100 Bai Hinh Hoc Chon Loc

8/9/2019 100 Bai Hinh Hoc Chon Loc

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100 Bai toan Hinh hoc Lp 9 Chon loc

1 LVC

Bien soanLU VAN CHUNG

100 BAI TOANHNH HOC 9


2 LVC


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3 LVC

PHAN I---------o0o---------

MOT SO BAI TOANON THI VAO LP 10 THPT


4 LVC

A N

x

E

M D

H

B K C

F I

AM

xI K D

EN H O

B F C

J Q

Bai 1Cho BE , CD la hai ng cao , AI la ng knh cua (O), Ax la tieptuyen . Chng minh :

1. BDEC noi tiep va MN // DE2. BH = BF va DM = DH3. BHCI la hnh bnh hanh4. BCIF la hnh thang can5. KH.KA = KB.KC 2BC

4

6. H la tam ng tron noi tiep DEF---------------------------------------------------------------------------------------------Bai 2Cho BD , CE la hai ng cao , Ax la tiep tuyenAQ la ng knh I la trung iem AH.Chng minh :1. BEDC va AEHD noi tiep2. AE.AB = AD.AC3. MAN can4. KDCQ noi tiep5. AH = 2OF6. IOFJ la hnh thang can

7. BC = R 3 . Tnh DE theo R

8. BC = R 3 . Khi A chay tren cung ln BC th H di chuyen tren

ng nao ?

9. Xac nh tam P cua ng tron ngoai tiep BHC va chng minhAOPH la hnh thoi .

---------------------------------------------------------------------------------------------Bai 3Cho H la trc tam cua ABC , M la trung iem BC . I la iem oi xngvi H qua K ; N la iem oi xng vi H qua M. Chng minh

1.I (O) va N (O) va A , O , N thang hang2.AF la tia phan giac chung cua ABC va HAN 3.AB.AC = AH.AD4.Dien tch AHM bang 2 lan dien tch AOM


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5 LVC

A

D

O

E H

K

B C

M

I

F

AE

OH

SC K M D

B

BI

H O

AC E

D

K

S

D

NM

C

I K

A O B

Bai 4Cho SA , SB la hai tiep tuyen , M la trung iem CD .Chng minh :

1. SO AB va SAOB noi tiep2. SA2 = SC.CD = SH.SO3. SAOM noi tiep4. CHOD noi tiep5. AE // CD6. CH cat ng tron tai F.

Chng minh SO la phan giac cua FSC

--------------------------------------------------------------------------------------------Bai 5Cho AB la tiep tuyen , ACD la cat tuyen , BI // CD, EC = ED

1.Chng minh ABOE va AHES noi tiep2.Chng minh AK la tiep tuyen cua (O)3.Chng minh OE.OS = R24. AO = 3R , CD = R 3 .

Tnh dien tch AOS theo R5. A, B co nh , tm v tr cua cat tuyen

ACD e S AID ln nhat

---------------------------------------------------------------------------------------------Bai 6Cho AC , BD , CD la cac tiep tuyen

1.Chng minh AC + BD = CD2.Chng minh COD vuong3.Chng minh AC.BD = R24.Chng minh ACMO va BDMO noi tiep5.Chng minh CN = CA6.Chng minh IK // AB va IK = R


6 LVC

Y

DX

M

CI

E A H O B

AE

C IH

M O

KF

B

A

K

H

C I

E O

N

D

B

Bai 7 Ax , By , ED la cac tiep tuyen . EO = 2R1. Chng minh ACDB la hnh thang vuong2. Chng minh AC.BD = R23. Chng minh CM.DE = CE.DM4. Chng minh MI AB va IM = IH5. Chng minh AB tiep xuc vi ng tron ng knh CD6. Tnh chu vi va dien tch ACDB theo R

-------------------------------------------------------------------------------------------------Bai 8Cho OM = 2R , MA va MB la hai tiep tuyenEF tiep xuc vi (O) tai C.

1. Chng minh EF = EA + FB2. Tnh chu vi MEF theo R3. Tnh EOF 4. Chng minh EIKF noi tiep5. Chng minh CH 2EF

4R

---------------------------------------------------------------------------------------------Bai 9Cho NA , NB la hai tiep tuyen . BK NA , CI AB , CD NB.

1.Chng minh cac t giac sau noi tiep : AKCI , BDCI , NKCD , NDIA2.Chng minh CI2 = CK.CD3.Chng minh CHIE noi tiep4.Chng minh EH // AB5.Chng minh 2

2

KI CK=

DI CD


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7 LVC

K

x

M

I

E F

A C B

A

FD

C

M O

E

B

x y

D

M

CE F

A I O B

Bai 10

Cho Ax , By la hai tiep tuyen . M AB , I AO , CD qua M vavuong goc vi IM . Chng minh :

1.CAIM va BDMI noi tiep2. CID vuong3.

EF // AB4. AC.BD R2

5. Khi M co nh , I chay tren AO.6. Tm v tr I e AC.DB ln nhat

---------------------------------------------------------------------------------------------Bai 11AB = 2R co nh , Ax , By AB , I Ax ,C AB , IC CK. ng tron ng knh ICcat IK tai M. Chng minh :

1. KBCM noi tiep2. AI.BK = CA.CB3. AMB vuong

4. Neu C co nh , tm v tr cua I tren Ax e SABKI nho nhat-------------------------------------------------------------------------------------------------

Bai 12Cho OM = 3R , MA , MB la haitiep tuyen ,AD // MB , MD cat (O)tai C , BC cat MA tai F , AC catMB tai E. Chng minh :

1. MAOB noi tiep2. EB2 = EC.EA3. E la trung iem MB4. BC.BM = MC.AB5. Tia CF la phan giac cua MCA 6. Tnh S BAD theo R7. Tnh khoang cach gia hai ng thang AD va MB

---------------------------------------------------------------------------------------------


8 LVC

IA D

HM

N

B CK

A EM

HD

C

K

B

F

Bai 13 ( Hnh ve nh Bai 12 )Cho MO = 3R , MA , MB la hai tiep tuyen . Ela trung iem MB. EA cat(O) tai C. Tia MC cat (O) tai D , BC cat MA tai F .

1.Chng minh MA2 = MC.MD2.Chng minh ME2 = EC.EA3.Chng minh AD // MB4.Chng minh DAB can

-------------------------------------------------------------------------------------------------Bai 14Cho DH AC , DK BC , KH cat ABtai I . MA = MB , NH = NK. Chng minh

1.DHKC noi tiep2.DI AB3.DI.DK = DA.DC4. DHK ~ DAB5. DMN vuong

-------------------------------------------------------------------------------------------------Bai 15Cho MA , MB la hai tiep tuyen , CD AB .CE MA , CF MB

1. Chng minh cac t giac sau noi tiep : DAEC , DBFC2. Chng minh CE.CF = CD23. Chng minh CHKD noi tiep4. Chng minh HK //AB

-------------------------------------------------------------------------------------------------Bai 16Cho ng thang d cat (O;R) tai C va D. M la iem di ong tren d ( Mngoai ng tron va MC < MD ) MA , MB la hai tiep tuyen , H la trungiem CD


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9 LVC

F

A

d M C E H D

I O

B

A

H O

M B D I C

E K

N

1.Chng minh MIHF noi tiep2.Chng minh OHEI noi tiep3.Chng minh MA2 = MC.MD4.Chng minh CIOD noi tiep5.Chng minh 4 IF.IE = AB2

6.Chng minh khi M di ong th ng thangAB luon iem qua iem co nh

-------------------------------------------------------------------------------------------------Bai 17

Cho (O;R) va day BC = R 3 .M la iem di ong tren tia oi cua tia BC .Ve tiep tuyen MA vi (O). Ve ng tron tam M ban knh MA cat BCtai H va cat (O) tai E. AH cat (O) tai K, OK cat BC tai I va cat AE tai N.Chng minh :

1.AK la tia phan giac cua BAC 2.MAOI va MHIN noi tiep3.ME la tiep tuyen cua (O)4.BHOC noi tiep5.ng thang AE i qua iem co nh

-------------------------------------------------------------------------------------------------Bai 18Cho (O;R) va day BC = 2a co nh. M tia oi tia BC. Ve ng tron

ng knh MO cat BC tai E , cat (O) tai A va D ( A cung ln BC). ADcat MO tai H , cat OE tai N. Chng minh :

1. MA la tiep tuyen cua (O) va MA2 = MB.MC2. MHEN noi tiep3. Tnh ON theo a va R4. Tia DE cat (O) tai F. Chng minh ABCF la hnh thang can


10 LVC

A

M H O

EB C

D

N

M

E

B O AK H

CI

DN

-------------------------------------------------------------------------------------------------Bai 19Cho ng tron (O ;R) va iem A sao cho OA =3R . T A ve hai tieptuyen AM , AN vi (O) ( M , N la tiep iem )

1. Chng minh t giac MANO noi tiep .Tnh AM theo R2. Ve cat tuyen ACD vi ng tron ( C nam gia A va D ; CN < CM

) . Chng minh : AM2 = AC.AD3. ng tron ng knh OA cat CD tai I.Chng minh I la trung

iem CD4. Ve day cung CB OM , CB cat MD tai K , cat MN tai H. Chng

minh HINC noi tiep5. Chng minh ng thang DH i qua trung iem E cua AM.

-------------------------------------------------------------------------------------------------


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11 LVC

Bai 20Cho tam giac ABC co ba goc nhon noi tiep ng tron (O;R); hai ngcao AD va BE cat nhau tai H ( D BC ; E AC ; AB < AC )

1.Chng minh cac t giac AEDB va CDHE noi tiep2.Chng minh CE.CA = CD.CB va DB.DC = DH.DA3.Chng minh OC vuong goc vi DE4.ng phan giac trong AN cua AC cat BC tai N , cat ng tron

(O) tai K. ( K khac A). Goi I la tam ng tron ngoai tiep CAN.Chng minh KO va CI cat nhau tai iem thuoc ng tron (O).

------------------------------------------------------------------------------------------------


12 LVC

PHAN II

MOT SO BAI TOAN LUYEN THI VAOCAC TRNG CHUYEN


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13 LVC

C

A H O B

IE

DJ

FM

Bai 21Cho ng tron (O) va day CD vuong goc vi ng knhAB tai H . tren tia oi tia DC lay iem M. ng thang MB

cat (O) tai F. AF cat CD tai I1. Chng minh : ID.MC = IC.MD2.

Tiep tuyen vi (O) tai F cat DM tai J. Ch/ minh IJ = JM3. MA cat (O) tai E. Chng minh JE la tiep tuyen cua (O).

---------------------------------------------------------------------------------------------- Hng dan giai:

1. Chng minh ID.MC = IC.MDFA , FM la phan giac trongva ngoai cua CFD nen :

ID MD

IC MC ID.MC = IC.MD

2. Chng minh IJ = JM

SAFJ S

AF

S DIF= S AC SdDF

= S AD SdDF

= S AF

AFJ JIF IJ = JFMa IFM vuong FJ = JM IJ = JM

3. Chng minh JE la tiep tuyen cua (O)Chng minh E,I,B thang hang

IEM vuong tai E JE = JF. Suy ra OEJ = OFJ

JE OE JE la tiep tuyen cua (O).----------------------------------------------------------------------------------------------Bai 22Cho ng tron (O) va day cung AB quay quanh mot iemco nh P.Ve ng tron tam C iem qua A va P va tiepxuc vi (O). Hai ng tron nay cat nhau tai N

1. Chng minh CODP la hnh bnh hanh2. Chng minh ABN ~ DCO3. Tm quy tch N khi AB quay quanh P


14 LVC

NO

CD

A P B

K

4. Neu AB co nh, con P la iem di o ng tren AB th N diong tren ng nao? Chng minh NP i qua mot iemco nh

---------------------------------------------------------------------------------------------- Hng dan giai:

1.

Chng minh CODP la hnh bnhhanh 0180 2 AOB B , 0180 2 PDB B

AOB PDB OC // DPTng t ta co OD // CP

CODP la hnh bnh hanh .

