bdt dien tich

A

RN LUYN MT S HOT NG TON THNG QUA

MT BI TON BT NG THC V DIN TCH

I - CON NG I N BI TON V CC CCH CHNG MINH:

Chng ta bt u t bi ton sau: cho tam gic ABC u cnh a, khi tam gic ABC c din tch S c tnh theo cng thc S = , suy ra:, tc l: , vy ny sinh vn : trong tam gic

bt k th ta c kt qu nh th no ?. Th vi tam gic c bit nh tam gic vung, tam gic cn ta s hng dn hc sinh a ra bi ton sau :

BT 1.Cho tam gic ABC c cc cnh a, b, c v din tch S.

Chng minh : a2 + b2 + c2 4S

(1)

Sau khi d on c (1), ta yu cu hc sinh vn dng cc kin thc hc chng minh. Thc t li gii bi ton ny c trnh by nhiu ti liu tham kho, tuy nhin y chng ti hng dn hc sinh chng minh theo10 cch khc nhau da vo cc ni dung kin thc ca cc lp 10, 11, vi mc ch l rn luyn tnh linh hot ca t duy ng thi cc phng php chng minh cn dng cho cc bi ton sau ny. Sau y l mt s cch chng minh (1)

Cch 1. S dng cng thc Herong v BT Cosi

p dng cng thc Hrng: S = , theo BT Csi ta

c: (p - a)(p - b)(p - c)

EMBED Equation.3 p(p - a)(p - b)(p - c)

EMBED Equation.3 , do : (a + b + c)2 12S, mt khc d chng minh c BT : a2 + b2 + c2 (a + b + c)2 , nn t cc BT trn ta suy ra:

a2 + b2 + c2 4S. Du ng thc xy ra khi tam gic ABC u.

Cch chng minh ny chng ta c th trnh by phn p dng BT Csi chng minh BT lp 10 .

Cch 2. S dng nh l csin v BT Csi, BT Bunhia

p dng nh l cosin : c2 = a2 + b2 - 2ab.CosC v cng thc tnh din tch S = ta c (1) a2 + b2 + a2 + b2 - 2ab.CosC 2ab.SinC

a2 + b2 ab.CosC + ab.SinC

EMBED Equation.3

EMBED Equation.3 (1'). p dng BT Csi ta c 2, p dng BT Bunhia ta c:

EMBED Equation.3

EMBED Equation.3 = 2, nn (1') ng do (1) ng. Du ng thc xy ra khi tam gic ABC u.

Cch chng minh ny chng ta c th trnh by phn chng minh BT bng cch p dng BT Csi , BT Bunhia lp 10 .

Cch 3. S dng cng thc cng cung v nh l cosin

p dng nh l Cosin: c2 = a2 + b2 - 2ab.CosC v cng thc tnh din tch S = ta c (1)a2 + b2 + a2 + b2 - 2ab.CosC 2ab.SinC

a2 + b2 - ab.CosC ab.SinC a2 + b2 - ab.CosC - ab.SinC

EMBED Equation.3 0

(a - b)2 + 2ab[1-()] 0 (a - b)2+ 2ab[1- Cos(C-600)] 0. Do (a - b)2 0 v 2ab[1 - Cos(C - 600)] 0 nn (1) c chng minh. Du ng thc xy ra khi tam gic ABC u.

Cch chng minh ny chng ta c th trnh by phn cng thc cng cung lp 11 .Cch 4. S dng cch dng hnh v cng thc cng cungTrong trng hp tam gic ABC u th (1) ng. Gi s tam gic ABC khng u , ta c th coi A l gc ln nht, suy ra A > 600 , dng vo pha trong tam gic ABC cc tam gic cn AMB, APC sao cho cc gc AMB = APC = 1200.

Khi AM = , AP = . p dng nh l cosin trong tam gic MAP ta c:

MP2 = AM2 + AP2 - 2AM.AP.CosMAP =

EMBED Equation.3 MP2 = =

= , do MP2 0 nn

.

Cch chng minh ny chng ta c th trnh by phn cng thc cng cung lp 11 .

Cch 5. S dng BT ph v cng thc Herong

Ta c: a2 a2 - (b - c)2 = 4(p - b)(p - c); b2 b2 - (c - a)2 = 4(p - c)(p - a)

c2 c2 - (a - b)2 = 4(p - a)(p - b), t suy ra:

a2 + b2 + c2 4[(p - a)(p - b) + (p - b)(p - c) + (p - c)(p - a)].

