ) 16 os os os2 2 2 2 2 2 2 2 2A B C A B C A B C

R R c c c

3tanos os os

2 2 2tan tan

2 2 2 1 sin sin sin2 2 2

A B Cc c cA B C

A B C

Vy, ta c bt ng thc sau: 3tanos os os

2 2 2tan tan2 2 2 1 sin sin sin

2 2 2

A B Cc c cA B C

A B C

y l mt bt ng thc kh.Bi ton 5.5: Trong ABC, chng minh:

p 3 32

R.

Chng minh:Ta c: 4( 2 2 2a b cm m m )=2 2 23( )a b c (a+b+c)2

M 2 2 2a b cm m m 227

4R

Suy ra: (a+b+c)2227

4R 4p2

2274R pcm

Trong nhng phn trn chng ta chng minh c bi ton sauBi ton 5.6:Trong ABC, chng minh:

p3 3 rBi ton ny d dng dn n bi ton sau:
http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/


13/116

- 12 -

Bi ton 5.7:Trong ABC, chng minh:

a) S2

3 3p

b) S2 2 2a b c

4 3

Sau y chng ti xin gii thiu mt bt ng thc c coi l mnh nht i vip, R, r trong tam gic (bt ng thc Bludon)Bt ng thc Bludon: Chng minh rng trong mi ABC ta c:

2 2 2 2 22 10 2( 2 ) ( 2 ) 2 10 2( ) ( 2 )R Rr r R r R R r p R Rr r R r R R r Chng minh

K hiu O l tm ng trn ngoi tip v I l tm ng trn ni tip ABC.Gi E l tip im ca ng trn ni tip vi cnh AB

t AI = x th R d x R + d (d = OI)Ta c: AE = 2 2x r v AE = p aT tam gic vung AEI ta c:

sin2A r

, cos2 2

2A x r

x

sinA=2 sin2A cos

2 2

2

22A r x r

x

Theo nh l hm s sin ta c:

A = 2RsinA =2 2

2

4Rr x r

T : p = (p a) + a = 2 2 24

1 Rrx r

Xt hm s: p = p(x) = 2 2 24

1Rr

x rx

(R d x R + d)

Ta c: p(x) =2 2 3

1 2 1 2

3 2 2 3 2 2

( )( )( )( )4 8 x x x x x x xx Rrx Rr

x x r x x r

Trong : x1= 2 ( )r R d , x2= 2 ( )r R d D dng chng minh c 0 < R d x1 x2 R + dBng bin thin ca hm s p(x) nh sau:

x R d x1 x2 R + d

p(x) + 0 - 0 +

p(x)

rng d = 2 2R Rr ta c th chng minh c

p(x1) = p(R + d) = 2 22 10 2( 2 ) ( 2 )R Rr r R r R R r v

p(x2= p(R - d) = 2 22 10 2( 2 ) ( 2 )R Rr r R r R R r
http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/


14/116

- 13 -

Vy2 22 10 2( 2 ) ( 2 )R Rr r R r R R r p 2 22 10 2( 2 ) ( 2 )R Rr r R r R R r

Ta c: 2 22 10 2( 2 ) ( 2 )R Rr r R r R R r

= 4R2+ 4Rr + 3r2 [(R-2r) - ( 2 )R R r ]2 4R2+ 4Rr + 3r2v 2 22 10 2( 2 ) ( 2 )R Rr r R r R R r

= [(R-2r) - ( 2 )R R r ]2+ 16Rr 5r2 16Rr 5r2Mi ng thc trn xy ra khi v ch khi R = 2r

ABC l tam gic u

Chng ti va trnh by mt s bt ng thc v h thc lng trong tam gic. Ccbn c th vn dng chng gii cc bi ton khc.

III. Bi tp cng c:1.

Cho ABC c bn knh ng trn ngoi tip R=1. Chng minh rng:sin sin sin

3a b c

A B C

m m m

Chng minh:

Tac:2 2 2sin sin sin

2 2 2 2 2 2a b c a b c a b c

A B C a b c a b c

m m m Rm Rm Rm am bm cm

M,2 2 2

2 2 22 2 2 2

3 32 3 (2 2 )a

a a a

am a b ca b c a

(theo C-si) (1)

Tng t,2 2 2

2 2 22 2 2 23 32 3 (2 2 )bb b bbm a b cb a c b

(2)

2 2 2

2 2 22 2 2 2

3 32 3 (2 2 )a

c c c

cm a b cc a b c

(3)

T (1), (2), (3) ta c: sin sin sin 3a b c

A B C

m m m

Du = xy ra khi v ch khi a=b=c hay ABC u.