2. Chng minh ABN~ DCO

PN CD CD la phan giac NCP

1

2 DCP NCP

Ma 1

2 NAP NCP ( goc noi tiep )

DCP NAP NCP ODC

Tng t cm NBP OCD ABN~ DCO3. Quy tch iem M

Ta co : AOB ANB ( ong dang ) A , O , N , B thuoc ng tron AB la day cung

NOB NAB ( cung chan

NB )

NOB ODC ON // CD ON NP 090ONP N ng tron ng knh OP co nh

4. Neu AB co nh , P di ong tren AB th N di chuyen tren cungcha goc AOB = ( khong oi ) dng tren oan AB co nh.

NP i qua iem co nhGoi O la tam ng tron (AOB) , NP cat OO tai K.Do ONK vuong tai N OK la ng knh cua (O).Suy ra K co nh ( do O , O co nh ).Vay NP iem qua iem K co nh.


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15 LVC

I

A P

H

B

O K

O1

C

----------------------------------------------------------------------------------------------Bai 23Cho ng tron (O,R). T iem P co nh ngoai ng

tron ve tiep tuyen PA va cat tuyen PBC vi ng tron . Hla trc tam cua ABC. ng thang AH cat (O,R) tai K.

1.Chng minh K oi xng vi H qua BC2. Goi O1 oi xng vi O qua BC. Chng minh O1H = R vaAOO1H la hnh bnh hanh

3. Tren ng vuong goc vi PA tai P lay iem I sao choPI = R va I , O nam cung pha oi vi BC. Chng minhHIPO1 la hnh bnh hanh

4. Tm quy tch cua H khi cat tuyen PBC quay quanh P.----------------------------------------------------------------------------------------------

Hng dan giai1. Chng minh K oi xng vi H qua BC

CK BAK ( chanBK)

CH BAK ( goc co canh )

CH BCK HCK can tai C K oi xng vi H qua BC

1. Chng minh O1H = R va AOO1Hlahnh bnh hanh

O i xng vi O1 qua BCKi xng vi H qua BC O1H = OK = R

Ta c 1 1

H K ( t/c i xng )

1 1

K A ( AOK cn )

1 1A H HO1 // OA

AH BC ; OO1 // BC AH // OO1 . Suy ra AOO1H l hnh bnh hnh2.Chng minh HIPO1 l hnh bnh hnh

HO1 // OA v OA AP HO1 AP m PI AP HO1 // PIM O1H = PI = R HIPO1 l hnh bnh hnh

3.Qu tch ca H khi PBC quay quanh P


16 LVC

E F

N S

D C

H KI

A BM

P cnh , PI AP ; PI = R I cnh . PH = PO1 = PO ( khng i )Suy ra qu tch ca H l ng trn (I ; PO )

-------------------------------------------------------------------------------------------------Bi 24

Trn on AB ly im M .Trn AM , BM dng hai hnh vung AMCD vBMEF v cng mt pha i vi ng thng AB . Hai ng trn (I) v

(K) ngoi tip hai hnh vung ct nhau ti N .1. Chng minh N , E , A thng hng2. Tm qu tch N khi M di ng trn on AB3. Chng t trung im H ca IK lun chy trn mt ng c nh

khi M di chuyn trn on AB-------------------------------------------------------------------------------------------------

Hng dan giai1. Chng minh A , N , E thng hngGi N l giao im ca BC v AE . AC ct EB ti S . Ta c :

045SAB SBA AS EB . Suy ra C l trc tm ca AEB BN AE . 090ENB N ( K ) ;

090ANB N ( I )Vy A , E thng hng vi giao im N ca ( I ) v ( K )2. Qu tch ca N khi M di ng trn ABQu tch ca N l nang trn ng knh AB.Gii hn : M A N S ; M B N S

3. Qu tch ca H khi M chy trn ABMI AC , MK BE , AS SB ISKM l hnh ch nht . H l trung

im IK H l trung im SM . M S cnh (imchnh gia cung AB ).Suy ra H chy trn ng trung bnh ca ASB

-------------------------------------------------------------------------------------------------


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17 LVC

P

K I

M

A O B

N

Q

Bai 25Cho AB la ng knh co nh , MN la ng knh di ong cua ngtron (O) .Tiep tuyen tai B cua ng tron cat AM tai P , cat AN tai Q1. Chng minh AMN ~ AQP2. Chng minh trung tuyen AK cua APQ vuong goc vi MN3. Tm quy tch tam ng tron ngoai tiep MNP.-------------------------------------------------------------------------------------------------

Hng dan giai1. Chng minh AMN ~ AQP

S ANM =1

2s AM

S 1

2APQ (s AB s MB )

1

2 s AM

ANM AQP

AMN ~ AQP

2. Chng minh AK MN PAQ vuong tai A AK = KP AKP can

PAK APQ MNA

Ma NA MA AK MN3. Quy tch tam ng tron (MNP)

T giac MPQN noi tiep ( P ANM )ng tron (MNP) la ng tron (MPQN)

co tam la I OI MN ( k dc) OI // AK . Ma IK PQ IK //AB

Do IK = OA = R nen quy tch tam I cua ng tron (MNP) la ng thangsong song vi PQ va cach PQ mo t oan bang R-------------------------------------------------------------------------------------------------Bai 26Cho ABC eu noi tiep ng tron (O;R). M la mot iem tuy y thuoc(O;R).1. Chng minh MA + MB + MC 4R2. Tm v tr cua M AB e MA + MB ln nhat-------------------------------------------------------------------------------------------------


18 LVC

A

MN

B

C

D

C

MN

A B

Hng dan giai1. Chng minh MA + MB + MC 4R

Tren oan MC lay iem N sao cho CN = AM BCN = BMA BM = BN BMNeu BM = MN MA + MB + MC = NC + MN + MC

= MC + MC= 2MC

Ma MC 2R MA + MB + MC 4R

2. Tm v tr M e MA + MB ln nhatTheo chng minh tren ta co MA + MB = MCDo o MA + MB ln nhat MC ln nhat MC = 2R M la iem gia cung AB nho

-------------------------------------------------------------------------------------------------Bai 27

Cho na ng tron ng knh AB . C la iem chnh gia cung AB , Mla iem di ong tren cung AC . Tren BM lay iem N sao cho BN = AM.1. Chng minh CM = CN2. Chng minh ng thang vuong goc vi BM tai N luon i qua iem

co nh-------------------------------------------------------------------------------------------------

Hng dan giai1. Chng minh CM = CN

CMA = CNB ( c- g c) CM = CN

2. Chng minh ng thang MB tai Ni qua iem co nh

Ve tia Nx BM tai N cat tia AC tai D t giac DCNB noi tiep

0DBC DNC 45 ( do MCN vuong can)

0DBA 90 Suy ra BD la tiep tuyen cua (O) tai B. Vay D la giao iem cua tia AC va tieptuyen cua (O) tai B. Vay ng thang vuong goc vi BM tai N i qua iemco nh D.-------------------------------------------------------------------------------------------------


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19 LVC

C

KI

GA B

O

Bai 28Cho ng tron (O;R) ng knh AB . C la iem di ong tren ngtron Goi I , K la hnh chieu cua O len AC va BC.1. Chng minh AK2 + BI2 khong oi2. BI cat AK tai G . Tm quy tch cua G khi C di ong tren (O)3. Tm v tr iem C tren (O) e tch GA.GB ln nhat . Tm gia tr ln

nhat o-------------------------------------------------------------------------------------------------

Hng dan giai1. Chng minh AK2 + BI2 khong oi

AK2 + BI2 = AC2 + BC2 + CI2 + CK2= BC2 + IK2 = 4R2 + R2 = 5R2

2. Quy tch cua G

Ta co OG =1

R3

. Suy ra quy tch cua G la

ng tron (O ;1

R3

)

3. Tm v tr iem C e GA.GB ln nhat

Ta co GA.GB 2 2GA GB

2

Dau = xay ra khi GA = GB AC = BC C la iem chnh gia cung AB

Max(GA.GB) =2 2GA GB

2=

2 22

4(AK BI ) 10R9

2 9

-------------------------------------------------------------------------------------------------

Bai 29Cho ng tron (O;R) , t iem P trong ng tron dng hai day cungAPB va CPD vuong goc vi nhau1. Tnh PA2 + PB2 + PC2 + PD2 theo R2. Cho P co nh , khi hai day AB va CD quay quanh P va vuong goc vi

nhau. Chng minh AB2 + CD2 khong oi-------------------------------------------------------------------------------------------------

Hng dan giai1. Tnh PA2 + PB2 + PC2 + PD2 theo R

Ve ng knh AE. Suy ra CB = DETa co : PA2 + PB2 + PC2 + PD2


20 LVC

A

C DP H

KO

B E

A

OM N

B CK

B D C

= CB2 + AD2 = DE2 + AD2 = AE2 = 4R22. Chng minh AB2 + CD2 khong oiVe OH CD , OK AB . Ta co :AB2 + CD2 = 4AK2 + 4HD2

= 4( OA2 OK2 + OD2 OH2 )= 4( 2R2 OP2 ) khong oi

-------------------------------------------------------------------------------------------------Bai 30

Cho ABC can tai A co 0BAC 45 noi tiep ng tron (O;R).

1. Chng minh AO la phan giac cua BAC 2. Tnh cac canh cua ABC theo R3. Neu cach dng ng tron tiep xuc vi OB va OC. Tnh ban knh

ng tron o theo R-------------------------------------------------------------------------------------------------

Hng dan giai 1. Chng minh AO la phan giac BAC AOB = AOC

2. Tnh cac canh cua ABC theo R

BOC vuong can BC = R 2

OK =R 2

2 AK = R +

R 2

2

AC = R 2 2

3. Cach dng ng tron (I)

Goi x la o dai ban knh (I) ta coCMIN la hnh vuong

OI = x 2 , ID = x

x + x 2 = R x = R( 2 1 )

Cach dng:Dng tiep tuyen tai D cua (O) cat OB , OC tai B va C . I la giao iemcua hai ng phan giac cua OBC.

-------------------------------------------------------------------------------------------------


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21 LVC

A

M

OI

N B C

DK

E

Bai 31Cho ABC eu noi tiep ng tron (O;R). M la iem thuoc cung nhoAB . AM cat ng thang BC tai N1. Chng minh ABM ~ ANB2. Chng minh ng tron tam I ngoai tiep MBN tiep xuc vi ng

thang AB

3. Goi D la iem chnh gia cung BC. Chng minh tam K cua ngtron ngoai tiep MNC thuoc ng thang CD

-------------------------------------------------------------------------------------------------- Hng dan giai

1.Chng minh ABM ~ ANB

1 1ANB (sdAC sdMB) sdAM2 2

1MBA sdMA2

ANB MBA ABM~ ANB

A chung

2. Chng minh AB tiep xuc vi ng tron ngoai tiep MBN

ABM~ ANB AB2 = AM.AN AB tiep xuc vi ng t ron (MBN)

3. Chng minh tam K cua ng tron (MNC) thuoc ng thang CD

Ve ng knh CE cua ng tron (K) 0EMC 90

Ma 0AMC ABC 60 0NME 30

0NCE 30 ( cung chan cung NE )

Mat khac D la iem chnh gia cung BC 01

BCD 30 ( sdBD )2

NCE NCD K,C, D, E thang hang.Vay K thuoc ng thang CD

-------------------------------------------------------------------------------------------------


22 LVC

A

E

F

BD M C

DA

MN

B C

Bai 32Cho ABC co trung tuyen AM va phan giac AD.ng tron ngoaitiep ADM cat AB , AC tai E va F. Chng minh BE = CF.-------------------------------------------------------------------------------------------------

Hng dan giaiTa co

DB AB

DC AC (tnh chat phan giac)Ta chng minh : BE.BA = BD.BM (1)

CF.CA = CM.CD (2)

BE BD BA

: 1CF CD CA

BE = CF

-------------------------------------------------------------------------------------------------Bai 33Cho ABC noi tiep ng tron (O) co AB < AC. M va N la hai iem

chay tren AB va AC sao cho BM = CN. Chng minh trung trc cua MNluon i qua iem co nh-------------------------------------------------------------------------------------------------

Hng dan giaiGoi D la iem chnh gia cung ln BC BMD = CND MD = ND MDN can tai D

ng trung trc cua MN i qua iem Dco nh

---------------------------------------------------------------Bai 14

Cho ng tron (O;R) co hai ng knh AB vaCD vuong goc. Ke day CE i qua trung iem Icua OB . Ke ng cao AH cua ACE.