Ta d chng minh c BT: (xy + yz + zx)2 3xzy(x + y + z), suy ra: nn p dng BT ny ta c:

a2 + b2 + c2 4S. Du ng thc xy ra khi tam gic ABC u.

Cch chng minh ny chng ta c th trnh by phn chng minh BT theo phng php tng ng lp 10 .

Cch 6. S dng BT ph v cnh v bn knh ng trn ngoi tip v BT Csi

D chng minh c a2 + b2 + c2 9R2 bng phng php vct. p dng BT Csi ta suy ra: 9R2 a2 + b2 + c2

EMBED Equation.3 abc 3R3, t ta c: a2 + b2 + c2 =

EMBED Equation.3 4S. Du ng thc xy ra khi tam gic ABC u.


Cch 7. S dng nh l cosin m rng v ng thc lng gic p dng nh l cosin m rng ta c: cotgA + cotgB + cotgC =, mt khc d chng minh c: cotgA.cotgB + cotgB.cotgC + cotgC.cotgA = 1, t ta c : cotgA + cotgB + cotgC

EMBED Equation.3 , cho nn suy ra: a2 + b2 + c2 4S. Du ng thc xy ra khi tam gic ABC u.

Cch chng minh ny chng ta c th trnh by phn cng thc bin i lng gic lp 11.

Cch 8. S dng BT lng gic v BT Csi

p dng nh l sin v cng thc tnh din tch S = 2R2.SinA.SinB.SinC ta c: (1)Sin2A + Sin2B + Sin2C 2SinA.SinB.SinC. p dng BT Csi: Sin2A + Sin2B + Sin2C

EMBED Equation.3 = , mt khc ta c BT c bn trong lng gic: Sin2A + Sin2B + Sin2C

EMBED Equation.3 ,

p dng BT Csi ta c:

EMBED Equation.3

EMBED Equation.3 , nn Sin2A + Sin2B + Sin2C

2SinA.SinB.SinC. Do (1) c chng minh. Du ng thc xy ra

khi tam gic ABC u.

Cch chng minh ny chng ta c th trnh by phn cng thc bin i lng gic lp 11 .

Cch 9. S dng cch k ng cao v BT Csi.

Gi s A l gc ln nht, t A k ng cao AH, khi trong cc tam gic vung ABH, ACH ta c: AB2 = AH2 + BH2 , AC2 = AH2 + CH2 nn

a2 + b2 + c2 = AB2 + AC2 + BC2 = 2AH2 + (BH2 +CH2) + BC2

2AH2 + + BC2 = 2AH2 + , p dng BT Csi ta c:

a2 + b2 + c2 2AH.BC =4S.


Cch 10. S dng cng thc ng trung tuyn v BT Csi.p dng cng thc ng trung tuyn : , khi , p dng BT Csi ta c:

EMBED Equation.3

EMBED Equation.3

EMBED Equation.3 ma.a

EMBED Equation.3 ha.a = 4S, t suy ra: a2 + b2 + c2 4S. Du ng thc xy ra khi tam gic ABC u.


Nh vy vi cc cch chng minh trn chng ti rn luyn cho hc sinh tnh nhun nhuyn ca t duy, tm nhiu gii php gii quyt mt vn , gii bi ton di nhiu cch nhn khc nhau.

II - MT S BI TON TNG T BI TON (1):

T cch chng minh 3 ta c BT: (a - b)2 + 2ab[1- Cos(C - 600)] 0 (*)

nu thay Cos(C - 600) bng Cos(C - 300) th (*) vn ng, tc l:

(a - b)2 + 2ab[1 - Cos(C - 300)] 0 a2 + b2 - ab.CosC. - ab.SinC 0

a2 + b2 - ab.. 2S 2(a2 + b2 ) - (a2 + b2 - c2).

EMBED Equation.3 4S

(2- 3)a2 + (2- 3)b2 + 3c2 4S. T ta c bi ton mi:

BT 2.1. Cho tam gic ABC c cc cnh a, b, c v din tch S. Chng minh

(2- 3)a2 + (2- 3)b2 + 3c2 4S. Du ng thc xy ra khi tam gic ABC cn ti C v C = 300.

Tng t nu thay Cos(C - 600) bng Cos(C - 450) th :

(a - b)2 + 2ab[1 - Cos(C - 450)] 0 a2 + b2 - 2ab.CosC. - 2ab.SinC. 0 a2 + b2 - 2ab..

EMBED Equation.3 4S .

EMBED Equation.3 (- 1)a2 + (- 1)b2 + c2 4S. (- )a2 + ( - )b2 + c2 4S.

