Tuyển tập 500 bài toán hình không gian chọn lọc phần 1 nguyễn đức đồng

• BOI Dl/ dNG HQC SINH GIOI BOG... HtfdTng din Doc gi a tu gia i, xe m hinh ben... Co so ciia phu ong phap mot la sii du ng d in h ly phu ong giao tu yen song song... lidc le cung la du

• BOI Dl/ dNG HQC SINH GIOI

BOG

• PHAN LOAI VA PHU dNG PHAP G IAI THEO 2 3 CHU YEN

• B oi difdng hoc s inh gioi

• C hu a n b i t h i Tii ta i, D a i hoc va Cao da ng

(Tdi ban i dn thvt ba, c6 svCa chUa bo sung)

N H A X U A T B A N D A I H O C Q U O C G I A H A N O I

A N

Q I

16 H an

g C hu

oi

- H

ai Ba Tr cfn

g H

-a N

pi

Die

n t ho ai:

B ien ta

p C he b an : ( 04 ) 3 97 14 89

-6

H an ln ch inli : ( 04 ) 3 97 14 89 9;

T o ng B ien ta p:

(0 4) 39 71 50

11

• Fa

x : (0 4) 39 71 48 99

*

* *

Ch iu tr dc

h nh ie

m xu at

b an :

Gid

m do

c T on

-g bie

n ta p:

TS

HA

M TH

I TR

AM

Bie

n ta p:

THU

Y HO

A

Sa

a bd i:

THA

Nha sac

h

HONG

AN

Tr in

h ba

y bia :

THAI

AN

SACH

0 B

A I

T O

A N H

J NH K

H O NG G

I A

N C

H O

N L OG

M a so: 1

L 195OH2014

-In 1.000 cuon, kh

o 1

7 x

24c

m t

ai Con

g t

i C

o phan V3

n h

o a

V Sn La

- 2014/C

X B/

01

- 127/

O H

Q G

H N ng

ay 10/03/201

4

Q uy

et din

h xuat ba

n so:

198LK

ay 15/04/201

4

in xo

ng

v a nop

IIA J

chieu qu

y

il

n Sm 201

4

NhSm phuc vu cho viec r e n luyen va on t h i vao D a i hoc b k n g phucrng phdp t i m

hieu cac de t h i dai hoc da ra, de tiT nang cao va chuan b i k i e n thiJc can t h i e t

De phuc vu cho cac do'i tUcfng t\ i hoc : Cac bai g i a i luon chi t i e t va ddy d u , p h a n

nho tCrng loai toan va dua vao do cac phucfng phap hop l i

Mac du chiing t o i da co g^ng het siic t r o n g qud t r i n h bien soan, song vSn k h o n g

t r a n h k h o i nhiJng t h i e u sot Chiing t o i mong don n h a n moi gop y, phe b i n h tii quy

dong nghiep ciing doc gia de Ian xuat ban sau sach ducfc hoan t h i e n hcfn

Cuoi Cling, chiing toi x i n cam cm NIlA X U A T B A N D A I H O C Q U O C G I A H A N O I da

NGUYEN DtfC DONG

3

BANG KE CAC KI HIE

U TOA

N HO

C vA

CAC Tl/ VIE

g

(il

• [(ABC)

; (EFG)] :

• =

> : (i) ke

tUdn

g dilcfn

g

• C > :

Phep tin

h tie

n vectc

• d>

g tru

e A

• = : don

g nha

t

• D

o : Phep

doi xiiTng tru

e

0

•

i : khon

g don

g nha

t

• Q(

0;

cp ) : Phe

= S(ABC) =

dt(ABC) : dien.tic

0;

k) : Phep

AH

c =

V(S.ABC

) : the tich

hin

h cho

p

• VT(

0;

k) : Phep

• Stp

ly

• Sxq

•

V :

The tich

• CM

R : chiJng

minh rin

g

• A' =

''7(a

i A

: A' l

g (a)

• TH

i : trudng

hop

xuong

dtfcfn

g thftn

g (d)

• V

T :

ve tr

• BD

T :

bat di ng

thijfc

• d[M

; (

ABC)

I :

khoang each

• dpc

m :

dieu

phai chuCng min

h

• (a

; P):

goc

nhi die

n tao

a (P)

(AB) :

nhi die

n ca nh

AB

• K

L : ket lua

n

• (3r3

^ :

goc

tao bd

i ha

i

dUomg thin

g d

• D

K : dieu kien

va d'

• P

B : phan ban

• [HTCABCT

n ba

n

va mp(ABC)

4

Chuyen de 1 : T O N G Q U A N V E C A C K H A I N I E M

T R O N G HINH H O C K H O N G G I A N

• H i n h hoc k h o n g gian la m o t mon hoc ve cac v $ t t h e t r o n g k h o n g g i a n ( h i n h h i n h hoc

t r o n g k h o n g gian) ma cac d i e m h i n h t h a n h nen v a t the do t h u d n g thiTcrng k h o n g ciing n f t m

t r o n g mot m a t phang

• Nhif vay ngoai d i e m v a d i i d n g t h d n g k h o n g drfoTc d i n h n g h i a nhiT t r o n g h i n h hoc

phAng; mon h i n h hoc k h o n g g i a n con xay di/ng t h e m mot doi tuong can n g h i e n ciifu nCfa la

k h a i n i # m m g t p h a n g c u n g k h o n g difoTc d i n h n g h i a K h i noi tori k h a i n i e m nay t a

lien tuang den m o t m a t ban b a n g phang, m o t m a t ho nildc yen l a n g , m o t tb giay dat d i n h

sat t r e n mot m a t da di/gc l a m phang No duoc k y hieu b d i cac chCf i n L a T i n h n h a : (P), (Q), (R), hoac cac chCf t h u d n g H y L a p nhU (a), ((5), (y),

• M a t phang k h o n g ducfc d i n h n g h i a qua mot k h a i n i e m k h a c ; n h i f n g thifc te cho thfi'y mSt

ph&ng CO nhutng t i n h chat cu t h e sau, goi la cac t i e n de :

O T I E N D E 1: C o it n h a t b o n d i e m t r o n g k h o n g g i a n k h o n g t h ^ n g h a n g (nghia la luon luon c6 i t n h a t 1 d i e m d ngoai m o t m a t p h ^ n g tiiy y)

O T I E N D E 2: N e u m p t dtfdng th&ng v a m p t m a t p h ^ n g c 6 h a i d i e m c h u n g t h i dUcTng th&ng a y se n S m t r p n v ^ n t r o n g m a t p h a n g n e u t r e n

O T I E N D E 6: M o i d o a n th&ng t r o n g k h o n g g i a n d e u c 6 dp d a i x a c d i n h : t i e n de neu

len sU bao toan ve dp dai, goc va cac t i n h chat lien thuoc da biet t r o n g h i n h hoc p h i n g

• TiT do chung t a c6 m o t so each xac d i n h m a t p h 4 n g n h i / sau :

• Dong t h d i t a phai hieu t h e m r k n g mot m a t phang se r o n g k h o n g bien gidi va dUcmg t h ^ n g c6

do dai v6 t a n mac du t a se bieu dien no mpt each h i n h thiifc hflu h a n va k h i e m t o n nhU sau:

• De thuc h i e n dirge phep ve c h i n h xdc m 6 t h i n h h i n h hoc t r o n g k h o n g g i a n ngoai cac dudng

t h a y ve l i e n net, t a can p h a i n a m chac di/pc k h a i n i e m di/dng k h u a t ve b k n g net dijft doan:

Mpt dtfdng b i k h u a t t o a n bp h a y c h i k h u a t m p t d o a n c u e bp n a o do k h i v a c h i

k h i t o n t a i i t n h a t m p t m a t p h S n g du'ng p h i a t r vC6c h o ^ c p h i a t r e n c h e n o m p t

e a c h t o a n bp h o a c c u e bp ti^cAig uTng

5

• Muon x

ac din

h nh

^n

h mo

t m

at ph

^n

g trong khong gia

n ta con cho

n thu thua

c, tii

" gia

c hoac d

a giac ph&n

g (khong genh), dUcfn

h mp

t m^

t phSn

g tron

g khon

g gian Ta g

pi ca

g hinh

th vCc

v

di ca

c ky hip

u

(ABC), (ABCD),

(C),

tx ic tn

g vtn

g

• Ma

t phdn

g hinh thu^

c h

i khua

t ne

u c6 mpt hay nhieu m

^t ph&n

g nao

t

dvi ctn g

thdng n

^m trong m

^t ph&

ng hin

h thd'c m

a ma

t do h

i kh

p va khi dUcTn

g th

^n

g do khong l

a bien cua mat phdng b

g do cun

g tii'oTn

g vl

ng

khuat cue b

p ha

y toan b

p

• Mp

t diem

n hm

trong mpt m

$t ph&n

g hin

h thuTc b

i khua

t th

i g

oi la die

i diem m

a it nha

t c6 mpt die

m khua

t th

i dUp

c mp

t dUcfn

g khua

p : neu h

ai diictag d

o khon

g la bie

n cu

a cac m

^t phAn

g hinh thufc

C HINH ANH MIN

H HQ

A

\(d)

• (d)

bi (a) ch

e khua

t c

ue bo,

do (d) c6

1 doa

n

ve

ne

t dijft doa

n nk

m dud

i (a)

S

• (d)

bi ma

t ph^

ng (SAC) ch

e khua

t c

ue bo, d

e duf

t doan nkm sa

u (SAC) (h

ie

n

nhie

n (d) cung d sa

u c

ac ma

t (SAB), (SBC))

• Canh A

C

bi ha

i m

at phan

g (SBC)

v £l (SBC) ch

e

khua

t toa

n b

o,

do

ca doan A

C xe

m nh

u hoa

n to

an

d

sau

don

^n

g (SAB), (SBC)

-A A.

c /—

1—

^V FJ L^

• A]

H

bi ch

e toa

n b

o d

o c

a doan A]H nkm sa

u

ma

t

phin

g (AiADDi), m

Sc dij n

o

d

t rU

<Sc ma

t ph

an

g

(ABBjAi) va d tre

n m

at phan

g (ABCD)

• (d)

bi ch

e khua

t c

ue bo v

i c

6 doa

n EF v

e net

d

uT

t

doan nam sa

u ha

i ma

t phan

g (ADDiAj);

(CDDjCj

),

ma

c dij doa

n EF

d

phia t rU cJ e h

ai ma

t ph

ang

(ABBjAi);

(BCCiBi);

v

a

d tre

n m

at phan

g (ABCD)

•

CAC

K

Y HIEU

AN ^fHd

Thiir tr

f

Ky hie

u

Y nght

em A thuo

c ducfn

g thing (d) ha

y dadng

thin

g (d) chu

ra

A

Ha

y vie

2

A

i

(d) Di

em A or ngo

ai difdn

g thin

g (d) ha

y

dUcfn

g thing (d) khon

g chti

fa

A

Ha

y vie

d)

3 (d)

c (a) DU

&ng thin

g (d) nk

m trong m

at phin

g (a)

ha

y (a) qu

ay quanh (a) ne

u (a) lu

u don

g

Ha

y vie

e (a)

4 (d)

/ /

(a) Dif

crn

g thing (d) son

g son

g

\6

\t ph in

g

(a )

Ca ch v ie

n (a)

=

0

5 (d)

n (a)

=

A Difdn

g thing (d) ei

t m

at phin

g (a) ta

i A each vie

t k

hd

c :

(d)

n (a)

=

{A

}

6 (d,) n (da) = A H a i dUcfng t h i n g ( d i ) , (da) dong quy t a i A Cach v i e t khac :

Cling chijfa 3 d i e m A , B, C p h a n biet k h o n g

t h i n g hang

Cach v i e t khac : (a) = (p)

Co so cua phiiong phap t i m giao tuyen cua

hai m a t p h l n g (a) va (P) can thUc h i e n 2 budc

• B , : T i m h a i d i e m chung A , B cua (a) va (P)

n Ba : Difdng t h i n g A B l a giao tuyen can t i m

hay A B = (u) n (P) (ycbt)

n P H i r O N G P H A P ,

• Ti/ong t u nhtr phaong phap 1 k h i chi t i m ngay dtfoc 1 d i e m chung S

• Luc nay t a c6 h a i trifcfng hap :

> H a i m a t p h l n g (a), (P) thuf tif chiJa h a i difdng

t h i n g ( d i ) , (da) ma (dj) n (da) = I

=> S I la giao tuyen can t i m

> H a i m a t p h l n g (a), (P) thuf t i f chtifa h a i difdng

t h i n g ( d i ) , (da) ma ( d i ) // (da)

S _

D i f n g xSy song song v d i (dj) hay (da)

=> xSy la giao t u y e n can t i m

7

m C AC B AI T OA

N CO B AM

oi ABCD c

6 ca

c canh d

oi khon

g song son

g va die

m

S

d ngoa

n cii

a :

a/ (SAC) va (SBD)

hi

(SAB) va (SDC)

; (SAD) va (SBC)

Gia

i

a/ Xe

t ha

i ma

t phan

g (SAC) va (SBD), c

6 :

•

S la die

m chun

g

th

uf

nhat

(1)

•

Tron

g tuT gia

c lo

i ABCD, ha

i ducm

g cheo A

C

n

BD = O : diem chung

thijf

nhi

(2) ^

Ti/

(1 ) v

a

(2)

suy r

a :

(SAC)

o

(SBD)

= SO (ycbt)

hi

Xe

t ha

i ma

t phan

g (SAB) va (SDC) cung c

6 :

•

S la mot die

m chun

g

•

Ha

i canh ben A

B va C

D cu

a ti

l gia

c ABCD

theo gia thiet khong son

g song

^

AB ^ C

D

=

E : l

a diem chung

o : (SAB)

n

(SDC)

= SE (ycbt)

Tucfn

g tif:

(SAD)

n

(SBC)

= SF (ycbt);

vd

i

F

= AD ^ BC;

do AD/

/ BC

Ba

i 2

Ch

o ti

l die

n ABCD Go

i Gj,

Ga la trpng tar

n ha

i ta

m gia

c BC

D va ACD La

y th

m cii

a BD, AD, CD.Tim cac gia

c tuye

n :

aJ

(G

1 G2C)

o

(ADB)

hi

(G

1 G2B)

n

(ACD) c/

(ABK

n

(ABD)

= IJ

hi

(GiGaB)

n

(ACD)

= Gg

K

hoSc

A

K

d

(ABK)

^ (CIJ)

= G,G

cho

p

S

ABCD

c

6 da

y ABCD l

a

hinh

bin

h hanh tam O

aJ

Ti

m giao tuyen cua h

ai ma

t phSn

g (SAD) va (SBC)

hi

Ti

m gia

o tuye

n cu

a ha

i mS

t phin

g (SAB) va (SCD)

c/ Ti

m gia

o tuye

n cii

a ha

i ma

t ph

^n

g (SAC) va (SBD)

i ma

t phSn

g (SAD) va (SBC), c6 :

•

S la die

m chun

g thur nhat

•

De y A

D

c (SAD);

BC c (SBC) ma A

D //

B

C

Ta dun

g xS

y //

A

D hoac B

C

[(SAD)

= (xSy;

AD)

^

|(SBC)

= (xSy;

BC)

=^ (SAD) n (SBC)

= xS

y (ycbt)

hi

Tifang tir, difng uSv //

A

B hoftc C

D

8

Giai

• D

e y den K

D

Do do trong (BCD) KN

o CD = I

Ma K

N c (MNK) CD

(MNK) =

I (ycbt)

• Taon

g t

a xe

t IM

c (MNK), tron

g (ADC)

Ta

C O : AD n I

M =

E

= > AD

n (MNK) =

E (ycbt)

Ba

i 7

Cho tiJ die

n ABCD La

y diem

M tren A

C v

a ha

i die

m N va

m cu

a C

D v

a AD

\ di

(MNK)

HtfdTng din

Doc gi

a tu gia

i, xe

m hinh ben

a/ CD (MNK) =

P (ycbt)

b/ AD

n (MNK) =

h cho

p tu

f gia

c S.ABCD

La

y tren

m

M,

N, P theo

P khon

g th

m gia

o die

m cu

a S

C v

a AC v

di (MNP)