1. Chng minh cac tam giac COI , EOC va AHI ong dang2. Tnh CE , AH va S ACE theo R3. Chng minh AC la tiep tuyen cua ng tron ngoai tiep AEI------------------------------------------------------------------------------------------------


12/38


23 LVC

C

H

A O I B

K E

D

M

I A

N

EI J

BK

H C

Hng dan giai1. Chng minh cac tam giac COI , EOC va AHI

ong dangBan oc t chng minh

2. Tnh CE , AH va S ACE theo R

CI = R 5 ( nh ly Pitago )

OCI ~ ECD

OC CI

EC CD EC =

2R 5

5

AHI ~ COI AH AI

CO CI

AH =3R 5

10

2

ACE

1 3RS AH.CE

2 10

3. Chng minh AC la tiep tuyen cua ng tron ngoai tiep AEI CAI va CEA co :

0

CAI CEA 45 va

C chung CAI ~ CEA AC2 = CI.CE AC la tiep tuyen cua ng tron ngoai tiep AEI

-------------------------------------------------------------------------------------------------Bai 34Cho ABC vuong tai A , dng hai ng tron (I ) va (J ) ng knh ABva AC , chung cat nhau tai H1. Chng minh H thuoc ng thang BC2. Qua A ve ng thang d cat hai ng tron tai M va N ( M (I) )

Tm quy tch trung iem E cua MN3. Tm v tr cua ng thang d e chu vi t giac BCNM ln nhat-------------------------------------------------------------------------------------------------


24 LVC

B M H

C O A

Hng dan giai1. Chng minh H thuoc ng thang BC

Chng minh 0AHC AHB 90 H B2. Quy tch iem I

BMNC la hnh thang .Ve IK MN K la trung iem BC. ( co nh ). AIK vuong tai I I thuoc ng tron ng knh AK

3. Tm v tr ng thang d e chu vi t giac BCNM ln nhatChu vi BCNM = BC + BM + MA + NA + NC. Trc het ta tm v tr Me MA + MB ln nhat . Ta co : (MA + MB)2 = MA2 + MB2 + 2MA.MBSuy ra MA + MB ln nhat MA.MB ln nhat MH.AB ln nhat MH ln nhat M la iem chnh gia cung AB

Ta chng minh khi M la iem chnh gia AB th N la iem chnh giaAC . That vay : M la iem chnh gia AB 0MAB 45

0NAC 45 N la iem chnh gia AC.

Vay chu vi t giac BCNM ln nhat khi ng thang i qua iem chnh giaAB va AC .-------------------------------------------------------------------------------------------------Bai 35Cho ng tron (O :R) va ng thang xy tiep xuc vi (O) tai A. T iemtuy y tren (O) ve BH xy tai H.

1. Chng minh BA la phan giac cua OBH 2. Chng minh phan giac ngoai cua OBH i qua iem co nh3. Phan giac AOB cat BH tai M. Tm quy tch M khi B di ong tren

(O;R)

------------------------------------------------------------------------------------------------- Hng dan giai

1. Chng minh BA la phan giac cua OBH

2. Chng minh phan giac ngoai cua OBH

i qua iem co nhAB AC AC la ng knh cua (O) C co nh

3. Tm quy tch MChng minh AOMB la hnh thoi

AM = R M (A; R)


13/38


25 LVC

B

D

I O N AK

H M

E

C

-------------------------------------------------------------------------------------------------Bai 36Cho ng tron (O;R) va iem A co nh vi OA = R. BC la ng knh

quay quanh O (A BC ). ng tron ngoai tiep ABC cat ng thangAO tai I.1. Tnh OI theo R. Suy ra I co nh2. Trng hp AB , AC cat (O) tai D va E . DE cat OA tai K.

a. Chng minh t giac KECI noi tiep .b. Tnh AK theo Rc. Chng to tam ng tron (AED) thuoc mot ng tron co nh.

4. Tm v tr BC e dien tch ABC ln nhat5. Tm v tr BC e ban knh ng tron ngoai tiep ABC nho nhat

------------------------------------------------------------------------------------------------- Hng dan giai

1. Tnh OI theo R. Suy ra I co nhOI.OA = OB.OC OI =

2R R

2R 2 I co nh ( do O co nh )

2.a. Chng minh t giac HECI noi tiep .

BDEC noi tiep (O) DBC DEA BICA noi tiep (O) ngoai tiep ABC

DBC AIC . Suy ra : AIC DEA Suy ra : AKCI noi tiepb. Tnh AK theo RECIK noi tiep AK.AI = AE.AC

BDEC noi tiep AE.AC = AD.AB = OA2 R2 AK.AI = OA2 R2 = 3R2

Ma AI = AO + OI = 2RR

2 =

5R

2 AK =

6R

5

c. Chng to tam ng tron ngoai tiep AED thuoc mot ng tron co nh.

Goi M la tam ng tron (ADE), goi N la giao iem cua (M) va OA.

Ta co DNA DEA ( chan DA )

Ma : DEA DBO ( cmt ) DNA DBO BDNO noi tiep


26 LVC

AN.AO = AD.AB = AK.AI. Do A, O , K , I co nh N co nh M thuoc ng trung trc cua NA co nh3. Tm v tr BC e dien tch ABC ln nhat

ABC

1S BC.AH R.AH

2 . Do o S ABC ln nhat AH ln nhat

AH AO BC OA4. Tm v tr BC e ban knh ng tron ngoai tiep ABC nho nhat

Ban knh ng tron (ABC) la OA. Ve OO OA , ta co OA OAMa O co nh ( O la trung iem IA ).Nen OA nho nhat OA = OA O O.Ma OB = OC BC OA hay BC OA

Khi o : OA =IA 5R

2 4

-------------------------------------------------------------------------------------------------Bai 37Cho hnh vuong ABCD co nh canh la a. E la iem di ong tren CD( E D ). ng thang AE cat ng thang BC tai F. ng thangvuong goc vi AE tai A cat ng thang CD tai K.

1. Tnh s AFK . AFK 2. Goi I la trung iem FK. Chng minh I thuoc ng thang co nh khi

E di chuyen tren canh CD

3. Chng minh : AFB AIB 4. Goi N la giao iem AF va BD. Chng minh INCF noi tiep .5. Tnh o dai cac canh AEK theo a va x , vi x = DE (0 < x a )6. Tm v tr cua K e EK ngan nhat .------------------------------------------------------------------------------------------------- Hng dan giai1. Tnh s AFK ADK = ABF AK = AF KAF vuong can AFK = 4502. Chng minh I thuoc ng thang co nh

T giac ADIK noi tiep 0KDI KAI 45

Ma 0ADB 45 0IDB 180 Suy ra : I , D , B thang hang I thuoc ng thang BD co nh.

3. Chng minh : AFB AIB : Chng minh ABFI noi tiep

4. Chng minh t giac INCF noi tiep


14/38


27 LVC

A B

N a

K D x E C

IF

Ta co IC = IA va NC = NA INA = ICA ( c- c-c )

IAN ICN

Ma IABF noi tiep 0IAN IBF 45 0ICN IFA 45 Suy ra t giac INCF noi tiep .

5. Tnh o dai cac canh cua AEK theo a va x

CF //AD EC FC a x FC a(a x)FCED AD x a x

BF = BC + CF = a +a(a x)

x

BF =

2a

x

Ta co DK = BF =2a

x KE = KD + DE =

2a

x+ x2 =

2 2a x

x

AE = 2 2 2 2AD DE a x

AK2 = KE2 AE2 =22 2 2 2 2

2 22

a x a (a x )(a x )

x x

Suy ra AK = 2 2a a xx

6. Tm v tr iem E e KE ngan nhat

EK =2 2a x 2ax

2ax x

EK min= 2a .Dau = xay ra khi a = x E C. Khi o KD = a.

-------------------------------------------------------------------------------------------------Bai 38Cho ABC vuong tai A (AB < AC).Dng ngoai tam giac hai hnh vuongABHK va ACDE .1. Chng minh H , A , D thang hang2. ng tron ngoai tiep ABC cat AD tai F. Ch/ minh FBC vuong

can3. BF cat ED tai M. Chng minh K , B , C , E , M cung thuoc mot ng

tron4. Chng minh MC2 = MB.MF5. Biet 0ACB = 30 .Tnh dien tch t giac HBCD theo BC.

--------------------------------------------------------------------------------------------


28 LVC

E

M

K D

F

A

H

B O C

Hng dan giai1. Chng minh H , A , D thang hang

0HAB DAC 45 H , A , D thang hang2. Chng minh FBC vuong can

0

FAC 45 FC FB FB FC

3. Chng minh K , B ,C , E , M cung thuoc ng tron 0BMC BKC 45 BKMC noi tiep 0BKC BEC 45 BKEC noi tiep K , B , C , E , M cung thuoc ng tron tam F la trung iem BM

4. Chng minh MC2 = MB.MF

0MCB 90 MC latiep tuyen cua (O) MC2

= MB.MF5. Biet 0ACB 30 .Tnh dien tch t giac HBCD theo BC.

at BC = 2a , ta co 0ACB 30 ABC la na tam giac eu

AB = a va AC = a 3SBHDC = SABH + SABC + SACD

=1 1 1

AB.BH AB.AC AC.CD2 2 2

= 2 21 1 1

R R.R 3 (R 3)2 2 2

= 21

R (4 3)2

-------------------------------------------------------------------------------------------------Bai 39Cho ABC vuong tai A. Lay D thuoc canh AC (DC < DA). Ve ngtron (D) tiep xuc vi BC tai E. Ve tiep tuyen BF vi ng tron (D) catAD tai I , BD cat AE tai K.1. Chng minh A , B , E , D , F thuoc mot ng tron. Xac nh tam.2. Chng minh IF.BK = IK.BF3. Trung tuyen AM cua ABC cat BF tai N. Chng minh NA = NF.-------------------------------------------------------------------------------------------------

Hng dan giai1. Chng minh A , B , E , D , F cung thuoc mot ng tron


15/38


29 LVC

A F

IN

DK

B C

M E

xE

M

O B A

IE

K

F N

0BAD BFD BED 90 2. Chng minh IF.BK = IK.BF

AFD FBD DBE DAF

AD la phan giac FAE

AB la phan giac ngoai cua FAE

IK BK IK.BF IF.BKIF BF

3. Chng minh NA = NF

AM la trung tuyen cua ABC vuong MAC MCA (MC = MA)

Ma FAM MAC FAC va AFB AEB ACE CAE

Do FAC CAE (cmt) ANF can tai N NA = NF.

-------------------------------------------------------------------------------------------------Bai 40

Cho x Oy = a ( khong oi ). iem O co nh . M Ox , N Oy sao cho

OM + ON = 2a ( khong oi ).1. Chng minh trung iem I cua MN thuoc mot oan thang co nh.2. Chng minh ng trung trc cua MN i qua mot iem co nh3. Chng minh ng tron ngoai tiep OMN i qua iem co nh khac

iem O4. Xac nh v tr MN e MN ngan nhat------------------------------------------------------------------------------------------------

Hng dan giai

1. Chng minh trung iem I cua MN thuoc mot oan thang co nhLay E Ox , F Oy sao cho OE = OF = a.


30 LVC

Do MO + NO = 2a ME = NF.Ve MK // EF (K Oy) KF = ME FK = FNSuy ra I la trung iem MN ( ng trung bnh )

Vay I EF co nh.2. Chng minh ng trung trc cua MN i qua mot iem co nhVe phan giac Oz cua xOy . Ve Et Ox cat Oz tai A ( co nh)

AF Oy tai F MEA = NFA AM = AN A thuoc trung trc cua MN , ma A co nh. pcm3. Chng minh ng tron (OMN) i qua iem co nh khac O

T giac MEAI noi tiep AEI IMA (chan cung AI)

Ma AEI EOA goc co canh tng ng vuong goc )

AEI AOF AOF AMN ( AMI ) MOAN noi tiepVay ng tron (MON) i qua iem co nh A

4. Xac nh v tr MN e MN ngan nhatXet MAN can tai A co : AMN ANM AON

2

( khong oi )

Ma AI MN . Vay MN AI nho nhat . Do tgAIB vuong tai B nen :AI nho nhat I B MN OA MN EF.