2. Cho tam gic ABC c 3 gc nhn. Gi a,b,c l di ba cnh v ha, hb, hc l

di cc ng cao tng ng. K hiu r, R ln lt l bn knh ng trn ni, ngoi tiptam gic . Chng minh:

2 2 2

9 1 1 1 12

a b c b c a c a b

R

a b c r h h h h h h h h h

Chng minh:Ta c:

1 1 1 1 1 1 1 1 14 4 2a a b ca b c b ch h h hh h h h h

Tng t:
http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/


15/116

- 14 -

1 1 1 1 1 14 2b c ab c a h h hh h h

; 1 1 1 1 1 14 2c a bc a b h h hh h h

Do :1 1 1 1 1 1 1 1

2 2a b ca b c b c a c a b h h h r h h h h h h h h h

(1)

Ta c:1 1 1 9

a b c b c a c a b a b c b c c a a bh h h h h h h h h h h h h h h h h h

9 9 9 91 1 12 4 ( )4a b c

abc R

h h h S bc ab ac ab bc caSa b c

2 2 29R

a b c

(2)

T (1),(2) suy ra pcm.

3. Chng minh rng:1 1 1 6

a b cRm m m

Chng minh:Ta c:

P= 1 1 1 3 3323

a b c a b cm m m m m m R

Suy ra : P 6R

4.Chng minh rng:

P=2 2 2

3 3 3

1b c a

a b c

h h h

h h h r

Chng minh:

p dng bt ng thc:3 3 3

2 2 2

a b ca b c

b c a

Ta c: P 1 1 1 1

a b ch h h r (pcm)

5. Chng minh rng:

a) ma.mb.mc327

8R
http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/


16/116

- 15 -

b) (la+lb)(lb+lc)(lc+la)38

3 3p

Chng minh

a) Ta c:ma.mb.mc

2 3 33 273 2 8

a b cm m m R R

ma.mb.mc327

8R

b) Ta c:

(la+lb)(lb+lc)(lc+la)32( )

3a b cl l l

3 32 83 3 3p p

(pcm)

6. Chng minh rng: ha.hb.hc 27r3

Chng minhTa c:3

3

1 1 1 1 127.

a b c a b cr h h h h h h

Suy ra: ha.hb.hc 27r3 (pcm)

7. Chng minh rng:

P= 181 1 1 2 9

a b c

a b b c c a

h h h r

h m h m h m Rr

Chng minhTa c:

P= 1 1 1 91 1 1 1 1 1

b c a a b ca b c a b c

m m m m m mh h h h h h

9 189 1 9 22

rR Rr

r

(pcm)

8.Chng minh rng:

2 2 2 21 1 1 1 27

93 2a b c a b c

m m m RR m m m

Chng minht a= 1

3 am

R, b= 1

3 bm

R, c= 1

3 cm

R. Ta c: a + b + c 3

2

BT tr thnh:

a + b + c + 2 2 21 1 1 27

2a b c

Ta c: 3 13 2

a b cabc

VT 3 3 3 33 32 2 2 2 2 2

3 13 3 8 8 15

3

abc abc abc abca b c a b c

3 15 273 3 642 2
http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/


17/116

- 16 -

9. Trong ABC chng minh rng:a) 2 2 2 2 2 2a b c a b cr r r l l l

b)rala+ rblb+ rclc p 2

Chng minha)Ta c 2 2 2a b cr r r = p

2 ( tan 22A

tan 22

B + tan 22C)

V, tan 22A

tan 22

B tan 22C tan tan tan tan tan tan 1

2 2 2 2 2 2A B B C C A

Nn: 2 2 2 2a b cr r r p (1)Mt khc: 2 2 2 ( ) ( ) ( )a b cl l l p p a p p b p p c = p

2 p(a+b+c-2p)=p2 (2)T (1),(2) suy ra 2 2 2 2 2 2a b c a b cr r r l l l

b) Ta c:

rala+ rblb+ rclc ( ) ( ) ( )S S Sp p a p p b p p cp a p b p c

= ( )( ) ( )( ) ( )( )p b p c p a p c p a p b p ( p b + p c + p a ) = p2

Vy rala+ rblb+ rclc p 2 (pcm)10. Trong ABC chng minh rng:

9

a b c

a b c R

r r r p

Chng minh

Ta c: 2 sin 2 sin 2 sin. tan . tan .tan

2 2 2a b ca b c R A R B R C A B Cr r r p p p

= 2 2 24 cos cos cos2 2 2

R A B Cp

= 2Rp

(1 + cosA +1 + cosB + 1 + cosC)

= 2Rp

(3 + cosA + cosB + cosC)

M: cosA + cosB + cosC

3

2 (bt ng thc c bn trong tam gic)Do : 9

a b c

a b c R

r r r p (pcm)

IV. Bi tp t gii:1. Cho ABC, chng minh rng:

2 2 2

9a b c

a b cR

r r r r r r

2. Cho ABC l tam gic nhn. Gi R v r l bn knh ng trn ni tip v ngoitip tam gic. Gi H l trc tm tam gic, da, db, dctng ng l khong cch t H ti ba

cnh BC, CA, AB. Chng minh rng:
http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/http://www.vietmaths.com/ht

Documents

[Vietmaths.com] Tonghop Chuyen de Boi Duong Hsg Toan Cua Khoi 11