Gi

ai

ThUdng thudn

g d

o ycb

t ti

m gia

o die

n (MNP) =

K (ycbt)

Trong mp(SAC)

MK

o A

C = H1

ma MK

c (MNP) |

= > AC r>

(MNP) =

H (ycbt)

Ba

i 9

Cho m

ot ta

m gia

c AB

C v

a mo

t diem

S d ngoai ma

t phin

g chil

a ta

m giac

a tron

g ma

t phin

g (ABC) t

a la

y mo

t die

m O Dinh ro giao

diem

cua

(MNO) v

di ca

c dudn

g thing A

B, BC, AC va

u, do

c gi

a tu gia

i (xe

m hinh ben)

AB

n (MNO) =

E (ycbt)

BC

o (MNO) =

F (ycbt)

AC

n (MNO) =

G (ycbt)

SC

n (MNO) =

H (ycbt)

10

Loal 3 : Cf l l /NG MWfl B A D I £ M T R O N G K H O N G G I A N T H A N G H A N G

I pm ro NG P H A P

Co so cua phiiong phap can phai chufng min h ba diem

trong yeu cau b ^ i toan la diem chu ng cua 2 mSt phSng nao

do (chfing b a n A, B, C nSm tr en giao tu yen (d) cua h a i ma t

phSng do nen A, B , C th a n g hang)

O day khong loa i triJ kh a n& ng chiJng min h difoc difdng

thang AB qua C => A, B, C t h i n g hang

n C A C B A I T O A N C O BA M

Ba i 10

Xet ba diem A, B , C kh on g thuoc ma t p h i n g (u) Goi D, E, F Ian lu ot la giao diem ciia AB ,

EC, CA va (g) ChCifng m i n h D , E, F th a n g hang

G i a i

De y tha y D, E, F viTa a tr on g mp(AB C ) vifa d trong mp(a)

Do A, B, C g (a), nen (a) va (AB C) pha n b iet nhau

to al 4 :

C mi UG

M Wf

l

MQ

T D

tf CiN G

T H AN G

T RO NG K HO NG G IA

N

QU

A MO

I.

P Hi rO NG

P HA P,

Ca

sd

cua phucfn

g phap chuTng min

h diXcrn

g thing (d)

qu

a mo

t diem co din

h :

Ta can tim tre

n (d) ha

i diem tuy y A

;

B va chuTng min

h

2

diem d

o thin

g hang vdi mot die

m

I co din

h c6 sSn trong

khon

g gian

=> (d) qua

I

C O din

h (dpcm)

IL

P Ht fO NG P HA

P,

Co s

d cu

a phiTcfn

g phap can thu

c hien b

a bifdc cc

f ba

n :

n

Bi : Ti

m dUctn

g thin

g

a co din

h

d

ngoa

i mS

t ph5n

g co

dinh (a) m

a (a) chil

a

d (li

Oi dong)

• B2 : Ti

m

giao

die

m co din

h ma d d

i qu

a

m C A

C BA

I

TO AN

C

O B AN

m co din

h tron

g khon

g gian

d

v

e ha

i phia

kha

c

nhau cua

t diem M lu

u dong trong khong gia

n sa

o ch

o MA n a =

I va M

g thin

g IJ luo

n di qua mot die

m co dinh

> O co din

h (v

i

A,

B co din

Ta CO : mp

(P )

= (MA;

MB)

n (a)

= IJ

De y tha

y : O e I

J

=>

O, I, J thing han

g

Nghi

a la dacfn

g thin

g IJ d

i qu

a

O co din

h (dpcm)

ABC

D

(A

B //

C

D va A

B

>

CD)

Xe

t diem S

e

(ABCD) va mat

C vd

i

a '-^

S

B

= M, a n S

D

= N

ChuTn

g minh difdn

g thin

g MN luo

De tha

y

dx ia

c

ngay

M

N

c (SBD) va

AC

c (SAC) va M

N

o AC = O thi O e

BD = (SBD) n (SAC)

=> M

N qu

a

O co din

h (dpcm)

12

Bai 15

Ch o h a i d i f d n g t h f t n g d o n g q u y O x , O y v a h a i d i e m A , B k h o h g n S m t r o n g m a t p h i n g ( x O y) M o t m a t p h a n g l i f u d o n g ( a ) q u a A B l u o n l u o n c a t O x , O y t a i M , N C h i i f n g t o M N q u a

O

Ca ch

h rS ng

F q ua O = I

G n C

F

do ng

q uy

g ia

c

AB C, A B

C sa

o ch

o AB c at

A 'B

t A

C

d F;

C ca

g mi nh

b

a

di em

E , F ,

G

th Sn

g ha ng

b/ C hi Jn

g mi nh d if cf ng t ha ng

A A' ,

BB ',

C

C

do ng

q uy

E , F ,

G la b

a

di em c hu ng

c ua h ai m at

p h^

ng p ha

n bi et

(a )

^

(A BC )

va ( P) =

( AB 'C ).

Do

d

o : E , F ,

G

e (A ) = ( a)

n

( P)

b/ N ha

n

xe

t nh

u s au :

: A

A',

B B'

cr

( EA A' );

A A'

o

BB ' #

0

^ B

B' ,

CC

c

(G BB ')

; BB

' r

^

CC *

0

Ice

,

AA ' c

( FC C)

;

CC

n

A A'

#

0

^

AA ', B' , C

C

do ng

q uy t ai O

( dp cm ).

Chuyen de

2 :

Q UA

N H

E S ON

G SO NG

to ai 1: C Ht JN

G M Wf

l HA

I D LfC JN G

T

H A

N G

S ON

G SO NG

I.

PHirONG

p

ca

n

th ii

c hi en

h ai

h xid c

CO b an c ho

d in

h ng hi

a

a //

b

ja ,b

c:

( a)

Ki em

t ra h ai

d if dn

g th an

g

a

cu ng t ro ng

m ot m at

ph an

g

ha

y

hi fe

u ng am r an

g hi en n hi en d ie

u

do x ay

a

ne

u ch un

g tr on

g

1

hi nh p ha ng

n ao d o.

( 1)

•

B2 :

Du ng d in

h

ly

T ha le s,

t am

g ia

c do ng d an g, t in

h ch at

b ac c au

( ti nh c ha

\

6i

di fd ng t hi

J

ba ) la h ai

c an

h

cu

a

hi nh t ha ng ,

ha

y ha

i

ca nh

d oi c ua

h in

g di nh

h ai

d if cf ng t h^

ng

d

o

kh on

g

c6

d ie

m ch un g.

( 2)

=

>

(y cb t)

p S.

AB CD

c

6 Gj , G2 ,

G3

,

G, I an

lu cf

t

la

t ro ng

t am c ac t am

g ia

S DA C

hu mg m in

h

tiJ

f

gi ac G iG aG gG ,

la

h in

h bi nh h an h.

14

Cho diem S d ngoai ma t phSng h in h b in h h a n h AB C D Xet mS t phdng a qua A D c^t SB

va SC Ian lucft d M va N Chiirng m i n h A M N D la h in h tha ng

G i a i S D6 y thay ha i mSt phSng (a) va (P) c6 2 diem M vfl N 1^ d i^ m chu ng

Cho tuT dien AB C D G oi M , N Ia n li^gt la tr u n g diem cua B C va B D G gi P la diem

tren canh AB sao cho P ?t A va P # B Xet 1 = PD A N va J = PC o A M

ChiJng min h rSng : I J // C D

tu y y

G i a i Xet ha i ma t pha ng ( A M N ) va (PCD ) c6 h a i diem chu ng la I va J

Co so ciia phu ong phap mot la sii du ng d in h ly phu ong giao tu yen song song

De chiing m i n h d // a ta can thUc h ien h a i bade CO b a n chufng m i n h : d

• E l : Chufng m i n h d = y o p ma

y r- a = a

p n a = b a//b

1 5

n PHOONG PHAP^

Ca sd

ciia phifcn

g phap la st

f dun

g die

u kie

n ca

n

va du

chijf

ng minh di/d

ng thin

g (d) son

g song vc

Ji ma

t phan

g (a)

ban

g ha

i b

t fd

rc :

• Bi

: Qua

n sat va qua

n ly gia thie

t tim dudng thing o

u

vie

t (

A) cz

(a) va chiJ

ng minh (d) / / (A)