-------------------------------------------------------------------------------------------------Bai 41Cho BC la day cung co nh cua ng tron (O;R). A cung ln Bcsaocho O nam trong ABC .Ve ng cao AD , BE , CF cat nhau tai H.1. Chng minh AEF ~ ACB2. M la trung iem BC.Chng minh OM = 1 AH

2

3. N la trung iem EF .Chng minh R.AN = AM.ON4. Chng minh : 2S ABC = (EF + FD +DE).R5. Tm v tr iem A tren (O;R) de chu vi DEF ln nhat ?-------------------------------------------------------------------------------------------------

Hng dan giai3. Chng minh R.AN = AM.ON

AEF ~ ACB ( t so k ) AN AE

kAM AB

( AN , AM la trung tuyen tng ng )Mat khac : BAE ~ BMO ( g- g)


16/38


31 LVC

A

E

F K

H

O

B D M C

A F

E

I

B C

D

AE OM OM

AB OB R

AN OM

AM R R.AN = AM.OM

4. Chng minh : 2S ABC = (EF + FD +DE).R

Ta co :

EF AN AE OM

BC AM AB OB EF.R = OM.BC = 2S BOC

Tng t ta chng minh : FD.R = 2S AOC va DE.R = 2S BOASuy ra : R(EF + FD + DE) = 2(S BOC + S COA + S AOB) = 2S ABC5. Tm v tr iem A tren (O;R) de chu vi DEF ln nhat ?

Theo cau 4 th chu vi DEF ln nhat S ABC ln nhat AD ln nhat A la iem chnh gia cung ln BC.

-------------------------------------------------------------------------------------------------Bai 42

Cho ABC vuong tai A , AI la trung tuyen . D BC ( D B , C )E , F la tam ng tron (ABD) va (ADC). Chng minh A, E , I , D , Fcung thuoc mot ng tron .-------------------------------------------------------------------------------------------------

Hng dan giaiTa co : AFD 2ACD ( goc tam F va goc noi tiep )

AIB 2ACD (AI la trung tuyen tam giac vuong)

AIB AFD


32 LVC

P BA

I

Q

D C

Suy ra : A , F , D , I cung thuoc ng tron (1)

Ta co : 1

ABD AED2

( goc noi tiep va goc tam E )

Ma 1

FED AED2

( EF la trung trc cua AD) FED ABD

Ma 0ADE FED 90 ( EF AD ) 0ACD ABD 90 ( ABC vuong )

AFE ADE A, F , D , E cung thuoc ng tron (2)T (1) va (2) ta co A , E , D , I , F cung thuoc mot ng tron-------------------------------------------------------------------------------------------------Bai 43Cho hnh vuong ABCD co canh bang 1 n v. Tren canh AB , CD lay haiiem P va Q sao cho chu vi APQ = 2 ( n v )

1. Chng minh PB + QD = PQ2. Chng minh 0PCQ 45

-------------------------------------------------------------------------------------------------

Hng dan giai1.Chng minh PB + QD = PQ

Ta co : PB + QD= AB AP +AD +AQ = AB + AD ( AP + AQ )= 2 ( AP + AQ ) = ( AB + AP + PQ ) ( AP + AQ ) = PQ

2.Chng minh 0PCQ 45


17/38


33 LVC

A B

E

M

I H

F

D CN

Ve ng tron tam C ban knh 1 v AB , AD la hai tiep tuyen cua (C ) .T iem P tuy y thuoc AB ve tiep tuyen vi (C) tai I cat AB tai Q. Ta co :PB + QD = PQ P va Q thoa man gia thiet cau 1. , tc la chu vi APQ = 2 (v). Ta lai co CI PQ CI la tiep tuyen chung cua haing tron (P; PB) va (Q ; QD) .

Suy ra : 1 2C C va 3 4C C

01 2 3 4C C C C 90 02 3C C 45 hay 0P'CQ' 45

Vay neu hai iem P va Q thoa man chu vi APQ = 2(v) th 0PCQ 45 .

Chu y1. Co the thay oi gia thiet la hnh vuong canh a , chu vi APQ = 2a2. Cau 2 co the hoi :a. Chng to ng thang PQ luon tiep xuc vi mot ng tron co nhb. Chng to chan ng cao CI cua CPQ thuoc ng tron co nh.

-------------------------------------------------------------------------------------------------Bai 44Cho hnh vuong ABCD canh la a .Tren canh AD , CD lan lt lay 2 iem

M , N sao cho 0MBN 45 . BM , BN cat AC tai E , F.1. Chng minh MEFN noi tiep2. MF , NE cat nhau tai H. BH cat MN tai I. Tnh BI theo a.

Suy ra AM + NC = MN( Hoac chng to MN luon tiep xuc vi mot ng tron co nh)-------------------------------------------------------------------------------------------------

Hng dan giai1.Chng minh MEFN noi tiep

0EBN ECN 45 BENC noi tiep

0ENF BCE 45 0MBF MAF 45

ABFM noi tiep 0BMF BAF 45

Suy ra : 0EMF ENF 45 MEFN noi tiep

2.Tnh BI theo a. Suy ra AM + NC = MN 0EBN ENB 45 ( cmt) EBN vuong can NE BM

Tng t : MF NB . Suy ra : BH MN tai I


34 LVC

A

I

E

D

B CM

Suy ra : MBI MNE (goc co canh tng ng vuong goc)

Ma : MNE MFE ( MEFN noi tiep ) EFM ABM ( ABFM noi tiep )

ABM MBI Suy ra : ABM = IBM ( canh huyen goc nhon )

BI = AB = a Suy ra : AM = MITng t : NC =NI MN = AM + NC MN BI tai I

MN luon tiep xuc vi ng tron (B; a) co nh.

-------------------------------------------------------------------------------------------------Bai 45Cho ABC can tai A noi tiep ng tron (O). Tren BC lay iem M. VeMD // AB va ME // AC ( D AC ; E AB ). Chng minh iem doixng vi M qua DE thuoc ng tron (O).-------------------------------------------------------------------------------------------------

Hng dan giai M oi xng vi I qua ED EID EMD

AEMD la hnh bnh hanh EMD EAD

EAD EID EAID noi tiep

ADI AEI (1)Ta co DM =DI ( oi xng )

DM = DC ( DMC can )

DI = DC DIC can ADI 2ACI (2)

Tng t : AEI 2ABI (3)

T (1) , (2) va (3)

ACI ABI

ABCI noi tiep I (O)-------------------------------------------------------------------------------------------------Bai 46Cho hnh vuong ABCD tam O.Ve ng thang d qua O cat AD , BC tai Eva F. T E , F ve cac ng thang song song vi BD va AC cat nhau tai I.

1. Tm quy tch I khi E chay tren AD2. Ve ng cao IH cua IEF. Tm quy tch cua H3. Chng minh ng thang HI i qua iem co nh

-----------------------------------------------------------------------------------------


18/38


35 LVC

K

A I B

E

H

N

O

F

D C

yA

xA1 D A2

EH

B C

Hng dan giai1. Tm quy tch I khi E chay tren ADOE = OF ( tnh chat oi xng )Ma EI // OM MI = MFDo FI BD I oi xng vi F qua BD I AB

2.Quy tch cua HIHFB noi tiep 0BHF BIF 45

0BHO 45 .Do B va O co nh nen H di chuyen trencung cha goc 450 dng tren oan BO( co gii han )

3. IH iem qua iem co nhVe Ox AB cat IH tai K. Ta co : 0KHB IFB 45 ( HIBF noi tiep )

0

KHB KOB 45 KHOB noi tiep 0

BKO BHF 45 (=

BIF ) OKB vuong can tai B OK = a K co nh

-------------------------------------------------------------------------------------------------Bai 47

Cho ng tron (O) day BC co nh . A BC ln .1. Tm v tr cua A e ABC co ba goc nhon.2. Tm quy tch trc tam H cua ABC khi A chay tren (O)

thoa man ieu kien cau 13. Chng minh ED OA

-------------------------------------------------------------------------------------------------

Hng dan giai1.Tm v tr cua A e ABC co ba goc nhon.Ta co A < 900 , e ABC co ba goc nhon

th B < 900 va C < 900 A 1 2A A

2.Tm quy tch trc tam HTa co 0BHC 180 A ( khong oi )

H cung cha goc 0180 A dng trenoan BC co nh

3. Chng minh ED OA


36 LVC

T E

DO

A B C

Ve tiep tuyen xAy cua (O) , chng minh xy // DE DE OA

-------------------------------------------------------------------------------------------------Bai 48Tren ng thang d lay 3 iem A, B , C co nh. Biet AB = a, BC = b.ng tron (O) di ong i qua B va C. Ve tiep tuyen AT cua (O) (T(O) )

1. Tm quy tch cua T2. Ve ng knh BE cua (O). AE cat (O) tai D. Chng minh :

AD.AE = OA2 OC23. Tm quy tch D va E .

----------------------------------------------------------------------------------------------- Hng dan giai

1. Quy tch cua TChng minh : AT2 = AB. AC = ab

AT = ab T (A; ab )

2. Chng minh AD.AE = OA2 OC2Chng minh AD.AE = AT2 = OA2 OC2

3. Tm quy tch D va E . ADB vuong tai D D ng knh AB ACE vuong tai C CE AC E ng thang d AC tai C

-------------------------------------------------------------------------------------------------Bai 49Cho ng tron (O;R) co hai day cung AC va BD vuong goc tai I nam

trong ng tron ( I O ).1. Chng minh IA.IC = IB.ID2. Ve ng knh CE cua (O) .Chng minh :

a. AB2 + CD2 = 4R2 b. AB2 + BC2 + CD2 + AD2 = 8R23. T A va B ha ng vuong goc vi CD lan lt cat BD tai F , cat

AC tai K. Chng minh ABKF la hnh thoi4. Goi M la trung iem CD. Chng minh AB = 2MO5. Goi P la trung iem OI, IH la ng cao cua ICD. Chng minh6. MO2 + MI2 2MP2= 2OI

2


19/38


37 LVC

F

DH

N M

A I C

KP

O

E

B

-------------------------------------------------------------------------------------------------Hng dan giai

1.Chng minh IA.IC = IB.IDChng minh IAB ~ IDC

2a.Chng minh AB2 + CD2 = 4R2

AE // DB AD EB ADB EBD

ADBE la hnh thang can AB = DE AB

2+ CD

2= DE

2+ CD

2= CE

2 = 4R2

2b.C/m AB2 + BC2 + CD2 + AD2 = 8R2BC2 + AD2 = BC2 + EB2 (AD = BE)

= EC2 = 4R2 AB2 + CD2 + BC2 + AD2 = 8R2

3. Chng minh ABKF la hnh thoi DCA DBK (goc tng ng )

ACD ABD ( cung chan AD )

DBK ABD ABK can tai B AI = IK (1)

Ta co : FBK BFA (so le trong )

Ma DBK ABD (cmt) AFB ABF FAB can tai A IF = IB (2)T (1) va (2) ABKF la hnh bnh hanhMa AK BF ABKF la hnh thoi

4.Chng minh AB = 2MOO la ng trung bnh cua DCE DE = 2OM ,

ma DE = AB AB = 2OM5.Chng minh : MO2 + MI2 2MP2=

2OI

2

IMO co : IO2 = IM2 + OM2 2IM.OM.cos IMO (ch/ minh cong thc)

= IM2 + OM2 2IM.OM.cos HIM

= IM2 + OM2 2IM.OM.HI

IM(cos HIM=

HI

IM)

= IM2 + OM2 2OM.HI= IM2 + OM2 [(OM + HI)2 OM2 HI2 ]= IM2 + OM2 ( 4PN2 OM2 HI2 )