T ta c bi ton mi:

BT 2.2. Cho tam gic ABC c cc cnh a, b, c v din tch S. Chng minh: (- )a2 + ( - )b2 + c2 4S. Du ng thc xy ra khi tam gic ABC cn ti C v C = 450.

By gi nu ta thay Cos(C - 600) bng Cos(C - 1200) ta c:

(a - b)2 + 2ab[1 - Cos(C - 1200)] 0 a2 + b2 + ab.CosC - ab.SinC. 0

a2 + b2 + ab.

EMBED Equation.3 2S 2(a2 + b2 ) + (a2 + b2 - c2)4S

3a2 + 3b2 - c2 4S, t ta c bi ton sau:

BT 2.3. Cho tam gic ABC c cc cnh a, b, c v din tch S. Chng minh : 3a2 + 3b2 - c2 4S. Du ng thc xy ra khi tam gic ABC cn ti C v C = 1200.

Ta li thay tip Cos(C - 600) bng Cos(C - 1350) ta c:

(a - b)2 + 2ab[1 - Cos(C - 1350)] 0 a2 + b2 + 2ab.CosC. - 2ab.SinC. 0 a2 +b2+2ab..

EMBED Equation.3 4S.

EMBED Equation.3 (+ 1)a2+(+ 1)b2 - c2 4S. (+)a2 + (+)b2 -c2 4S. T ta c bi ton mi:

BT 2.4. Cho tam gic ABC c cc cnh a, b, c v din tch S. Chng minh :(+)a2 + (+)b2 -c2 4S. Du ng thc xy ra khi tam gic ABC cn ti C v C = 1350.

Tip tc nu ta thay Cos(C - 600) bng Cos(C - 1500) th ta c:

(a - b)2 + 2ab[1 - Cos(C - 1500)] 0 a2 + b2 + ab.CosC. - ab.SinC 0

a2 + b2 + ab.. 2S 2(a2 + b2 ) + (a2 + b2 - c2).

EMBED Equation.3 4S

(2+ 3)a2 + (2+ 3)b2 - 3c2 4S. T ta c bi ton mi:

BT 2.5. Cho tam gic ABC c cc cnh a, b, c v din tch S. Chng minh :(+3)a2 + (+3)b2 - 3c2 4S. Du ng thc xy ra khi tam gic ABC cn ti C v C = 1500.

Nu thay Cos(C - 600) bng Cos(C - ) th: (a - b)2 + 2ab[1 - Cos(C - )] 0

a2 + b2 - 2ab.CosC.Cos - 2ab.SinC.Sin 0

a2 + b2 - 2ab..Cos 4S.Sin

(1 - Cos)a2 +(1 - Cos) b2 + Cos. c2 4S.Sin

EMBED Equation.3 . T y ta c bi ton:

BT 2.6. Cho tam gic ABC c cc cnh a, b, c v din tch S; l gc bt k khc 00. Chng minh :

T bi ton ny nu thay bi cc gc c bit th ta c cc bi ton trn.

Nu ta thay gc C bi cc gc A, B th ta s c mt lot cc bi ton tng t nh trn.

p dng cch chng minh nh trn cc em a ra mt s bi ton sau:

BT 2.7. Cho tam gic ABC c cc cnh a, b, c v din tch S. Chng minh: . Du ng thc xy ra khi tam gic ABC cn ti C v C = 450.

Chng minh: p dng nh l Cosin: c2 = a2 + b2 - 2ab.CosC v cng thc tnh din tch S = ab.SinC ta c (2.5) a2 + b2 + a2 + b2 - 2ab.CosC 2ab.SinC +

a2 - 2ab + b2 - ab.CosC - ab.SinC +ab 0 (a - b)2 + ab[- ()]0 (a - b)2 +ab[1- Cos(C- 450)] 0. Do (a - b)2 0 v ab[1 - Cos(C - 450)] 0 nn (2.5) c chng minh.

Tng t ta cng c cc BT sau:

;

Nh vy trong phn ny vi hot ng tng t, chng ti hng dn hc sinh vn dng cch gii 3 ca bi ton (1) a ra cc bi ton khc cng dng vi bi ton ban u.III - MT S BI TON CHT HN CA BI TON (1):

T BT: a2 + b2 + c2 4S (1), ta hng dn hc sinh hy tng qut bi ton trn theo hng lm cht hn BT (1), tc l thay v tri bi i lng nh hn hoc v phi bi mt i lng ln hn m (1) vn cn ng.