• B2

: Ke

t luan (d) / / (a) theo die

u kie

n ca

n va dii

m cA

c BA

I TO AN C

O BA

M

Bai2

1

Tron

g

t uf die

n AB

CD, chufn

g min

h rSn

g dean no'i ha

i tron

g ta

m Gi

hi son

g song

v6 [

(ACD)

Giai A

Go

i Ai

,

A2

la trung die

m BC v

a BD theo th

ut tiT d

o, ta c6 :

AG)

AG2

2

AA, ' AA

g

3

Theo din

h ly Thales, t

a c6 :

'0,02/

2

//C

D (tinh ch

at dU

c rn

g tru

ng binh)

Theo tin

h bS

c ca

u

=>

G1 G2

//

C

D c:

(ACD)

=j

G1 G2

//

(ACD) (dpcm)

Ba

i 22

Cho hin

h chop S

ABCD day la hin

h binh han

h AB

g minh : MN //

(SCD) va A

B / / (MNCD)

Gi

ai

Theo tin

h ch

at dudn

g trun

g bin

h tron

g ta

m gia

c

=>

MN //

A

B, ma A

B / / CD

=>

MN //

CD

cz

(SCD)

Theo die

u kie

n ca

n va d

u

=>

MN //

(SCD) (ycbt)

O Cach kha

n

(SAB) va trong h

ai ma

t phan

g

do

chiJa theo thijf tiT ca

c doan thing C

D / / AB D

MN //

B va C

D

=>

MN //

(SCD)

3

C

D (ycbt)

TifOng

tyl :

A

B / / MN

c

(CDMN)

=>

AB //

CDMN) (dpcm)

Ba

i 23

Xe

t ha

i hinh bin

h hanh ABC

D

va ABEF khong don

g phln

C va B

N

= B

-F Chufn

g min

h ring M

N / / (DEF)

3

3 Gi

ai

De

y tha

y

M,

N la trong tam cu

a ba

i ta

m gia

c ABD v

a

AB

E theo thijf tu d

o

Keo d

ai th

a trun

g die

m AB

PM P

X

1

^ PD P

E

3

Theo din

h ly Thal

es

^ MN //

E

D

c

(EFDC) ^ (DEF) (dpcm)

D

16

c kc

B AI

b

a t

ia

c un

g ch ie u, s on

g so ng

v

a

Ic ho ng d on

g ph

^n

g

Ax , By ,

B' , C s ao c ho

c kh on g.

C hO fn

g mi nh ( AB C)

A'B' II

AB

c

(ABC)

= C

C'

=j

A'C

//

AC

c

(ABC)

Ne

n

ta c

6 ha

i d

Uc rn

g th

^n

g do ng

q uy

A B' ,

A

C

tr on

g mp (A 'B 'C ') t ho

a di eu k ie

n (I ).

=> ( AB 'C )

//

AB C) ( dp cm )

h ha nh A BC D.

T ir

A

a C k

e A

x c

a Cy

on

g so ng c un

g ch ie

g

ma

t

ph Sn

g (A BC D) Ch ii

g mi nh

( B;

x) //

( D;

C y)

G i&

i

Ti ra ng

t

u x et

ai

m at

hi ng

( B;

A x)

th uT

t

a

ch uT

a

ca

c c ap

d ud ng t hi ng

do ng

q uy

fAB/

/C

D IAx/

/C

y

=>

( B;

x) //

( D;

y)

( dp cm )

F

d t

ro ng

h ai m at

p h^

ng k ha

c nh au

//

BC E)

v

a (

BC E) th iif

ti

T c

hu Ta

c ac

ap

d ir dn

g

th dn

g do ng

q uy

/ A;

iA F/

/B

E

AD//

BC

(A DF )

//

BC E) ( dp cm )

D an

g 2 : C

H UfN G

M

I N H

CA

C D

l /d

N G TH

A NG

D 6N G P

H A

N G

LP Bi rO NG PB AP

g ph ap c hi if ng m in

h

ca

c

du dn

g th in

g

di , d2 , dg

d on

g ph in

c hi

^n

h ai

b i/<

g mi nh

d ], d g, d s,

d oi m ot c at

n ha

u

va

c un

g so ng s on

g

vd

i mp

t ma

1 8

• B2 : Ket luan d], d2, ds, c (a) // (P) => d i , d2, d^j, dong p h i n g trong (a); (a) phai

chufa cac giao diem cija d,, da, ds,

Goi A t i , At2, Ata la ba diTdng phan giac

ngoai ciia goc : fiAfc, CXt), I5A6 theo thuT t u do

Do cac tam gidc can tai dinh A nen cac

phan giac ngoai song song vdi canh day, nen :

At, / / B C c (BCD)

A t a Z / C D e (BCD)

; A t 3 / / B D c ( B C D )

=> A t , , At2, Ata dong ph^ng (trong (P) // (BCD) \k (P) qua A) (dpcm)

Htfdng d i n

Doc gia t u giSi iMng t\l hai bai toan tren

19

DE SH L i"

HA IJ ES T RO N

G KH 6 N

G GI AN

•k

Din

h ly

i (thu|ln)

: Ha

i dit an

g th in

g tu

y

y d,, d

2 tr on

g

kh on

g gia

n ch dn tr en c dc in

^t ph dn

g so ng s on

g nh au (a )

II (P ) II (y ) t ao r

a cd

c do an th an

g tU cm

g l in

g ty le :

A,A.

-k

Dinh

l y2

( da o) :

• Tr Ud

c kh

i xe

t di nh l

y da o, t

a qu an t ar

n de

n ha

i kh ai

ni em s au

hi

et d en c ac d ay t

y so , c hi ng h an :

A,

A

B1 B9 (*

• (A i;

]) l

a ca

p go

c cu

a d ay t

y so (

*)

• (A 2;

B2 ) v

a (A 3;

B3 ) la c ac c Stp n go

n cu

a d ay t

y so

*)

• Do an n oi c ap g oc v

a cd

c c ft

p ng on l

a ( do an ) ba

c t ha ng c ua d ay t

y so

*)

• Dinh l

y

: N eu c

6 da

y ty s

o tr on

g kh on

g gia

n :

A,A

(*

) da , x dy r

a tr

g (d ,) (d 2) th

i cd

c ba

c th an

g AiB ,, A2B

2, A3B

3 cu ng s on

g so ng v

ai mo

t mg

i ch

u

: Ta

c

6

ph at b ie

u kh ac c ua d in

h ly

Th al es d ao n hi

f sa

u :

Va

i die

u kie

n c6 d ay ty s

o (*

) da x dy r

a tr en h

ai

du dn

g th dn

g (d j)

fd

^)

th

i mo

t tr on

g

3 ba

c

th an

g A,B ,, A2B

2, A^B

^ se s on

g so ng v

di mo

t

ma

t ph dn

g ch da h

ai ba

c th an

g co

n la

i

"A ,B

i (a )

= ( A2 B2 ; A3 B3 )

A2 B2

//(P)

-(

A3 B3

;A iB i) ( A)

A3 B3 //

(Y )

S (A ,B

i

;

A2 B2 )

(d i) (g oc )A

i

(n gp

n tr en ) A

2

(n go

n dif di) A

3

(d

z)

^ B,

o d in

h)

\

Dan

g 3 : CH

U fN

G M

I N

H D JC

i N

G TH AN

G S ON

G S ON

G M AT P HA NG

BA NG D

I N

H L

Y T HA LE

S

L

PH ir ON

G

PH AP ,

Co s

d cu

a ph uo ng p ha

p ch uf ng m in

h dU dn

g th in

g so ng s on

g vd

i mS

t ph in

h ly T ha le

s da

o tr on

g kh on

g gi an g om

ai b ud

c ca b an s au d ay :

• Bi

: Xa

c d in

h tr en h ai d ud ng t hi ng t ii

y

y ch an

g ha

n ( di ), ( d2 ) d

e ti

m t re

n do d

1 B

2

Aj As B ,B

3

Xd

c di nh c Sp ( Ai

; B j) l

a cS

p go c, c dc c Sp ( Aj

; B 2) v

a (A

3

B3 ) la h ai

Sp n go n.