38 LVC

A

E

D M

I

B C

F

= IM2 + OM2 + OM2 + HI2 4(PM2 + NM2)= IM2 + OM2 + OM2 + HI2 4PM2 4NM2

= IM2 + 2OM2 4PM2 + 42

2HM HI4

= IM2 + 2OM2 4PM2 + IM2= 2IM2 + 2OM2 4PM2

Vay :2OI

2= MO2 + MI2 2MP2 (pcm)

--------------------------------------------------------------------------------------------------- Bai 50Cho ABC co 3 goc nhon noi tiep ng tron (O). B va C co nh , D va

E la iem chnh gia AB, AC . DE cat AB , AC tai H va K.1. Chng minh AH = AK2. Ve tia Ax HK tai M cat BE tai I. Chng minh C , I , D thang hang

va tia AI luon i qua iem co nh

3. Chng minh t soHK

AH khong phu thuoc vao v tr iem A.-------------------------------------------------------------------------------------------------

Hng dan giai1. Chng minh AH = AKChng minh AHK AKH (goc co nh trong ng tron ) AH = AK2. Chng minh C, D , I thang hangCD la phan giac C , AF la phan giac A , BE la phan giac BSuy ra C , I , D thang hang

AI i qua iem co nh F (iem chnh giua BC )

3. Chng minh HKAH khong phu thuoc vao v tr iem A

HK HM 1 1 A

sinHAM sinAH 2AH 2 2 2

khong oi ( do B , C co nh)

-------------------------------------------------------------------------------------------------Bai 51


20/38


39 LVC

M

F A I O K B

C H

N L D

E

Cho ng tron (O) ng knh AB .Tren AB lay hai iem I va K oixng nhau qua O. M la iem (O) ( M A va B). MI , MO , MK cat(O) tai C , E , D. Qua D ke ng thang song song vi AB cat ME tai L,cat MC tai N.1. Chng minh LN = LD2. Ve OH CD tai H . Chng minh LHDE noi tiep3. CD cat AB tai F. Chng minh FE la tiep tuyen cua (O).]-------------------------------------------------------------------------------------------------

Hng dan giai

1. Chng minh LN = LDIO OM

LN ML

( do IO // NL )

IO OK

NL LD LN = LD ( do OI = OK )

OK OM

LD ML ( do OK // DL )

2. Chng minh LHDE noi tiepH la trung iem CD , L la trung iem DN HL // MN

CHL MCD (slt) ma MCD MED (chan MD ) CHL LED HLED noi tiep ( goc ngoai bang goc trong oi dien )

3. Chng minh FE la tiep tuyen cua (O)

HDL HEL

(t giac HLED noi tiep )Ma HDL BFD (slt) OFH OEH OFEH noi tiep

0FEO FHO 90 EF OE FE la tiep tuyen cua (O)-------------------------------------------------------------------------------------------------Bai 52Cho ABC eu .Tren cac canh AB , BC , AC lay cac iem M , N , R saocho AM = CN = BR .Goi O la tam ng tron ngoai tiep ABC. E , F , Kl;an lt la hnh chieu cua O len AB , AC , MN.1. Chng minh K , F , N , O cung thuoc mot ng tron


40 LVC

A

M

E K F

O N

BR C

2. Chng minh K la trung iem MN . Suy ra K thuoc ng thangco nh khi M di ong tren AB

3. Chng minh E , K , F thang hang4. Tm v tr M , N , R e chu vi MNR nho nhat-------------------------------------------------------------------------------------------------

Hng dan giai1. Chng minh K , F , N , O cung thuoc mot ng tron

0OKN OFN 90 2. Chng minh K la trung iem MN

Chng minh MNR eu , chng minh AMO = CNO (c-g-c) OM = ON K la trung iem MN

3. Chng minh K , E , F thang hang OEM = OFN ( OM = ON , OE = OF ) EOM FON

Ma MKE MOE va FKN FON ( t giac noi tiep )

MKE FKN

Do M , K , N thang hang E , K , F thang hang4 Tm v tr M , N , R de chu vi MNR nho nhat

Chu vi MNR min KN min

Ma tg OK

KNCKN

KN =0

OK

tg30.

Do o KN min OK minMa K EF co nh , nen OK min OK EF MN = EF .

Vay chu vi MNR nho nhat khi M , N , Rla cac trung iem cua AB , BC , AC

-------------------------------------------------------------------------------------------------Bai 53

Cho ABC can tai B , noi tiep ng tron (O). Tren cung AC khong cha B lay 2 iem K va M ( K , M A , C ). BK cat AM tai E ,

KC cat BM tai D. Chng minh ED // AC-------------------------------------------------------------------------------------------------

Hng dan giaiT giac KEDM noi tiep ( BKC BMA )


21/38


41 LVC

B

A CE D

KM

A

B

H M

B A M C

EDK AMK

Ma AMK ACK EDK ACK ED // AC

-------------------------------------------------------------------------------------------------Bai 54Cho ABC co 3 goc nhon . M la iem trong ABC. H la trc tam .Chng minh :

MA.BC + MB.AC + MC.AB HA.BC + HB.AC + HC.AB-------------------------------------------------------------------------------------------------

Hng dan giaiTa co AA AM AM + MM AH + AA AM + MM

BC( AH + AA) BC( AM + MM)

AH.BC + S ABC BC.AM + S BMC

Tng t : BH.AC + S ABC AC.BM + S AMC CH.AB + S ABC AB.CM + S AMB

Cong theo tng ve 3 bat ang thc tren ta cieu phai chng minh

Chu y: Ta co the thay oi cau hoi bai toan nay theo hai cach sau:1. Tm v tr cua M e MA.BC + MB.AC + MC.AB nho nhat2. Chng minh MA.BC + MB.AC + MC.AB 4S ABC

1. Dau = xay ra khi M H MA.BC + MB.AC + MC.AB nho nhat khi M la trc tamcua ABC

2. Ta co AA AM BC.AA BC.AM 1/2BC(AM + MM) S ABC 1/2BC.AM + BC.MM S ABC BC .AM + S BMC

BC.AM S ABC S BMC (1)Tng t : AC.BM S ABC S AMC (2)

AB.CM S ABC S AMB (3)Cong (1) , (2) , (3) theo tng ve ta c : MA.BC + MB.AC + MC.AB 4S ABC

-------------------------------------------------------------------------------------------------


42 LVC

A

DE

H

B K C

A

1 2 3 4

E

F I

J D K

1 1

2 2

B N M C

Bai 55Cho ABC co 3 goc nhon . Hai ng cao BD va CE cat nhau tai H.

Ve ng cao AK. Chng minh : KH.KA 2BC

4

------------------------------------------------------------------------------------------------- Hng dan giai

Ta chng minh AKB ~ CKH AK.HK = KB.KC

Ma theo BT Cauchy :KB KC

KB.KC2

KB.KC 2(KB KC)

4

=

2BC

4

Vay AK.KH 2BC

4

-------------------------------------------------------------------------------------------------Bai 56Cho ABC vuong tai A . ng cao AH. Goi I , J , K lan lt la tamng tron noi tiep ABC , ABH , AHC .1. Chng minh AI JK.2. Chng minh BJKC noi tiep .-------------------------------------------------------------------------------------------------

Hng dan giai1.Chng minh AI JKTa co

00

1 190

A B 452

Ma 02 3A A 45 1A + 3A = 450

1B = 3A 2B = 3A Ma 3A + M = 900 2B + M = 900 BE AM JE AK ABM can tai BTng t : KF AJ I la trc tam AJK AI JK2.Chng minh BJKC noi tiep

EJK DAK ( goc co canh tng ng ) 0JAK IAJ 45 IAJ

Ma 2C = 2A = 1A ( goc co canh tng ng )

= 0IAB IAJ 45 IAJ

Suy ra 2EJK C BJKC noi tiep


22/38


43 LVC

A

I

O K

B H C

-------------------------------------------------------------------------------------------------Bai 57Cho ABC can tai A. H la trung iem BC. Ve HI AC. Goi O la trungiem HI.

1. Chng minh AO BI2. Chng minh OA.B = 1

2AH.BI

------------------------------------------------------------------------------------------------- Hng dan giai1.Chng minh AO BI

Goi K la trung iem HK // BI va OK // BCMa BC AH OK AH O la trc tam cua AHK AO HK AO BI

2.Chng minh OA.B = 12

AH.BI

AOH va BIC co : AHI ACH ( cung phu vi HAC) OAH IBH ( goc t/ng ) AOH ~ BIC

AO AH

I BC

AO.BC = AH.BI

AO.BH =1

2AH.BI

-------------------------------------------------------------------------------------------------Bai 58Cho ABC noi tiep ng tron (O). Mot ng tron (I) tuy y i qua Bva C cat AB , AC tai M , N. ng tron tam K ngoai tiep AMN cat (O)tai iem th hai la D.

1. Chng minh AKIO la hnh bnh hanh2. Chng minh 0ADI = 90

------------------------------------------------------------------------------------------------- Hng dan giai1. Chng minh AKIO la hnh bnh hanh


44 LVC

A xy

DK N

M O

B CI

Ve tiep tuyen Ax cua (O) tai ATa cm c Ax // MN OA MNMa IK MN ( ng noi tam ) IK // OA (1)Ve tiep tuyen Ay cua (K) tai A

Ta co AN = AMN ( chan AN)

Ma AMN ACB (BMNC noi tiep )

AN ACB Ay // BC

Ma BC OI ( ng noi tam ) Ay OIMa Ay AK (tiep tuyen ) OI // AK (2)T (1) va (2) AKIO la hnh bnh hanh

2. Chng minh 0ADI = 90

Hai ng tron (O) va (K) cat nhau tai A va D OK la trung trc ADGoi E la giao iem cua KO va AI ta co : EA = EI ( hnh bnh hanh )Ma E OK EA = ED ( t/c trung trc )

Suy ra : ED = AI DAI vuong tai D 0ADI = 90

-------------------------------------------------------------------------------------------------Bai 59

Cho ng tron (O;R) co day AB co nh . iem M di chuyen tren AB ln . Cac ng cao AE , BF cua AMB cat nhau tai H.

1. Chng minh OM EF2. ng tron (H;HM) cat MA , MB tai C , D. Ch . minh ng

thang ke t M vuong goc vi CD i qua mot iem co nh

3. Chng minh ng thang ke t H vuong goc vi CD cung i quamot iem co nh------------------------------------------------------------------------------------------------- Hng dan giai

1. Chng minh OM EF

Ve tiep tuyen xMy cua (O). Ta co : xMA MBA ( chan cung AM )

Ma MFE MBA (AFEB noi tiep)

xMA MEF xy // EF OM EF2 . Ta co HE MD ( k-dc )


23/38


45 LVC

yM

x EF

H D

C O

A B

O

A

BC

MK

BH C

A

1

2MHE MHD

Ma 1

2MCD MHD (goc nt- tam)

MHE MCD

Ma MFE MHE (t giac noi tiep)

MFE MCD EF // CDDo OM EF OM CD

Vay ng thang ke t M vuong goc vi CD i qua iem co nh O3. Goi O oi xng vi O qua AB . Ta chng minh c : OO = MH

Ma OO // MH MHOO la hnh bnh hanh OH // MO

OH CD. Vay ng thang ke t H vuong goc vi CD i quaiem O co nh-------------------------------------------------------------------------------------------------Bai 60Cho ABC noi tiep ng tron (O). Goi A , B , C la iem chnh gia

cua cac cung nho BC , AC , AB .

1. Chng minh cac ng tron (A;AC) ; (B;BA) ; (C;CB) cung iqua mot iem

2. Chng minh AA + BB + CC > AB + AC + BC-------------------------------------------------------------------------------------------------

Hng dan giai1.Ta co AA , BB , CC la ba ung phan giac cua ABC nen chung

ong qui tai M.