Chng ta bt u t mt BT quen thuc: a2 + b2 + c2 ab + bc + ca, t t ra vn l BT: ab + bc + ca 4S (2) c ng na khng ?, nu BT ny ng th ta c kt qu cht hn BT ban u.

Chng minh :(2)

EMBED Equation.3 .

p dng BT Csi : , m ta c :

SinA.SinB.SinC , nn , tc l (2) c chng minh. Vy ta c bi ton sau :

BT 3.1. Cho tam gic ABC c cc cnh a, b, c v din tch S. Chng minh:ab + bc + ca 4S. Du ng thc xy ra khi tam gic ABC u.

Ta c th chng minh bi ton trn theo cch 2 nh sau:

p dng BT Csi ta c:

EMBED Equation.3

EMBED Equation.3

EMBED Equation.3 , mt khc ta c:

a2 + b2 + c2 9R2 nn theo BT Bunhia ta c: a + b + c 3R, thay vo BT trn suy ra:

EMBED Equation.3 ab + bc + ca 4S.

T BT : a2 + b2 + c2 4S ta th lm cht hn bng cch cng vo bn phi mt i lng dng hay khng ?. Ta c a2 + b2 2ab, nhng BT sau cht hn :a2 + b2 2ab + (a - b)2 ( thc t y l ng thc ), t suy ra:

. Tng t ta cng c: , , cng cc BT trn ta c: a2 + b2 + c2 ab + bc + ca + , theo BT trn suy ra: a2 + b2 + c2 4S + .Vy ta c bi ton tng qut hn:

BT 3.2. Cho tam gic ABC c cc cnh a, b, c v din tch S. Chng minh: + . Du ng thc xy ra khi tam gic ABC u.

Vi s hng dn nh trn cc em a ra bi ton sau :

BT 3.3. Cho tam gic ABC c cc cnh a, b, c v din tch S. Chng minh:

. Du ng thc xy ra khi tam gic ABC c gc C = 600.

Chng minh: (3.3) a2 + b2 + a2 + b2 - 2ab.CosC 2ab.SinC + 2a2 + 2b2 - 4ab CosC + SinC 2

cos(C - 600 ) 1, BT ny ng nn (3.3) ng.

Tng t cc em cng a ra cc BT , , cng cc BT ny v rt gn ta c:

, do cc em a ra mt bi ton cht hn nh sau:


. Du ng thc xy ra khi tam gic ABC u.

T cch chng minh BT 3.3 ln lt thay C - 600 bng C - 300 C - 450, C - 1200, ... th c :

1/ (2- 3)a2 + (2- 3)b2 + 3c2 4S +2(a - b)2 , du ng thc xy ra khi C = 300.

2/ (- )a2 + ( - )b2 + c2 4S+ (a - b)2. Du ng thc xy ra khi C = 450.

3/ 3a2 + 3b2 - c2 4S +2(a - b)2, du ng thc xy ra khi C = 1200.

.......

Sau khi thay th nh trn ta cng cc BT trn li ( ch du ng thc xy ra) ta c:

3/++

EMBED Equation.3 4S+++, du ng thc xy ra khi tam gic ABC cn ti C v C= 1200.

4/ (- )a2 +b2 +c2 4S+ (a - b)2 +(a - c)2 , du ng thc xy ra khi B = C = 450 .

T vic thay th nh trn ta c cc bi ton sau:

BT 3.4. Cho tam gic ABC c cc cnh a, b, c v din tch S. Chng minh: (2- 3)a2 + (2- 3)b2 + 3c2 4S +2(a - b)2 , du ng thc xy ra khi tam gic ABC c C = 300.


(- )a2 + ( - )b2 + c2 4S+ (a - b)2, du ng thc xy

ra khi tam gic ABC c C = 450.


3a2 + 3b2 - c2 4S +2(a - b)2, du ng thc xy ra khi tam gic ABC c C= 1200.


(+)a2 + (+)b2 -c2 4S+(a - b)2, du ng thc xy ra khi tam gic ABC c C = 1350.


(+3)a2 + (+3)b2 - 3c2 4S + 2(a - b)2 , du ng thc xy ra khi tam gic ABC c C = 1500.


++

EMBED Equation.3 4S+++ du ng thc xy ra khi tam gic ABC cn ti C v C = 1200.


(- )a2 +b2 +c2 4S+ (a - b)2 +(a - c)2 , du ng thc xy ra khi tam gic ABC vung cn ti A.

Nhn xt :

- Nu ta thay gc C bi cc gc A, B th ta s c mt lot cc bi ton tng t nh trn.