n

B2 : Lu

c do c ac d oa

n ba

c th an

g Ai Bi , A2 B2 ,

A3 B3

d ir ac k et l ua

n cu ng s on

g (P ) ( xe

m • >

20

• pmroNG PHAPj

Ta chutng m i n h dUdng t h i n g (d ) n am tro ng m at p h i n g (a) / / (()) => (d ) / / (p )

m cAc B A I T O A N C O B A N

Bai 32

Cho tut d ien A BC D c6 A B = CD Go i M v a N la hai d iem lUu d o ng tre n A B v a C D sao cho

A M = CN Chutng m i n h M N luo n so ng so mg vdi mSt p h I n g co d inh^

Giai

Neu d at A B = C D = a; A M = C N = x De y thay tre n A B v a C D ta co d ay ty

A M ^ CN

A B CD

i(A; C) la cap go'c

|(M; N) va (B; D) la hai cap ngon tUcJng ufng

Ap d ung d i n h l y Thales dao tro n g kho ng g ian t h i ba bac

thang A C, M N v a BD ciing so ng so ng v<Ji mot mSt phIng (a)

due nay (a) chUa co d i n h v i d ay ty so — chUa la h k n g so)

a

Ta diTng (a) n h u sau : go i E, F, G la tru n g d i e m cac canh A B,

DC, CB theo thuf t i l do t h i (a) = (EFG) v a (a) tho a y eu cau la

Vay M N / / (EFG) = (a) co d i n h (d p cm)

Bai33

Cho hai h i n h binh h an h A BC D v a A BEF kho ng dong phIng; tren cAc dUOng cheo A C v a

BF Ian lucft lay cac d iem tuy y M , N sao cho

Cho h i n h v uo ng A BC D v a A BEF d tro n g hai m at p h I n g khac nhau Tre n cac d ifd ng cheo

A C v a BF, ta Ian lug t lay cac d i e m M , N sao cho A M = BN Chutng m i n h ran g M N / / ( CEF)

A p d ung d i n h ly Thales cho cac d o an bac thang :

A B, M N , CF voti de y EF cz ( CEF) ; A B / / EF c (CEF)

^ M N / / (CEF) (d p cm)

2 1

Bai

35

Tr en h

ai ti

a Ax v

a By ch6o n ha

u, ta I an

lua

t

la

y ha

i di em

M N sao

c ho A

h rS ng M

N lu on l uo

n son

g son

g vd

i mp

t mS

t ph Sn

g c

o di

nh

Tr ad

e he t:

HU oT ng

dim

By la'y d ie

m

N, di nh b

di : B

N, =

1

Ax l ay d ie

m

Mj di nh b

di : A M; =

k (v

i k > 0 , ch

o tn /d c)

Hi en n hi en 1^

h

ai di em M

j \k

N ] c

o di nh

Theo g ia t hi et v

a tii

each d an

g tr en h in

h ta

co :

AM , AM , AM

i BN

i

BN ,

BN AM

y da

o cu

a di nh l

y Th al

^s M

N

lu on l uo

n son

g son

g vd

i mS

t ph in

g c

o di nh ((5) =

(A

; Bd ) cha

a AB v

a da dn

g th in

g d q ua

B son

g

song v

di N, M]

(dpcm )

i da dn

g th in

g che

o nh au d

j va

d 2.

M la m ot d ie

m ch uy en dong t re

n don

g tr en

d 2.

im q uy t ie

h tr un

g di em

I cu

a doa

n MN

Hu&ng dan

Go

i AB l

a doa

n vu on

g go

e ch un

g cu

a

di va

d

2

(A

e dj , B

d2 );

O la t ru ng d ie

m eii

a AB

N'

OB I

X

Theo d in

h ly

T ha le

s da

o th

i 01

nkm

t ro ng m at p hl

ng

(P )

qu

a O song song v

di

di va

da, tiif

c la m at p ha ng x

ac

di nh b

ai ha

i da dn

g th in

O Ia

n lacr

t son

g

song v

di d]

va

d-^

Giai han

; M v

a N chay t re

n d]

va k ho ng

co ra

ng

huge n en

I cha

y

ti

y tr en

( P)

• Dao :

L ay m ot d ie

a I t

a dO ng dacfng t hi ng song

song v

di d

'2 ,

da dn

g th in

g na

y ci

'l

do'i xO ng vdri O q ua E

MI

c lt

d'2

d N' Di nh

l

y

da dn

g tr un

g bi nh c ho t ha

y I l

a

tr un

g di em c ua M 'N ' TC

f M' va N ' di fn

g ca

c da dn

g th in

g son

g

song v

di AB Ch un

g Ia

n la

gt el

t d] d

M va

d

2 d

N

Ha

i ta

giac O M' MA v

a ON 'N

B de

u la n hf ln

g hi nh c hS n ha

t :

=> , ^ ^ ^ _^

M N' NM ' la m ot h in

h bi nh h an

h d

o do

I la t ru ng d ie

m cu

y ti eh t ru ng d ie

m I c ua doan M

N la

m at p hi ng

( P)

i qu

a O song song v

di

dj

va

22

lidc le cung la du

Co so ciia phifong phap la sii du ng sii c^n th iet cua h a i tien de 5 va tien de 6 d4 xay diTng

va chufng min h mot so b ai toan co b a n tr on g khong gian k h i h in h th a n h nen cac va t the (hien

nhien 4 tien de d trirdc da duoc nga m hieu la lu on lu on di/gc sOf dung)

B a i 37

Cho a, b, c la ba difdng t h i n g khong ciing nkm tr on g mpt ma t phAng va doi mot cSt nhau

Chufng min h rSng : a, b, c dong guy

G i a i Tha t vay : gia sijf a, b, c kh on g dong quy, th i cac giao diem ciia chiing lap th a n h ba diem

khong th a n g ha ng va ba difcfng thftng cung nam tr on g mot ma t phang Tr a i vdi gia th iet

Theo phep chufng m i n h p h a n chiifng ycb t dUcrc chijfng m i n h xong

B a i 38

Cho 3 tia Ox, Oy, Oz doi mot vu ong goc

a/ Chufng min h r k n g ba tia do kh on g cung n k m tr on g mot ma t phang

b/ Ijay tren ba tia Ox, Oy, Oz Ian lifgt cac diem A, B, C (khac goc O) Chijfng min h r a n g :

(AB + BC + CAf ^ eiOA' + OB^ + OC^)

c/ Ky hieu a, p, y la ba goc tarn giac AB C, a, b, c la do dai OA, OB , OC Ti n h cosa, cosp, cosy

va chufng to r a n g a, [3, y nhon

G i a i a/ Tha t vay : gia sCf ba tia cijng thuge mot ma t

phang, vi Ox va Oy ciing vuong goc v6i Oz, nen Ox va

Oy cung n a m tr en mot du'dng tha ng D ieu do tr a i vdi

gia thiet

Do do ycbt di/gc chufng min h b a ng phep chufng

min h phan chufng

b/ Ap dung bat da ng thufc B u nhiacovky, ta eo :

(AB +BC +CA)^ < 3( AB ' + BC^ + CA^) = 3(0A^ + 0B ^+ OB^ + OC^ + OC^ +OA^)

(AB + BC + CAf <i 6(0A^ + OB^ + OC^)

c/ Ap du ng djnh ly h a m cos cho AAB C, ta c6 :

BC^ = AC^ + A B ' - 2AC.AB.cosa

2 3

cosa =

< =>

a nhon (dpcm)

>0

Tirang ti

f t

a c

6 : cos(i

= Va

^ + b^

Vb

^+

c^

Do d

o : P, Y cun

g nho

n (dpcm)

Ba

i 39

>

0;

cos

y = Vc

2 + b2 V a^

g khong gia

n ba tia Ox,

Oy,

Oz do

i mo

t ta

o vdr

i nha

u mo

t goc 120

Oy,

Oz ph

ai dong phSn

g

Gi

ai

Gia si

if

Ox,

Oy,

Oz khon

g dong phSn

g

va ta cho

n

sAn

tre

n Ox

; Oy ca

c die

o : O

A

= OB =

1 (dvcd)

Dong

th

di

tren tia d

oi Oz' cu

a tia Oz, ta cho

n die

m C sa

o ch

AC =

OC = OAcos60"

OC =

Dinh ly ham cosi

n tron

g A BOC c

ho ta :

BC

^

= OB' + OC

- 2OB.OCcos60°

« BC'

=

1 +

i

~ 2.1.-.i =

« BC =

Do d

o : AC = BC = (1)

Tifon

g tu : AB

^

= OA' + OB

^ 2OA.OBcosl20''

1

) =

3

« AB =

Va

(2)

Ttf (1)

va (2) ta difg

c : CA + CB = AB <=>

C

e AB <=>

Ox, Oy;

Oz dong ph^

ng (v

f ban dau)

ai dong phan

g

(dpcm)

Ba

i 40

Cho ba tia Ox,

Oy,

Oz sao ch

o xO

y ^ x

&

= 45

" v

a y(5

z = 90

"

ChiJ

ng min

c m

ot ma

t phang Hi^c

Tn

g da

n

Gia sU Ox,

Oy,

Oz khon

g dong phSn

g

va chon tre

n

do th

m A,

B,

C sao ch

o : O

a V

2

Do gia st

f =

> A

e (OBC)

Stf dung din

h ly ham cosi

n

=> AB = AC = VOC

^ + OA

^

- 20 C.