MCA co : 1

' ' ( ' ')2

C CA C B BA

1' ( ' ')2

A MC C A CA

Ma ' ' ; ' ' AC BC CA BA

' ' A MC MCA MAC can tai


46 LVC

A

NM

O

B Q CA

ND

Q

P N

B C

AM = AC M (A;AC)Tng t ta chng minh c: M (B;BA) ; M (C; CB)

Vay ba ng tron tren ong quy tai iem M2. Chng minh AA + BB + CC > AB + AC + BCKe AH AB ; AK AC AH = AK BHA = CKA BH = KC AB + AC = AH BH + AK + KC = AH + AK = 2AK

Ma AK < AA 2AK < 2AA AB + AC < 2AA

AA >2

AB AC

Chng minh tng t ta c : BB >2

BC ABva CC >

2

AC CB

Vay : AA + BB + CC > AB + AC + BC-------------------------------------------------------------------------------------------------Bai 611. Cho ABC ngoai tiep ng tron (O). Cac tiep iem M , N , Q lan

lt cua AB , AC , BC vi (O).Ch. minh 2AM = AB + AC + BC

2. Cho t giac ABCD. Hai ng tron noi tiep hai ABC va ADCtiep xuc vi AC tai M , N. Hai ng tron noi tiep BAD va BCDtiep xuc vi BD tai P va Q. Chng minh MN = PQ

3. Cho ABC vuong tai C, ng tron noi tiep ABC tiep xuc vi ABtai I. Chng minh S ABC = AI.BI

------------------------------------------------------------------------------------------------- Hng dan giai

1. AB + AC BC= AM + MB + AN + NC BQ QCMa AM =AN ; BN = BQ ; CN = CQ

AB + AC BC = AM + AN = 2AM2. Theo ket qua cau 1. ta co ;

2AM = AB +AC BC

2

AB AC BC AM

Tng t :2

CA CD ADCN

Ta co : MN = AC AM NC

= AC 2

AB AC BC

2

CA CD AD


24/38


47 LVC

C

B AP

D

C NH

M

EA B

O

K

MN =( ) ( )

2

AD BC AB CD

Tng t ta chng minh : PQ =( ) ( )

2

AD BC AB CD

Suy ra : MN = PQ3. Ap dung cau 1. ta co : 2AI = AC +AB BC

2BI = BC + BA AC 4AI.BI = (AC +AB BC )(BC + BA AC )

= 2AC.BC = 4S ABC S ABC = AI.BI

-------------------------------------------------------------------------------------------------Bai 62Cho ng tron (O;R) ng knh AB . C la iem (O). M , N la iem

chnh gia , AC BC.

1. Ke ND AC tai D. Chng minh ND la tiep tuyen cua (O)2. Goi E la trung iem BC .ng thang OE cat (O) tai K

(K N).Chng minh ADEK la hnh bnh hanh3. Chng minh khi C di chuyen tren (O) th MN luon tiep xuc vi mot

ng tron co nh-------------------------------------------------------------------------------------------------- Hng dan giai

1. ND la tiep tuyenON BC , AC BC ON //AC Ma DN AC DN ON tai N

Do N (O) DN la tiep tuyen

2. ADEK la hnh bnh hanh DEN CNE ( hnh ch nhat ) CNE AKN ( CK AN )

DEN AKN DE //AKMa AD // EK

Suy ra ADEK la hnh bnh hanh3. Chng minh MN tiep xuc vi ng tron co nh


48 LVC

A

O

B

HQ D

N

I

M J C

K

MON vuong can OH =2

2

R.

Vay MN luon tiep xuc vi ng tron (O; OH) co nh-------------------------------------------------------------------------------------------------Bai 63Cho ABC co nh can tai A. Goi D la iem di ong tren BC. Qua D vehai ng tron (I) va (J) tiep xuc vi AB tai B , tiep xuc vi AC tai C.Hai ng tron (I) va (J) cat nhau tai K ( K D)1. Chng minh ABKC noi tiep ng tron2. Chng minh DK luon i qua iem co nh va AD.AK khong oi khi D

di ong tren BC3. Chng minh tong hai ban knh cua (I) va (J) khong oi4. Tm tap hp cac trung iem M cua IJ------------------------------------------------------------------------------------------------- Hng dan giai

1. Trong (I) :

ABD BKD

( chan

D )Trong (J) : DKC ACD ( chan DC)

Ma 0180 ABO ACD A 0180 AKC BKD A

0180 BKC A ABKC noi tiep

2. Theo chng minh tren ta co : DKB DKC KD la phan giac BKC

Ma AKC AKB ( AB AC )

Suy ra KA la phan giac cua

KC K , D , A thang hang DK i qua A co nh

Ta co ADC ~ ACK AD.AK = AC2 ( khong oi )

3. Goi R la ban knh cua (I) va r la ban knh cua (J)

Ve IH , JN BC 2 2

D DC BC HN

(khong oi )

IH BKD DJN DKC


25/38


49 LVC

AD

K

IO

E F

B C

2

SdAB BIH DJN ( khong oi )

BKD DKC

Ta co : R = sin sinsin

BH BH HD

BIH ; r =

sinsin

DN DN

DJN

Suy ra : R + r =sin 2sin

DH DN BC

( khong oi )

4. Ke MQ BC ta co MQ la ng trung bnh cua hnh thang HIJN

MQ =2

IH JN

Ta co : IH = BH. cotg ; JN = DN.cotg

MQ = (BH + DN).cotg =2

C. cotg ( khong oi )

Vay tap hp M la ng thang // cach BC mot oan d =2

C.cotg .

-------------------------------------------------------------------------------------------------Bai 64Cho ng tron (O;R) ngoai tiep ABC eu. K la iem thuoc cung nhoAB. Ve ng tron (I; r) tiep xuc vi (O;R) tai K (R > r). ng tron (I)cat KA , KB , KC tai D , E , F1. Chng minh DE // AB va DF // AC2. Chng minh FC = AD + BE3. T A , B , C ve cat tuyen AM , BN ,CP vi (I).

Chng minh CP = AM + BN-------------------------------------------------------------------------------------------------

Hng dan giai1. Chng minh FC = AD + BE Ta co : 060 DEF AKF ( chan AC)

060 FDE BKF (chan C )Suy ra DEF eu.Theo bai 6 ta chng minh c :KC = KA + KBKF = KD + KE

KC KF = (KA KD ) + (KB KE) FC = AD + BE


50 LVC

C

EA

F

D

B

2. Chng minh CP = AM + BNAM , BN , CP la cac tiep tuyen cua (I) AM2 = AD.AK , BN2 = BE.BK AM2.BN2 = (AD.BK)(AK.BE)Ma CP2 = CF.CK = (AD + BE)(KA + KB)

= AD.AK + AD.KB + BE.KA + BE.KB

= AM2 + AD.KB + BE.AK + AN2

Mat khac : ED // AB AD BE

KA KB AD.KB = AK.BE

T AM2.BN2 = (AD.BK)(AK.BE) ( chng minh tren) (AM.BN)2 = (AD.BK)2 = (AK.BE)2 AM.BN = AD.BK = AK.BESuy ra : CP2 = AM2 + 2AM.BN + BN2 = (AM + BN)2Do o : CP = AM + BN

-------------------------------------------------------------------------------------------------Bai 65

Cho ABC co phan giac BE tao vi canh AC mot goc bang 450 0BEA= 45 Ve ng cao AD cua ABC. Tnh so o EDC ?-------------------------------------------------------------------------------------------------

Hng dan giaiT E ve Ex AC cat BC tai F.Suy ra FEAD noi tiep

Ta co : 045EAB ; 090AEF

Suy ra : 045BEF EFB = EAB ( c-g-c)

EF = EA

FEA vuong can 045EAF 045EDF ( cung chan EF )

Vay 045EDC ---------------------------------------------------------------------------------------------Bai 66Cho ABC vuong tai A , A va B co nh , C di ong tren tia At ABtai A. Goi I la tam ng tron noi tiep ABC vi P , Q , R la cac tiepiem tren aC , BC , AB. ng thang PQ cat tia AI tai D1. Chng minh B , D , Q , R cung thuoc mot ng tron


26/38


51 LVC

C D

Q

P I

A BR

C

H

A O I B

K E

D

2. Chng minh khi C chay tren tia At th ng thang PQ luon i quaiem co nh

---------------------------------------------------------------------------------------------Hng dan giai

1. T giac APIQ la hnh vuong

045PIA 045QDI DPI ( goc ngoai tam giac ) (1)

Ta co : IQR IRQ ( can )

ma 045 IQR IQP PQR ( = 1

2sdPR )

ma PQI QPI ( can ) 045 IRQ QPI (2)

T (1) va (2) IKQ QDI IQDR noi tiep

Ta chng minh RIQB noi tiep R , Q , D , B ( va I ) cung thuoc motng tron ng knh BI

2. Ta co D ng tron ng knh IB

0

90IDB hay 0

90ADB D ng tron ng knh AB (co nh)

Ma 045DAB D la iem chnh gia AB Vay PQ luon i qua D co nh

-------------------------------------------------------------------------------------------------Bai 67Cho AB va CD la hai ng knh vuong goc cua ng tron (O;R). DayCE i qua trung iem I cua OB. ng cao AH cua ACE

1. Tnh CE , AH , S ACE theo R2. Chng minh AC tiep xuc vi ng tron ngoai tiep AEI3. AE cat BD tai K . Chng minh : AK.AE + BK.BD = 4R 2-------------------------------------------------------------------------------------------------

Hng dan giai1. Xem bai 142. Xem bai 143. Ta co 045IBD va 045IEA

IBEK noi tiep

045 IKB IEB


52 LVC

A

K G O

NB I C

M

KIB vuong can KI AB AIK ~ AEB AK.AE =AI.AB BIK ~ BDA BK.BD = BI.AB AK.AE + BK.BD = (AI + BI ).AB

= AB.AB = 4R2

-------------------------------------------------------------------------------------------------Bai 68Cho ng tron (O ; R) va day BC co nh. iem A cung ln BC. Ve

AH BC . Phan giac BAC cat (O) tai M.1. Chng minh tam ng tron noi tiep ABC di ong tren ng

tron co nh2. Tm quy tch trong tam G cua ABC

3. Cho BC = R 3 .Tnh OI theo R. nh v tr iem A e dien tch

gii han bi BC , day AB va day AC ln nhat

-------------------------------------------------------------------------------------------------Hng dan giai 1. Goi K la tam tron noi tiep ABC

0

0 0 180180 1802 2

B C ABKC

0902

ABKC (khong oi )

Vay K di ong tren cung cha goc

( =

0902

A ) dng tren oan BC co nh .

2. Ta co : OM BC tai I . G la trong tam ABC G AI la trung

tuyen ABC va1

3

GI

GA

Ve GN // AO (N OI ) IN =1

3OI

Do O va I co nh } N co nh Va1 1

3 3

GNNG

AO R


27/38


53 LVC

A P Q B

M N

O

D C

Vay quy tch G la ng tron ( N ;3

R)

3. BC = R 3 OI =2

R. Khi A la iem chnh gia C ln th S ln

nhat . S =ABCS +

vp

S BC nho.

------------------------------------------------------------------------------------------------Bai 69Cho hnh vuong ABCD . M , N AC Sao cho AC = 3AN = 4 AMDM , DN cat AB tai P , Q .

1. Chng minh AMP , Q.2. Chnh minh MNPQ noi tiep .3. Chng minh BC tiep xuc vi ng tron ngoai tiep MNB.

------------------------------------------------------------------------------------------------

Hng dan giai 1. Ta co AC = 3AN AN =1

3AC

AC = 4AM AM = 14

AC = 12

AO

N la trong tam ABD Q la trung iem ABGoi a la o dai canh hnh vuong.