- BT 3.2 c v l mt mt bi ton mi nhng thc t chnh l mt dng khc ca bi ton BT 3.1, cn bi ton 3.4 l cht thc s ca cc bi ton trn. Tuy nhin t bi ton trn mt cu hi t nhin xut hin l: c th thay s bi s ln hn khng ? , c th liu BT: a2 + b2 + c2 4S+ (a - b)2 + (b - c)2 + (c - a)2 (*) c ng khng ?

Chng minh : Khai trin v phi v rt gn ta c: (*)2(ab + bc + ca) 4S+ a2 + b2 + c2. p dng nh l cosin m rng : cotgA + cotgB + cotgC = v cng thc tnh din tch tam gic ta suy ra:

(*) 4S. 4S + 4S.(cotgA + cotgB + cotgC )

+

EMBED Equation.3 (**) ta d

chng minh c trong tam gic ABC : v

EMBED Equation.3 nn (**) ng, do (*) c chng minh . Vy ta c bi ton sau :

BT 3.11. Cho tam gic ABC c cc cnh a, b, c v din tch S. Chng minh:a2 + b2 + c2 4S+ (a - b)2 + (b - c)2 + (c - a)2. Du ng thc xy ra khi tam gic ABC u.

Ta c th chng minh BT 3.11 theo cch 2 nh sau:

Ta c: a2 - (b - c)2 = 4(p - b)(p - c); a2 - (b - c)2 = 4(p - b)(p - c);

a2 - (b - c)2 = 4(p - b)(p - c), cng cc BT trn v kt hp vi cng thc Hrng ta c (3.11) a2 - (b - c)2 +a2 - (b - c)2 + a2 - (b - c)2 4S

4(p - b)(p - c) +4(p - b)(p - c) +4(p - b)(p - c) 4, t p - a = x , p - b = y , p - c = z, suy ra p = x + y + z , do

(3.11) 4(xy + yz + zx) 4 (xy + yz + zx)2 3xyz(x + y + z) (xy - yz)2 + (yz - zx)2 + (zx - xy)2 0, BT ny ng nn BT (3.11) ng.

Nhn xt:

Ta thy BT a2 + b2 + c2 4S+ (a - b)2 + (b - c)2 + (c - a)2 cht hn BT : a2+ b2+ c2 4S; a2 + b2 + c2 4S+ v BT: ab + bc + ca 4S .

p dng cch gii 2 ca BT 3.11 ta c hng lm cht hn nh sau:

BT 3.12. Cho tam gic ABC c cc cnh a, b, c v din tch S. Chng minh

a2 + b2 + c2 4+ (a - b)2 + (b - c)2 + (c - a) 2. Du ng thc xy ra khi tam gic ABC u.

Chng minh: p dng cch t nh trn ta a bi ton v dng:

(3.12) 4(xy + yz + zx) 4.

(xy + yz + zx)2 3.xyz.(x + y + z) +[(xy - yz)2 + (yz - zx)2 + (zx - xy)2] (xy)2+ (yz)2 + (zx)2 xyz.(x + y +z) [(xy - yz)2+ (yz - zx)2+ (zx - xy)2] (*)

Ta s chng minh (*). p dng BT: ta c:

(xy)2 + (yz)2 2xy2z +(xy - yz)2 ; (yz)2 + (zx)2 2xyz2 +(yz - zx)2 ;

(xy)2 + (zx)22x2yz + (zx - xy)2 , cng cc BT ny li ta c:

2[(xy)2 + (yz)2 + (zx)2 ]2xyz.(x + y + z)+[(xy - yz)2 + (yz - zx)2 + (zx - xy)2]

(xy)2 + (yz)2 + (zx)2 xyz.(x + y + z) +[(xy - yz)2 + (yz - zx)2 + (zx - xy)2] vy (*) c chng minh .

p dng cch gii 1 ca BT 3.11 cc em c hng lm cht hn bi bi ton sau:

BT 3.13. Cho tam gic ABC c cc cnh a, b, c v din tch S . Chng minh: a2 + b2 + c2

+ (a - b)2 + ( b - c)2 + (c - a)2 . Du ng thc xy ra khi tam gic ABC u.

Chng minh: p dng cch chng minh BT 3.11 ta a n BT sau:

3.13)

EMBED Equation.3 (*)

Ta s chng minh (*), tht vy p dng BT: ta c: ;

EMBED Equation.3 , cng cc BT trn v rt ngn ta

c:

+

EMBED Equation.3

EMBED Equation.3 (do ) suy ra :

EMBED Equation.3

Vi cch a ra BT ph nh sau: cng vi cch chng minh nh trn cc em chng minh c bi ton cht hn nh sau.