OA C OS 45

"

AB = AC

=

ha^

a '

- 2.a>/2 a ^

=

=

a

Ma : BC =

a

^/2.^/

2 = 2a

24

1/ G i a sd O x ; O y ; O z d o n g p h ^ n g t r o n g m p ( a ) n a o d o , t a co h a i k h a n&ng :

T i a O y h o a c n a m t r o n g m i e n goc xOz ( x e m h l ) h o a c m i e n n g o a i goc ic8z ( x e m h 2 ) t h i

xOy = 3 0 ° * 6 0 ° ( h 2 ) h o a c xOy = 1 5 0 ° * 6 0 ° ( h l ) ( v 6 l y v d i g i a t h i e t xOy = 6 0 ° )

c d

o, (1) + (2) ch

o t

a gia thi§

't:

2 c

os fli Ot)

1 OA O

'

=.1 oO

E

= 2cosf^

; OC co din

h trong khong gia

n ne

n tia pha

n gia

c O

D cung c6' din

h

Ha

i tia OB;

OD co din

h tron

g khon

g gia

n ne

n tia pha

n gia

c OE co din

h, ma

t ch

o t

a die

m

E

co din

h trong khong gian

Va

y kh

i A,

B,

C lo

u dong sa

o ch

o

nhun

g luo

n qua E co din

h

Ba

i 42

hi ma

t ph

^n

g (ABC)

h chop S

ABCD co day la hin

h binh han

h ABCD Tren ca

c can

g c

^c die

m A], B

i,

Cj sao ch

o SA

SC : k (k >

0,

k c

ho san)

ChuT

ng min

i

Ci thay d

oi th

i mp(AiBiCi) ca

t S

O ta

i 1 die

m

co din

h (vd

i

O

= AC

o BD)

Gi

ai

Go

i AM la trung tuye

n cu

a ZiABC tu

y y con

B', C tu

y y

thijf

ti

i tre

n A

B

va AC (h.2)

Kh

i AM

o B'C = M', ta co nga

y h

e thu

fc :

+ — = 2

— (*) (D

oc gia t

u chufn

g min

Do d

6, ne

u g

oi SO ^ A,C,

= O', th

SO

SAi SCi SO'

(1) (h.l)

NenSA SC ,

SAi SC]

(2)

< =>

„ SO , SO'

0, ch

o

< => O' CO

in

h (ycbt)

Ba

i 43

(h

.2

)

Cho hin

h chop deu co canh ben v

a can

h da

y de

u ban

g

a

Tim die

m

M

e SA sa

nho nha

t, ha

y ch

i r

a gia t

ri nh

o nha

n tic

h AMBD S =

i BD.M

O

=

|

aV2

M0 (1)

=>

min

S xa

y ra minMO xay r

a

Nhun

g minMO = dlO

; SA]

= OH

Vi

t Of die

n de

u ne

n AC n B

D

=

O th

i S

O la dudng cao

=> ASO

A vuong t

ai

O (2)

26

(3)

^ S = 4SO^ + AC'^

Be y trong (3) chi c6 SO la thay doi, do do S nho

nhat khi va chi khi SO nho nhat Trong mp(0; d) co

dinh ha OH 1 d tai H

=> OH = d[0; (d)] = minSO (do (d) co dinh)

=> minS = 40H^ + AB^ xay ra k h i S = H (ycbt)

Bai 45

Cho 3 diem A, B, C khong thuoc mat phang (P) Gia sd cac doan t h i n g AB va BC deu c l t

(P) Chufng minh rang doan t h i n g AC khong c l t (P)

• NhUng M , N , I la ba diem n l m trong ba canh cua AABC

ma t h i n g hang t h i dan den di6u v6 ly

Vay doan AC khong the nao c l t mp(P) difoc (dpcm)

a si

f A, B, C 1^

g

di qu

a

A, B va

C Dudn

g thin

g na

y

cung v

di diem

D xa

c dinh m

ot ma

t phftn

g (a) Ta c6 : / ^rv

De

y tha'

y (1)

t

"khong c6

4 diem nao /

tron

g n die

\

Vay khong the c6

3 diem nao trong

n diem ay thing han

g (dpcm)

2/ Ba

y gid t

a xe

t n die

m kha

c vd

i tinh chat l

a 4 diem bat ky nao trong chiing

deu

dong

phang

Goi n die

m a

y la

• Kh

i n = 4 t

hi ba

i an difang nhien dung

m Ai

; A 2;

3 xa

c dinh m

at phln

g (a) X6

t di

em

Aj (v(J

i 3 < i <

n) The

o gi

a thie

t 4 die

m Aj

a A

j e (a) vd

i m

oi i = 4

t c

a n die

m a

y thup

c mp(a) (dpcm)

Ba

i

47

Cho tiJ

ie

n ABCD Go

i I;

J Ia

n lug

t l

a trun

g diem cua h

ai can

h do

i AB v

m tuyy tre

n can

h AC Ma

t phln

g (IJM) ca

t can

h BD t

ai

N Chufn

g min

t die

n IMJN thanh h

ai phan dien tic

h ban

g nhau

Tron

g ma

t phing (IJM)

h ly Thales dao t

hi AC

; BD

; IJ nam trong

ba

ma

t phan

=> O la trung die

K tron

g tam giac MIJ

va

NI

J la ban

g nhau

dt (MIJ) = d

t (NIJ)

Vay I

J chi

a thie

t dien IMJ

N than

h ha

i pha

n die

n tic

h

bang nhau (dpcm)

Ba

i

48

Cho tuf die

n ABCD Ti

m diem

M tron

g khong gia

n sa

o ch

o I = MA

^ + MB^

i nh

o nhat

Gi

ai

Goi I,

, G Ian lifcrt la trung die

m cii

a AB;

CD va I

J

Dinh l

y difcfn

g trung tuye'n cho :

2

^ /c /- i 2 T T 2 A B ^ + C D ^ ^ A B ^ + C D ^ , , 2 L i

=> L = 4 M G ^ + IJ'' + > + IJ'' = h a n g so

2 2 Dang thufc xay ra k h i va chi k h i M = G => E = M A ^ + M B ^ + M C ^ + M D ^ d a t gia t r j nho nhat k h i va chi k h i M d G, t r o n g tarn cua tuT d i e n (ycbt)

Ta CO : VQ.MNP = VAO.NP + VA.OMP + VA.OMN

Khoang each tif A den ba m a t p h l n g (Oyz); (Ozx) va (Oxy) la :

Chuyen de

4 :

QUAN H

E VU

6

NG G

OC

Loai

1: DU CI NG T HA NG WONG

GOC fdJ^

T P HANG

Dang

1

: C

HQ NG

M IN H

Ol/

G

O C Vd

l MAT P

U KI EN

C

A N VA

DU

L PHiroN o

vdi ma

t phln

g a bSng dieu ki^n can va du

hai difcfn

g th^

ng

a, b dong

d_La (dpcm)

n

CA

C B

AI T OA

diy ABCD la tiJ gid

c loi Bie

CI

(SAD)

^ ^A B L DA c (SAD)

AB

1

(SAD) (dpcm)