AC = a 2 . Ta co :1

224

1 2

2

aAM

AQa

(1)

31 223

AP AP AP

AN DCa

ma AP AM DC MC

32

AP AM

AN MC

13. 2

1 243 222. 24

aAP

ANa

(2)

Do BAC chung nen t (1) va (2) ta co : AMP ~ ANQ ( c- g - c )

AQN AMP


54 LVC

A P B

KN

O

D CM

B

NM

A CD O

2. AQN AMP MNQP noi tiep ( goc ngoai bang goc trong oi dien )

3. Ta co CN.CM = 2 22 3

2. 23 4

a a a BC BC tiep xuc vi (MNB)

------------------------------------------------------------------------------------------------- Bai 70

Cho hnh vuong ABCD . M la iem tuy y thuoc canh CD. Hai ng tronng knh CD va AM cat nhau tai N ( N D ). K la giao iem DN vaBC. Chng minh AC KM-------------------------------------------------------------------------------------------

Hng dan giai Goi I la giao iem cua (O) va AB APMD la hnh ch nhat AP = DM PB = MCTa co : DN PN va DN NC P , N , C thang hang

PCB KDC (goc tng ng )

PBC = KCD KC = PB KC = MC AC AM

-------------------------------------------------------------------------------------------------Bai 71Cho ABC co ng cao BD. Ve ng tron co tam O canh AC tiep

xuc vi AB va BC tai M va N . Chng minh ADM = CDN -------------------------------------------------------------------------------------------------

Hng dan giaiChng minh B , M , D , N , O cung thuoc

mot ng tron BM BN

MDB NDB ADM CDN


28/38


55 LVC

CK

EI M

NA

O B

D

-------------------------------------------------------------------------------------------------Bai 72Cho ng tron (O) va hai ng knh AB va CD vuong goc . Ve day AMcat CD tai N. Goi I la tam ng tron ngoai tiep CMN1. Chng minh C , I , B thang hang2. K la giao iem cua MI va (O).Tm v tr cua M e IM.IK ln nhat------------------------------------------------------------------------------------------------ Hng dan giai

1. Chng minh C , I , B thang hangXet ng tron (I) ta co OI CM

1

2CIE EIM CIM

Ma 1

2CNM CIM (goc nt- goc tam)

CIE CNM

ma

ANO CNM ( ) CIE ANO

Ta co : 090 ECI CIE 090 NAO ANO ECI NAO ( MAB )

Ma CIE ANO ( cmt)

Mat khac trong (O) ta co : MCB MAB ECI MCB Suy ra : C , I , B thang hang2. Ta co IM.IK = IB.IC

IB.IC 2 2( )

4 4

IB IC BC . Vay IM.IK at gia tr ln nhat la

2

4

C.

Khi o IB = IC I la trung iem BCMa I thuoc trung trc cua CN N O M B ( K C )

-------------------------------------------------------------------------------------------------Bai 73Cho hai ng tron (O1) va (O2) tiep xuc ngoai tai D . T iem A tuy ynam tren (O1) ve tiep tuyen vi (O1) cat (O2) tai B va C. Chng minh Acach eu hai ng thang BD va CD-------------------------------------------------------------------------------------------------


56 LVC

M CB

A

O1 O2D

C

Hng dan giaiTa can chng minh A thuoc ngphan giac goc tao bi 2 ng thangBD va DC.Ta can chng minh DA la phan giac

cua 'BDC . Ve tia Dx AD cat (O2)

tai M . Ta co : 0 0

1 2 90 ( 90 ) ADO MDO MDA

01 2 180 AO D DO M

O2M // AO1 ma O2M BC MB MC

DM la phan giac cua BDC

DA la phan giac cua 'BDC A cach eu BD va CD---------------------------------------------------------------------------------------------Bai 74

Cho goc nhonxOy .Tren tia phan giac cua goc

xOy lay iem I. ng

tron (I) ba n knh thay oi cat hai tia OK Oy tai M, N. (M , N khong oixng qua OI.)1. Chng minh OMIN noi tiep .Suy ra quy tch tam ng tron ngoai

tiep OMN2. Hai ng thang vuong goc v i ox tai M va oy tai N cat nhau tai P.

Chng to P thuoc mot ng thang co nh .---------------------------------------------------------------------------------------------1. Chng minh OMIN noi tiepVe OE , OF ox . Ta co IEM = IFN ( ch cgv )

EMI INF OMIN noi tiep ( goc ngoai bang goc trong oi dien)Suy ra tam ng tron (OMN) thuoc ng d la trung trc OI .Do ban knh (I) OI . Do o Ve (I , IO ) cat d tai A va Boan AB la quy tch cua tam ng tron (ONM)2. Chng minh P thuoc ng thang co nhT giac OMNP noi tiep . P (OMN) co ng knh OP

090OIP IP OI P ng thang vuong goc OI tai I . Gii han: oan AB


29/38


57 LVC

A

A

d xE

MI

O

FN

B

PB

A

K

E

B M C

DI

Bai 75Cho ABC can tai A. M la trung iemBC . D la iem ng thang AM saocho DB AB . E la iem bat ky BCQua E ve ng thang cat AB , AC taiI , K . Chng minh :

ED IK DI = DK

Hng dan giai1. DE IK ID = KD

DE IK BEDI va KEDC la cac t giac noi tiep

EID EBD va EKD ECD

ma EBD ECD EID EKD IDK can tai D ID = KD

2. ID = KD DE IKID = KD IBD = KCDHai ng tron ng knh KD va ID cat nhau tai F. Ta co :

KFC KDC va BFI BDI

Ma KDC BDI ( 2 bang nhau ) FI KFC Suy ra : B , F , C thang hang E F

Ma DF IK DE IK

--------------------------------------------------------------------------------------------Bai 76Cho ABC .Phan giac ngoai goc Acat ng thang BC tai D. Trungtuyen AM cua ABC . ng tron ( ADM ) cat AB , AC tai E , F.Chng minh BE = CF.


58 LVC

x

A

F

D B M C

E

A

M

K

B

O C D

Hng dan giai BE.BA = BD, BM .BD BMBEBA

CF.CA = CM.CD .CDCM

CFCA

Ma DB CD

AB CA ( tnh chat phan giac )

Va MB = MC ( AM la trung tuyen )Suy ra : BE = CF

-----------------------------------------------------------------------------------------------Bai 77

Cho ABC noi tiep ng tron (O ; R) co AB = AC = R 2 .1. Tnh BC theo R2. M di ong tren AC. ng thang AM cat BC tai D. Chng minh

AM. AD khong oi3. Chng minh tam ng tron ngoai tiep MCD di ong tren mot

ng co nh khi M di ong tren AC-----------------------------------------------------------------------------------------------Hng dan giai1. BC = 2R2. AM.AN khong oi

DM.AD = DC.DB AD(AD AM) = (DO OC)(DO + OB) AD2 AD.AM = DO2 R2 AD.AM = AD2 DO2 + R2 AD.AM = DO2 + OA2 OD2 + R2 = OA2 + R2 = 2R2 khong oi

3. Tam ng tron (MCD) thuoc ng co nh

Ta co : AC2 = (R 2 )2 = 2R2AM.AD = 2R2 AC2 = AM.AD AC la tiep tuyen cua (CDM) Tam ng tron (CDM) thuocng thang d vuong goc vi ACtai C co nh

-----------------------------------------------------------------------------------------------Bai 78


30/38


59 LVC

AR

M

IK

N O

B C

Cho ABC can tai A ( A < 900 ). Mot ng tron (A;R) (R


31/38


61 LVC

B

O N

M

A

K

CM

BG

O OA E D

Bai 81Cho ng tron (O;R). iem A co nh nam trong ng tron . iem Bdi ong tren ng tron . Ve tiep tuyen Bx , ng thang qua O vuonggoc vi AB tai N . Ve MK OA.1. Chng minh OA.OK = R22. Tm quy tch iem M khi B di ong tren ng tron (O;R)--------------------------------------------------------------------------------------------Hng dan giai1. T giac ANMK noi tiep OA.OK = ON.OM

Ma ON.OM = OB2 = R2 OA.OK = R2

2. OA.OK = R2 OK =2

R

OA( khong oi ) K co nh

M ng thang vuong goc vi OA tai K

---------------------------------------------------------------------------------------------Bai 82Cho hai ng tron (O) va (O) tiep xuc ngoai tai A. B va C di ong tren

(O) va (O) sao cho 0BAC= 90 . M la trung iem BC

1. Chng minh OB // OC2. Tm quy tch iem M3. Tm quy tch trong tam G cua ABC-----------------------------------------Hng dan giai

1. 0180 2 AOB BAO 0' 180 2 ' AO C CAO

0 0' 360 2( ') 180 BOA AO C BAO CAO OB // OC


62 LVC

N

A I

MO O

D EB K

2. MA = MB , OA = OB OM ABTng t : OM AC OM OM

Ma AB AC

0' 90O MO

M ng tron ng knh OO , tam la D, ban knh R =1

'2

OO

3. Goi G la trong tam cua ABC .

Ve GE // MD ED =1

AD3

E co nh

Ta lai co :2 2

EG = MD = R3 3

. Vay G ng tron (E ;2

3

R)

-------------------------------------------------------------------------------------------- Bai 83Cho hai ng tron (O) va (O) cat nhau tai A va B ( O va O hai benAB ). Mot ng thang qua A cat (O) va (O) lan lt tai M , N

1.

Chng minh ng trung trc cua MN luon i qua iem co nh2. Tm quy tch trung iem I cua oan MN--------------------------------------------------------------------------------------------Hng dan giai

1. Ve ng knh AD , AE cua (O) va (O)* Chng minh : D, B , E thang hang* Chng minh MNED la hnh thang , ng trung trc cua MN cat DE tai

trung iem K K co nh

2. Ta co 090AIK I ng tron ng knh AK ( co nh )

--------------------------------------------------------------------------------------------Bai 84


32/38


63 LVC

A

EI

KD

B M H C

A

E

IF

C

B D M

K

Cho ABC can tai A . AH la ng cao . M la iem di ong tren canhBC . Ve MD // AC va ME // AB ( D AB , E AC ).1. Tm tap hp trung iem I cua DE2. ng tron (ADE) cat AH tai K. Chng minh K la tam ng tron

(ABC). Suy ra quy tch tam O cua ng tron (ADE)-----------------------------------------------------------------------------------------------

Hng dan giai1. Quy tch cua I la ng trung bnh cua ABC2. Ta co DK KE DK KE

ma EC = CM = AD va ADK KEC ADK = CEK ( c g c ) KC = KA = KB K la tam cua ng tron (ABC) K co nh O trung trc cua AK

------------------------------------------------------------------------------------------------Bai 85

Cho ABC , AD la phan giac va AM la trung tuyen .ng tron ngoaitiep AMD cat AB tai E va cat AC tai F. Goi I la trung iem EFChng minh IM // AD.

-----------------------------------------------------------------------------------------------Hng dan giai

Goi K la iem oi xng vi E qua MTa co : BECK la hnh bnh hanh

BE // CK BEM CKM

Ma BEM AFM ( AEMF noi tiep )

AFM CKM

MFCK noi tiep CMK CFK

Ma EMB CMK ( )

va EMB EAD ( chan ED)

CFK EAD

Ma EAD DAF ( AD la phan giac )

KFC DAF AD // FKMa IM // FK ( ng trung bnh EFK ) IM // AD

-------------------------------------------------------------------------------------------------


64 LVC

AI

E

K H

B C

N

M

A

H

B I C

K

Bai 86Cho ABC co ba goc nhon noi tiep ng tron (O). Trc tam la H. Lay

iem M cung nho BC .1. Xac nh v tr M e BHCM la hnh bnh hanh2. Lay M tuy y cung nho BC . Goi N , E la cac iem oi xng vi M

qua AB , AC . Chng minh N , H , E thang hang

3. Xac nh v tr M cung nho BCe NE co o dai ln nhat------------------------------------------------------------------------------------------------

Hng dan giai

M

Hnh 1 Hnh 2

1. BHCM la hnh bnh hanh MC // BHMa BH AC MC AC AM la ng knh cua (O)

Ve ng knh AK cua ng tron (O). Ta chng minh BHCK la hnh bnhhanh K M. Vay M BC sao cho AM la ng knh cua (O)

2. Goi I , K la giao iem cua BH va CH vi (O). Ta chng minh c:K oi xng vi H qua AB ; I oi xng vi H qua AC. Do o : HIM HIE va NHB MKB

Ma MKB HIM ( cung chan BM) NHB IHE H , H , E thang hang

3. Ta co NE = NH + HE = MK + MI . Do o :NE nho nhat MC + MI ln nhat . Ma KI khong oi nen :

MC + MI ln nhat MK = MI M la iem chnh gia IK AM IK AM la ng knh cua (O)

------------------------------------------------------------------------------------------------Bai 87


33/38


65 LVC

A

HM

IO

KB C

L

Cho ABC noi tiep (O;R) va ng tron (I;r) noi tiep ABC. AI cat(O) tai L.1. Chng minh LB = LC = LI va IA.IL = 2Rr2. Chng minh OI2 = R2 2Rr-----------------------------------------------------------------------------------------------Hng dan giai

1. Ta co BAL BCL CBL va AIB IBC

Ma BIL BAL ABI va IBL IBC C BL

Suy ra : BIL IBL LB = LI = LCVe ng knh BM cua (O)Ta co : AIH ~ MBL ( g-g) AI.BL = IH.BM AI.IL = 2Rr2. Ve OK AL AK = KL .