BT 3.14. Cho tam gic ABC c cc cnh a, b, c v din tch S . Chng minh: a2 + b2 + c2

+ (a - b)2 + ( b - c)2 + (c - a)2 . Du ng thc xy ra khi tam gic ABC u.

Vi cch dng thm hnh ph, cc chng minh c bi ton:

BT 3.15. Cho tam gic ABC c cc cnh a, b, c v din tch S. Chng minh a2 + b2 + c2 +8Rr + (a - b)2 + ( b - c)2 + (c - a)2 . Du ng thc xy ra khi tam gic ABC u.

Chng minh: Gi M, N , P l tm cc ng trn bng tip tng ng cc gc A, B, C ca tam gic ABC; gi R' , r' l bn knh ng trn ngoi tip, ni tip tam gic MNP , S' l din tch tam gic MNP. Trc ht ta chng minh cc kt qu sau:

1/ MN = 4R; NP = 4R;

PM = 4R;

2/ S' = .S

3/ MN2 + NP2 + PM2 = 8R(ra + rb + rc ).

4/ 4r( ra + rb + rc ) = a2 + b2 + c2 - (a - b)2 - (b - c)2 - (c - a)2.

Chng minh cc kt qu:

1/ Ta c MN = MC + CN = + = = 4R. Chng minh tng t ta cng c: NP = 4R; PM = 4R.

2/ Do tnh cht ng phn gic nn ta c:S' = 2R'2.sinM.sinN.sinP

= 2R'2., mt khc S' = = , nn suy ra R' = 2R do S' = 8R2. = = (do r = 4R.).

3/ T chng minh trn ta c: MN2 + NP2 + PM2 = 16R2(cos2 + cos2+ cos2). Mt khc: ra = MC.cos; rb = NC.cos suy ra: ra + rb = MN.cos= 4Rcos2, suy ra ; Tng t ta cng c: ; ; suy ra: MN2 + NP2 + PM2 = 8R(ra + rb + rc ).

4/ p dng cc cng thc: S = p.r ; S =(p - a)ra; S =(p - b)rb; S = (p - c)rc ta c: 4r( ra + rb + rc ) =

== a2 + b2 + c2 - (a - b)2 - (b - c)2 - (c - a)2.

Chng minh bi ton:

T cc kt qu trn p dng (3.11) ta c:

MN2 + NP2 + PM2 4S' + (MN - NP)2 + (NP - PM)2 + (PM - MN)2

8R(ra + rb + rc ) 16R2 +

+ a2 + b2 + c2 + (a - b)2 + ( b - c)2 + (c - a)2 +

8Rr .

Nh vy vi cch chng minh nh trn hc sinh a ra mt bi ton cht hn cc bi ton ban u.

By gi th tng thm gi thit ta c th lm cht bi ton nh th no , vi cu hi ny cc em a ra bi ton sau:

BT 3.16. Cho tam gic ABC c cc cnh a, b, c v din tch S. Vi gii thit abc, chng minh: + (a - b)2 +(b - c)2 + (c - a)2. Du ng thc xy ra khi tam gic ABC vung cn ti A.

Chng minh: p dng nh l Cosin: a2 + c2 = b2 + 2ac.CosB v cng thc tnh din tch S = ta c (3.14) a2 + b2 + c2 2ac.SinB + 4ac

+2 + 2a2 + 2b2 + 2c2 -2ab - 2bc - 2ca 2ab + 2bc - 2ca - b2 - (a2 + c2) - 2ac.SinB +2ac 0 ab + bc - ca - b2 - ac.SinB - ac.CosB +ab 0

(a - b)(b - c) +ab[1- Cos(B - 450)] 0. Do abc nn (a - b)(b - c) 0 v ab[1 -Cos(B - 450)] 0 nn (3.14) c chng minh.

By gi ta gi thit thm: tam gic ABC c 3 cnh lp thnh mt cp s cng, khi ta cng chng minh c bi ton sau:

BT 3.17. Cho tam gic ABC c cc cnh a, b, c lp thnh mt cp s cng v din tch S. Chng minh: a2 + b2 + c2 4S+. Du ng thc xy ra khi tam gic ABC u .