Ba

i 5

1

Cho hinh chop S.ABCD

c6 day ABCD la hinh binh hanh tam

O v

a S

A = SC

m gia

c ca

n ta

i S : ASA

tSO L BD c (ABCD )

^ is O i AC c (ABCD )

= > S

O 1

(ABCD) (dpcm) A

\k hin

h tho

i tS

m O

= > AClBDc(SBD) (1

AC

J _ S

O c (SBD) (2)

TC r (1) v

a (2) ch

o : AC

1

(SBD) (dpcm)

30

(Sau nay t a c6 t h e chufng m i n h (1) bSng d i n h l y 3 dadng

vuong goc se n h a n h h d n hoac b d n g t i n h c h a t giao tuyen

cua h a i mftt p h i n g vuong goc)

Tif (1) vk (2) cho : E A _L (SBC) 3 SC SC 1 E A (3)

Tif (3) va (4) => SC 1 ( A E F ) (dpcm)

B a i 54

Cho h i n h chop S.ABCD c6 day A B C D l a h i n h chuT n h a t , goi I , J l a t r u n g d i e m A B , C D v^

gia sii SA = SB ChOfng m i n h r S n g C D 1 ( S I J )

Cho tijf d i e n A B C D c6 H , K l a true tarn cdc tarn gidc A B C va D B C G i a sC( r k n g H K 1

(DBC) Chufng m i n h A H , D K va BC dong quy

I i ^ I Vfiy A H , D K v ^ BC dong quy t a i I

ai

a / Ch

o A

H o B

C

= A'.

D

e c hii ng m in

h S

; K;

A' th in

g

ha ng ta ch ufn

g m in

h : S A'

± BC

BC

^A A'

'B CI SA (d

o SA

X (AB O)

Ta

C

O :

Tif do ta su

y r

a : B

C 1 (S AA ')

= >

BC

1 SA '

Va

y C

O t

he no

i : A H;

SK v

a B

C d on

g q

uy (d pcm )

b/ Th eo g ia th ie

t

ta c6 : S

C 1 B

K (1

)

Ma tk ha ct ac

o :

\^^^^^

J BHXSA (doSAKABO)

ne

n : BH

1 (S AC )

^ BH

1 SC

Tif (1 ) va (2 )

ta su

y r

a : S

C ± ( BH K) (d pcm )

d

Th eo ca

u a /

ta c6 : BC ± (S AA ')

= >

BC ± H

K

Th eo c au b /

ta

c 6 : SC

_L

BH K)

= >

SC J_

H

K

Tif (3 ) v

a ( 4)

ta su

y r

a : H

K ± ( SB C) (d pcm )

Ba

i

57

(2 )

(3 )

(4 )

Ch

o hin

h v uo ng A BC

D na

m tro ng m

at ph An

g ( P).

Qu

a A d ita

g

n ijf

a d Ucm

g t hin

ot die

m lU

u d on

g tre

n Ax

Bad

n g

th an

g qu

a M v uo ng g

oc vd

i m p(M

ng th Sn

g q

ua

M vu on

g g

oc

\6i

m p(M CD ) c

at (P ) t

ai

S

1 / Ch ufn

g m in

h : A

; B

;

R t ha ng h an

g

2 / Ti

m q

uy tic

h t ru ng d ie

m I c

ua do an R

S k

hi

M l ifu d on

g t re

n

n ijf

a d ud ng t hi ng A

x

Gi

^i

1 / Th eo g ia th ie

t t

a c

6 : MR

1 (M BC ) MR

1 BC

Ma d a

C O

: AD

1 AM ^ A

D 1 (M AR ) =

>

AD

1 AR

Va

y AR

; A B;

AD c un

g

a

tr on

g mp (P )

ma AR v

a AB

cu ng v uo ng g

oc vd

g ha ng (d pcm )

TiJ on

g t

if tre

g

2 / Do M

R ± ( MB C)

^ MR ± M

B

Ta

m g ia

c M BR v uo ng d M c6 d ifc

r ng ca

o MA n en :

MR

1 AD (v

= AB A

R

Tifc

r ng ti f:

MA ' = A D.A

S

AB A

R = A D.A

S

^ AR = A

S

= j>

I th uo

c d ifd ng t hi ng A

C

Do R c ha

y t re

n tia A

u ( la ti

a d

oi cii

a

tia A B) va S c ha

y tre

n tia A

v ( tia d

oi tia A D) ne

n I

or

n go

ai hin

h v uo ng A BC

D

Va

y,

I c ha

y t re

n tia d

oi

At cu

a t ia A

C ( bo d ie

m A )

Va

y s au k

hi la

m p ha

n d

ao th

i q

uy tic

h c

ua

I l

a t ia A

t ( kh on

g k

e d ie

m A ) ( ycb t).

32

Dang 2 : C H J N G MI N H Dl/ dNG T H AN G V U ON G GO C Vdl MAT P H A N G

BA N G T R U C Dl/ CiNG T RO N

L PHirONG P H A P

Ca stf ciia phi/ang phap chiirng min h difdng t h i n g d vu ong

goc v6i mat pha ng a bftng van du ng d in h nghia t r u e dxidng

t ro n: l a dxtiing t h S n g v u o n g g o c vdri m a t p h a n g c h i i ' a

dUdng t r o n t a i t a m c u a n o b a ng h a i btfdp ca b an nhU sau :

n B i : Ti m mot diem S a d in h each deu cac d in h da giac

day ABC n h a sau : SA = SB = SC =

Tim diem O d day each deu cac d in h da giac day ABC

OA = OB = OC =

• Ba : No'i h a i diem S, O do th a n h tru e d cua di/crng tr 6n

No la dudng thftng vu ong goc vd i moi ma t pha ng ehiJa

duac di/cfng tr on (AB C)

n. C A C B A I T O A N C O B A N

B a i 58

Cho h in h vu ong AB C D canh a Ve cung ve mot phia (AB C D ); cac doan AA'' C C vu ong goc

(ABCD) sao cho AA' = C C = a Chufng min h : A'C 1 ( B C D )

B C = 2C I = 2 ^ = a V s (ASIC 1^ niJra A d e u)

Ta CO : •iCA = a (.\ASC d e u)

A B = a>/2 (AASB v u o n g c a n t a i S)

[CA^ - AB^ = a^ - 2a^ - Sa^

Theo d in h l y Pythagore dao => ACAB vu ong ta i A (dpcm)

3 3

b/ The

o ti nh c ha

t dUcfng

t ru ng t uy en

Cifn

g v

di ca nh h uy en cua t am

g'lAc

A BC v uo ng

B = I

C (1)

Ha

n nf fa d

a

CO

S

A = S

B = S

C (2)

ch

o : S

I la

tru

e ducrng t ro

n

ng oa

i ti ep zVABC =>

S

I 1 ( AB C) (d

pc

m)

Ba i6

0

Ch

o hi nh chop S A BC

D da

y AB CD l

a hi nh t ho

i c

6 IJ AC = 60

" v

a SA = SB =

g SG

x (A BC D) Vd

i G l

a tr on

g ta

m ta

m gia

c AB

: i

BA

- BC ( ca nh h in

h th oi ) AA BC d eu

i

g AC 6 0"

(gt)

=>

G la t am d ud ng t ro

n AB

C ng oa

i ti ep t am giac d eu A BC

Do d

o ; G

A = G

B = G

a du dn

g tr on ( AB C)

Gi

a th ie

t c

6 : S

A = S

B = S

C S

e (d )

To

m la

i SG

c (d ) ha

y SG

1 (A BC ) ha

y SG

1 (A BC D) ( dp cm )

D c

6 SA = SC = SD v

a AD

t = 9 0"

G

oi I l

a tr un

g di em

A

C ChuTn

g

mi nh r kn

g SI -L ( AB CD )

Gi

ai

De

y ti

f t am giac A DC (

A = I

C = I

D

Kc

t hcr

p gi

a th ie

t SA = SC = SD

=i> SI l

a tr ue d ud ng t ro

n (A CD ) ng oa

i ti ep

\A CD

=^ S

I 1 ( AC D) ^ (A BC D)

« S I

l (A BC D) ( dp cm )