Ta co 2Rr = AI.IL

= (AK + KI)(AKKI)= AK2 KI2Mat khac : OI2 = OK2 + IK2 ( OIK vuong tai K ) 2Rr + IO2 = AK2 + OK2 = OA2 = R2

--------------------------------------------------------------------------------------------Bai 88Cho ng tron (O) ngoai tiep ABC can tai A . ng cao AH . D latrung iem cua AB , G va E la trong tam cua ABC va ACD.1. Chng minh ADO ~ AHB2. Chng minh ADG ~ DOE3.

Chng minh OE CD .--------------------------------------------------------------------------------------------

Hng dan giai

1. Ta co : ADO ~ AHB ( g g ) DA AH

=DO BH

(1)

2. Ta co :2 2

;3 3

DE AG

DM AH

DE AG AG AH

DM AH DE DM (2)

Ma : DM = HB ( ng trung bnh ) (3)


66 LVC

A

N

D E M

OG

B H C

A

IT

O

B C

AI

TA O

C

B

T (1) ; (2) va (3) DA AG

DO DE

Xet ADG va DOE ta co : EDO GAD ( goc tng ng )

DA AG

DO DE ( cmt )

ADG ~ DOE (c-g-c)

3. Ta co : DEO AGD ( ong dang )Ma ED AG EO DC

--------------------------------------------------------------------------------------------Bai 89Cho hai ng tron (O;R) va (I;r) vi I nam tren ng tron (O). Mottiep tuyen cua (I) tai T (I) cat (O) tai A va B. Chng minh tch IA.IBkhong phu thuoc vao v tr cua iem T va tnh tch ay theo cac ban knhR va r. Xet ca hai trng hp T nam trong va ngoai ng tron (O).

--------------------------------------------------------------------------------------------Hng dan giaiKe ng knh IC coa (O).Hai tam giac IBC va ITA la cac tam giac vuong

tai B va T , co IAB ICB ( cung chan IB ) nen ITA ~ IBC IT IA

IB IC IA.IB = IC.IT

Ma IC = 2R ; IT = r . Vay tch IA.IB = 2Rr ( khong oi )

Khi T ngoai ng tron (O)ta van co : IAT ICB ITA ~ IBC pcm

--------------------------------------------------------------------------------------------


34/38


67 LVC

A

K

L B

H CA

G O

B I M C

Bai 90Cho ABC co trong tam G , trc tam H . AI , BK , CL la cac ng cao. Tren cac ng cao ta lan lt lay cac iem A , B , C sao choA'A B'B C'C

= = = 2A'I B'K C'L

.

1. Chng minh cac iem G , H , A , B , C cung thuoc mot ng tron2. Xac nh tam va ban knh ng tron o cho biet khoang cach gia

H va tam O cua ng tron ngoai tiep ABC la 6 cm.--------------------------------------------------------------------------------------------Hng dan giai

1. Ke trung tuyen AM cua ABC .Ta co 2

GA

GM

Ma theo gia thiet :'

2'

A A

A I

'

'

GA A A

GM A I Ta suy ra : GA // IM .Ma IM HA ( v AI la ng cao )

Vay 0' 90GA H A ng tron ng knh GHChng minh tng t , cac iem B va C cung thuoc ng tron nayVay 5 iem G , H , A , B , C cung thuoc ng tron ng knh GH

2. Tam ng tron noi tren la trung iem P cua GH. Ta co the chngminh c G , H , O thang hang va GH = 2GO . Ta suy ra :

2 2.6 4

3 3

HG HO cm

Do o ban knh cua ng tron can tm la 2 cm

-------------------------------------------------------------------------------------------------Bai 91Cho hnh vuong ABCD. Mot ng thang song song vi ng cheo BDcat cac canh AB , AD lan lt tai E va F. Goi H la hnh chieu cua A trenDE, M la trung iem cua EF1. Chng minh AHF ~ DHC2. Chng minh 5 iem C, D , F , M , H cung thuoc mot ng tron------------------------------------------------------------------------------------------------


68 LVC

B C

E H

M

A F D

B

D EA O M

C

Hng dan giai

1. Ta co HAD AEH ( goc tng ng vuong goc ) AEH HDC ( slt) HAD HDC (1)

AHD ~ EHA (g-g)

DH DA

AH AE

Ma DA DC DH DC

DE AF AH AF (2)

T (1) va (2) AHF ~ DHC (c-g-c)

2. Chng minh FDCH va FDHM noi tiep pcm-----------------------------------------------------------------------------------------------Bai 92Ben trong ng tron (O;R) lay hai iem A va M oi xng nhau qua O ,

ke day BC i qua M. Chng minh tong bnh phng cac canh cua ABC khong phu thuoc vao v tr day BC.Tnh tong o theo R va OA = a.-----------------------------------------------------------------------------------------------Hng dan giai ABM co BO la trung tuyen

BA2 + BM2 = 2BM2 +1

2AM2

( Ban oc t chng minh )

= 2R2 +1

2

(2a)2 = 2R2 + 2a2

Tng t : AC2 + CM2 = 2R2 + 2a2Ta lai co : BM.MC = R2 a2 ( t chng minh )Ta co : BC2 = ( BM + MC)2

= BM2 + 2BM.MC + MC2Do o : AB2 + AC2 + BC2

= AB2 + AC2 + BM2 + 2BM.MC + MC2= 4R2 + 4a2 + 2(R2 a2) = 6R2 + 2a2 (khong oi )

-----------------------------------------------------------------------------------------------


35/38


69 LVC

E

A K

BO

D C F

CB

NC I B

QO P

A D

M A

D

Bai 93Cho t giac ABCD noi tiep ng tron (O). ng thang AD cat ngthang BC tai E. ng thang AB cat ng thang CD tai F.Chng minh : AE.DE + FA.FB = EF2------------------------------------------------------------------------------------------------Hng dan giai

Ve ng tron ngoai tiep ABE cat EFtai K. Ta chng minh c t giac CBKFnoi tiep . Suy ra :

EA.ED = EB.EC = EK.EFFB.FA = FK.FE EA.ED + FA.FB = (EK + KF)EF = EF2

-------------------------------------------------------------------------------------------------

Bai 94Cho t giac ABCD noi tiep ng tron (O) ( 0 090 , 90B C ).

ng thang vuong goc vi AB ke t A va ng thang vuong goc viCD ke t D cat nhau tai M. ng thang vuong goc vi AB ke t B vang thang cug vi CD ke t C cat nhau tai N.1. Chng minh MAD ~ NCB2. Goi P va Q la hnh chieu cua M lan lt xuong BD va AC. Chng

minh MP.MA = MQ.MD3. Chng minh ba ng thang AC , BD va NM ong quy

-------------------------------------------------------------------------------------------------

Hng dan giai1. Chng minh MAD ~ NCB

090MAD BAD 0 0180 90CN NCD BAD BAD

CN MAD

Ma NC DMA ( goc co canh tg-ng //) MAD ~ NCB ( g-g)

2. Chng minh MP.MA = MQ.MD AC AMQ ( cung phu CAM )


70 LVC

CDB DMP ( cung phu DM)

Ma AC BDC PMD AMQ AMQ ~ PMD

AM.MP = MD.MQ3. Chng minh ba ng thang AC , BD va NM ong quy

Goi I la giao iem MN va BD ta co : 1 21 2

( )

( )NBD

MDB

S NI h h NI

S MI h h MI

( h1 va h2 la cac ng cao ha t B va D xuong ND)

Ma

. .sin .sin

. .sin .sin

NBD

MDB

S N BD NBD BN NBD

S DN BD BDM DN BDM

.sin

.sin

NI BN NBD

MI DM BDM (1)

Goi K la giao iem cua MN va CA ta co :

.sin

.sin

NK CN NCA

MK AM CAM (2)

Ta co: ; NBD NCA BDM CAM (3)

CBN ~ ADM ( cmt) BN NC

DM AM (4)

T (1) (2) (3) va (4) NI NK

MI MK I trung vi K

Suy ra AC , BD va NM ong quy tai I------------------------------------------------------------------------------------------------Bai 95Cho ABC (AB < AC) co BE la phan giac . D la iem tren AC sao cho

AB = AD . I la trung iem BD . ng tron (O) tiep xuc vi AB , AC ,BC lan lt tai M , N , P . Goi K la giao iem cua BE va NP1. Chng minh N , I , K thang hang2. T giac AMNK la hnh g ?3. Chng minh BO ( AB + BC + AC ) = BE ( AB + BC )------------------------------------------------------------------------------------------------- Hng dan giai

1. Chng minh N , I , K thang hangTa co B, M , N , O , I cung thuoc ng tron ng knh BO nen :

MN BIN


36/38


71 LVC

A

P

E

K

M D

O

I

B N C

A

Ec b

O

B CI a

H

F

Do MN OB 090 BMN ABO

T giac POID noi tiep POD PID

Lai co 090 POD ADO ABO ADO

T cac ang thc tren PID BIN

Vay N , I , K thang hang2. T giac AMNK la hnh g ?

T giac OIDP noi tiep AIK ODP

Ma ODP ABK AIK ABK t giac ABIK noi tiep

Suy ra 090 BKA BIA AK KBMa MN KB MN // AK AMNK la hnh thang

3. Chng minh BO ( AB + BC + AC ) = BE ( AB + BC )

Ve CF // BE cat AI tai H E FC

BO FH

Ta co AH la phan giac A HF a c HC b

(1)

FH HC a b c

HC b

FC a b c

HC b

(2)

Chia (2) cho (1) theo tng ve : FC a b c

HF a c

Suy ra : BE a b c

O a c

BO ( a + b + c ) = BE ( a + c )

Hay : BO ( AB + BC + AC ) = BE ( AB + BC )-------------------------------------------------------------------------------------------------Bai 96Cho hai tiep tuyen AB , AC cua ng tron (O;R). Goi P , Q la trungiem cua AB , AC. Lay M tren ng thang PQ va ve tiep tuyen MD vi(O). PQ cat OA tai I . BC cat OA tai H

a. Chng minh OI2 IH2 = R2b. Chng minh MA = MD

-----------------------------------------------------------------------------------------Hng dan giai


72 LVC

M

B

P

O H I A

D

Q

C

A

F

OM

B E H C

a. OI2 IH2= (OA IA)2 IH2= OA2 2IA.AO + IA2 IH2= OA2 AH .AO = OA2 AB2

= OB2 = R2b. MA2 = MH2 = MI2 + IH2

= MI2

+ OI2

R2

( cmt)= OM2 R2 = OM2 OD2 = MD2

Suy ra MA = MD-------------------------------------------------------------------------------------------------Bai 97

Cho ABC can tai A , AB =2

3BC , ng cao AE . ng tron (O) noi

tiep ABC tiep xuc vi AC tai Fa. Chng minh BF la tiep tuyen cua ng tron ngoai tiep ECFb. Goi M la giao iem cua BF vi ng tron (O) , chng minh

BMOC la t giac noi tiep-------------------------------------------------------------------------------------------------Hng dan giai

a. at BC = 6a . V AB =2

3BC nen BE = EC = CF = 3a

AB = AC = 4a ; AF = AC CF = aDe thay OECF la t giac noi tiep .Ta can chng minh : BF2 BE.BC

Ta co BE.BC = 3a.6a = 18a2 (1)

Ke FH BC , ta co FH //AE 3

4

CH CF

CE CA

3 9

4 4

CH CE a

BH = BC CH = 6a 9

4a =

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100 Bai Hinh Hoc Chon Loc