Chng minh: Ta c a2 + b2 + c2 - 4S = a2 + c2 + a2 + c2 - 2ac.CosB - 2ac.SinB= 2(a - c)2 + 2ac[2 - CosB - SinB] = 2(a - c)2 + 2ac[2 - CosB - SinB] = 2(a - c)2 + 4ac[1- Cos(B - 600)] . Gi s a, b, c theo th t lp thnh cp s cng vi cng sai d th a - b = d; a - c = 2d; b - c = d, do ta c

a2 + b2 + c2 - 4S - [(a - b)2 + (b - c)2 + (c - a)2]

= 2(a - c)2 + 4ac[1- Cos(B - 600)] -

= 8d2 + 4ac[1- Cos(B - 600)] - = 4ac[1- Cos(B - 600)] 0 suy ra

a2 + b2 + c2 4S+, v du ng thc xy ra khi tam gic ABC u .

Tng t cc em cng a ra bi ton :

BT 3.18. Cho tam gic ABC c cc cnh a, b, c lp thnh mt cp s cng v din tch S. Chng minh:

ab + bc + ca 4S+. Du ng thc xy ra khi tam gic ABC u.

Chng minh:

Ta c: ab + bc + ca - 4S= = ab + bc - ac - b2 + 2ac + b2 - 2ac.SinB

= (a - b)(b - c) + a2 + c2 + 2ac - 2ac.CosB - 2ac.SinB

= (a - b)(b - c) + (a - c)2 + 2ac[2 - CosB - SinB]

= (a - b)(b - c) + (a - c)2 + 4ac[1- Cos(B - 600)]. Gi s a, b, c theo th t lp thnh cp s cng vi cng sai d th a - b = d; a - c = 2d; b - c = d, do ta c

ab + bc + ca - 4S -

= d2 + 4d2 + 4ac[1- Cos(B - 600)] -

= 5d2 + 4ac[1- Cos(B - 600)] - 5d2 = 4ac[1- Cos(B - 600)] 0 , suy ra

ab + bc + ca 4S + .

Nhn xt: Nu xt trong lp cc tam gic c 3 cnh lp thnh cp s cng th cc bi ton 3.17 , 3.18 l cht hn cc bi ton nu, tuy nhin nu bin i tng ng (3.18) th ta s i n (3.17). Tht vy: (3.16)6(ab + bc + ca) 6.4S +10(a2 + b2 + c2 - ab - bc - ca)6(a2 + b2 + c2) 6.4S +10(a2 + b2 + c2 - ab - bc - ca) - 6(ab + bc + ca) + 6(a2 + b2 + c2) a2 + b2 + c2

4S+[(a - b)2 + (b - c)2 + (c - a)2], tc l (3.18)(3.17).

Nh vy trong phn ny vi hot ng tng qut ho bi ton theo hng lm cht hn, chng ti hng dn cc em a ra mt s bi ton mi cht hn cc bi ton trc .

Nh vy, t cc bi ton trn, chng ti rn luyn cho hc sinh mt s hot ng ton pht huy tnh sng to ca hc sinh, c th:

- Gii bi ton cho theo nhiu cch khc nhau, t cc cch ny c th tm thm cc bi ton khc.

- T mt bi ton no trong trng hp c bit, hy tm cch m rng cho cc trng hp khc.

- S dng cc hot ng ca t duy sng to nh: tng t , tng qut ho, c bit ho d on v chng minh cc bi ton mi.

1

_1232522026.unknown

_1263409935.unknown

_1263410046.unknown

_1263410200.unknown

_1263410312.unknown

_1263410338.unknown

_1263410449.unknown

_1263410503.unknown

_1263411287.unknown

_1263410678.unknown

_1263410450.unknown

_1263410368.unknown

_1263410443.unknown

_1263410344.unknown

_1263410326.unknown

_1263410333.unknown

_1263410319.unknown

_1263410283.unknown

_1263410300.unknown

_1263410307.unknown

_1263410295.unknown

_1263410227.unknown

_1263410257.unknown

_1263410207.unknown

_1263410142.unknown

_1263410180.unknown

_1263410190.unknown

_1263410196.unknown

_1263410185.unknown

_1263410154.unknown

_1263410166.unknown

_1263410175.unknown

_1263410160.unknown

_1263410147.unknown

_1263410115.unknown

_1263410125.unknown

_1263410133.unknown

_1263410138.unknown

_1263410120.unknown

_1263410105.unknown

_1263410110.unknown

_1263410093.unknown

_1263409998.unknown

_1263410020.unknown

_1263410037.unknown

_1263410041.unknown

_1263410026.unknown

_1263410009.unknown

_1263410015.unknown

_1263410005.unknown

_1263409957.unknown

_1263409973.unknown

_1263409981.unknown

_1263409962.unknown

_1263409968.unknown

_1263409948.unknown

_1263409952.unknown

_1263409940.unknown

_1232648082.unknown

_1263409831.unknown

_1263409917.unknown

_1263409925.unknown

_1263409931.unknown

_1263409920.unknown

_1263409898.unknown

_1263409911.unknown

_1263409837.unknown

_1263409798.unknown

_1263409812.unknown

_1263409822.unknown

_1263409805.unknown

_1232649000.unknown

_1232649178.unknown

_1232649427.unknown

_1233843028.unknown

_1263409765.unknown

_1263409793.unknown

_1263409751.unknown

_1233843128.unknown

_1232649342.unknown

_1232649157.unknown

_1232648649.unknown

_1232648666.unknown

_1232648397.unknown

_1232648508.unknown

_1232648096.unknown

_1232648361.unknown

_1232646967.unknown

_1232647383.unknown

_1232647508.unknown

_1232647587.unknown

_1232647660.unknown

_1232647027.unknown

_1232647003.unknown

_1232562743.unknown

_1232564345.unknown

_1232646486.unknown

_1232646547.unknown

_1232646558.unknown

_1232564353.unknown

_1232563894.unknown

_1232563925.unknown

_1232564039.unknown

_1232563920.unknown

_1232563258.unknown

_1232561738.unknown

_1232562570.unknown

_1232561957.unknown

_1232561199.unknown

_1232561218.unknown

_1232522202.unknown

_1232522064.unknown

_1232341478.unknown

_1232352494.unknown

_1232423036.unknown

_1232521614.unknown

_1232521984.unknown

_1232521818.unknown

_1232521893.unknown

_1232521776.unknown

_1232476942.unknown

_1232520648.unknown

_1232477410.unknown

_1232470141.unknown

_1232475789.unknown

_1232475882.unknown

_1232475987.unknown

_1232476004.unknown

_1232476044.unknown

_1232475969.unknown

_1232475808.unknown

_1232475769.unknown

_1232475781.unknown

_1232470760.unknown

_1232471184.unknown

_1232427173.unknown

_1232427231.unknown

_1232439028.unknown

_1232439073.unknown

_1232439408.unknown

_1232427265.unknown

_1232427188.unknown

_1232427224.unknown

_1232427179.unknown

_1232427153.unknown

_1232427167.unknown

_1232424331.unknown

_1232425349.unknown

_1232423137.unknown

_1232352545.unknown

_1232352560.unknown

_1232422512.unknown

_1232422542.unknown

_1232352597.unknown

_1232352608.unknown

_1232352614.unknown

_1232352602.unknown

_1232352592.unknown

_1232352549.unknown

_1232352521.unknown

_1232352540.unknown

_1232352516.unknown

_1232351605.unknown

_1232351626.unknown

_1232351637.unknown

_1232352140.unknown

_1232352239.unknown

_1232352247.unknown

_1232352163.unknown

_1232352176.unknown

_1232351915.unknown

_1232352081.unknown

_1232351937.unknown

_1232351740.unknown

_1232351632.unknown

_1232351615.unknown

_1232351621.unknown

_1232351610.unknown

_1232341980.unknown

_1232351534.unknown

_1232351563.unknown

_1232351568.unknown

_1232351559.unknown

_1232351519.unknown

_1232341573.unknown

_1232341942.unknown

_1232341564.unknown

_1231826956.unknown

_1232338986.unknown

_1232339329.unknown

_1232341346.unknown

_1232341382.unknown

_1232341228.unknown

_1232339235.unknown

_1232339008.unknown

_1232339024.unknown

_1232169911.unknown

_1232338788.unknown

_1232338930.unknown

_1232169917.unknown

_1232133536.unknown

_1232133574.unknown

_1232133503.unknown

_1226469928.unknown

_1226562121.unknown

_1228203041.unknown

_1228203064.unknown

_1228203090.unknown

_1228203098.unknown

_1228203083.unknown

_1228203050.unknown

_1226734752.unknown

_1227525321.unknown

_1227772270.unknown

_1226562318.unknown

_1226663550.unknown

_1226562179.unknown

_1226473455.unknown

_1226473470.unknown

_1226495598.unknown

_1226471461.unknown

_1226471527.unknown

_1226471567.unknown

_1226473439.unknown

_1226471484.unknown

_1226471289.unknown

_1226471361.unknown

_1226471343.unknown

_1226469967.unknown

_1226423213.unknown

_1226425307.unknown

_1226469760.unknown

_1226425262.unknown

_1226421110.unknown

_1226422298.unknown

_1226421043.unknown

_1226420345.unknown

_1226420412.unknown

_1226420344.unknown

bdt dien tich

Documents