D c

6 AB CD l

a nijf

a lu

c gia

c de

u c

6 SB t) - SCT) = 9 0"

m AD v

a SD Ch ij

g m in

h rf tn

g 01

1 (B CD ) va S

A 1 ( AB CD )

Gi

ai

De

y de

n ti nh c ha

t ci ia d ud ng t ru ng t uy en ufng

\'6i

c an

h

hu ye

n cu

a ta

m gia

c vu on

g, ta eo :

AS BD (

fi = 9 0"

) :>

I

B = I

D

AS CD (C 9 0°

) :r>

C = I

D

Xe

t nijf

a lu

c gia

c de

u AB CD ( la

m ci ia l uc giac d eu l

a O)

=>

O

B = O

C = O

D (2)

Ti

r (1)

va

(2 )

=>

1

0

la t ru

e di /d ng t ro

n (B CD ) ng oa

i ti ep t am giac B CD

=>

I

O 1 ( BC D) (d pc m)

IB = IC =

ID

(1)

Ma S

A / / - O

SA

1 (A BC D) = (B CD ) (d pc m)

34

tool I: DUdNG T H A N G W O N G G ^ C DUdJNG T H A N G

OUdNG THANG VUONG GOC Vdl MAT PHANG

L P B U O N G P H A P

Co sd cua phiTOng phap chOfng m i n h dudng t h i n g d vuong goc vdfi diTcfng

thSng a k h i t a sit dung d i n h n g h i a : d 1 a => d 1 a (tuy y t r o n g a),

qua 2 bade ca b a n :

• B i : Quan sat v a quan l y gia t h i e t t i m mp(a) chijfa dudng t h d n g

a can chufng m i n h no vuong g6c v d i d

Goi I la t r u n g d i e m c a n h C D v a de y h a i t r u n g tuyen cung l a

dudng cao t r o n g h a i tarn giac can cung ddy C D l a : A A D C v a ABCD

Cho h i n h chop S A B C D c6 A B C D l a niJfa h i n h luc giac deu v a S A 1 ( A B C D ) M o t m a t

phang qua A vuong goc v d i S D t a i D ' cat SB; SC t a i B', C ChuTng m i n h t i l giac A B ' C ' D ' n o i tiep difoc

Theo eac

h difng (a)

^ (AB'C'D')

=>

AB'

1 SD (5

)

Tif (4) va (5)

=>

AB'

1 (SBD)

; ma (SBD)

3

B'D'

=>

A B'

± B'D '

« AB TT = 9 0"

(6 )

Cung ti

f (5) va (6) =

> T

u T gia

c AB'C'D' no

i tie

p du

ac (dpcm)

n ABCD ChiJn

g minh

D IE

U KI EN

DA I

s

6 sa

u

de

tijf die

n c6 can

h do

i nha

- AD'

= BC'

- BD

l

Gia

i

Ta chiln

g minh die

u kien b

^ng h

ai trtfcrn

g hop :

• Die

u ki

#n

(=>

) : Gia si

jf AB

1 CD C

D

± (ABH)

H

la chan dUcm

g cao BH =>

C

D

± AH

Ap dun

g he thiJc luan

g tron

g ta

m gia

c vdri

M

la trung die

m CD

f

A C'

- AD

' -_

2 CD

.M

H

=> <i

iB C'

-B D^

- 2C D.

MH

=> AC'

- AD'

= BC'

- BD' (dpcm)

• Di^

u k i$

n

(<=) : Gia AC'

- AD'

= BC'

- BD' (1)

Gp

i

M

la trung die

m CD, AH

j, BH

2 l

a cac cha

n di/

dn

g cao

.MH7

( 2)

tuon

g tfn

g, ta

C

O :

\

;B C'

- BD '

= 2

CD M H2

( 3)

S ijf dun

g (1) ch

o (2) va (3) :

2C D.

MH,

= 2

CD M H2

=>

M

H,

= MH

2

<= >

Hi =

H

2

(4)

Nghia la

\.

\S

(4) ta c6 C

D

1 (ABHi)

^

(A BH 2) ,

ca h

ai ma

t phSn

g (ABHi);

(ABHj)

o CD

1 AB (dpcm)

O Ke

t lu gin

: Die

u kien ca

n va d

u (da

i so) de h

ai can

h do

i AB v

a CD cu

a tiJ d

g go

c nhau la AC'

- AD'

= BC'

- BD' (dpcm)

Ba

i

67

Chiifn

g minh rkn

g ha

i can

h do

i b

at ky ciia tu

f die

n de

u th

i vuong g

oc v<J

i nhau

m can

h AB Theo tin

h cha

t cua ta

m gia

c de

u

d

cac ma

t ti

l dien:

'A

B

^ C E

c

( CD E)

|A B D

E C ( CD E)

=> A

B

1 (CDE);

m

a CD

c (CDE)

AB

1 CD (dpcm)

Tuan

g tir t

a chiifng min

h difa

c B

C

1 AD v

a AC

L

B

D (dpcm)

O Ghi ch

u

: B

oc gid xe

m

eac

h

ch iin

a i

d ii

c tn

g th

nh

ch

a t

nay, ch un

g toi xi

n nh

d c : rdt ti

^n ich tr on

g qua tr

c til d ien dS

u

Doc gid cU

ng c6 the d un

la

B ai

66

36

Chufng m i n h rSng : T r o n g mot t i l di$n neu c6 2 cSp canh doi vuong goc nhau t h i c&p can h

doi thuT ba cung vuong goc nhau

Dang 2 : CHUfNG MINH HAI OUCiNG THANG VUONG GOC NHAU

L P H O D N G P H A P

Ca sd cua phuang phap can v a n dung d i n h l y ba dudng vuong goc n h u sau :

, » , T , > I A M la dildng xien (so vdi (a))

Gia stf A H 1 (a) => < ,

; H M la hinh chieu (ciia AM xuong (a))

t h i difcrng (d) n'km t r o n g (a) thoa :

(d) 1 A M (ducrng x i e n ) c* (d) J. H M ( h i n h chieu)

Do do philcfng phdp gom 2 budc thuc h a n h :

• B i : Xac d i n h dudng vuong goc v d i m o t m a t phSng (a) t ^

n cA

c BA

I T OA

N C OBA

1 (ABCD)

De

y tha

y : S

A

1 (ABCD)

Ma B

C

1 AB

Ch

o hinh chop S.ABC

D c6 SA

1 (ABCD) va ABC

D la hin

h ch

a nhat

Chuto

g min

u I

k

nhflng tarn gia

c vuong

JA

S A

B vuong d A (ycbt)

ISA

1

AD ^ [ASA

D vuong d A (ycbt)

SB : l

a d

U cJn

g xie

n

AB : l

a hin

h chieu

=>

B

C

1 SB (dinh l

y b

a dtfcrng vuong g6c)

=>

ASB

C vuong

a

B (ycbt)

Tiran

g t

ii DC

1 SD (dinh l

y 3 dudng vuong goc)

=>

ASD

C vuon

g d

D (ycbt)

To

m la

i hin

h chop c6

4

m &

t

ben deu la nhifng tam gia

c vuong (dpcm)

n ABCD c6 A

B _ L CD

vk

A

C

1 BD Go

i

H la hin

h chie

u cua

A xuon

g (BC

g

H la tr

Uc ta

m ta

m gia

c BCD v

a A

D 1 B

i :BH sa

CH saochoCHnBD = C

i

DH sa

o ch

o D

H n B

C = D

j Ta

C

O :

CD ± AB (g

ia thiet)

CD

1

AH (v

i A

H 1

(BCD))

=> CD1(AB

H) 3B

H

=> C

D _ L BH t

ai

Bi ha

y BB, la dUcfn

g ca

o ABCD

(1 )

Ti/on

g t

a B

D

± (ACH) z C

H BD ± CH t

ai Cj

=> C

H l

a dudng ca

o ABCD (2

)

Tii

(1) va (2) cho ta :

H la tr

Uc ta

m ABCD (dpcm)

AD

j

: \k

dudng xie

HD, : l

a hin

h chieu

Ma B

C

1 HD

i (

vi

H la trifc tam ABCD)

=>

B

C

J _ AD

j

=> B

C

1 (AHDi)

^ (ADDi) 3 A

D BC ± AD (dpcm)

ai dUdn

g vuong g

oc

v6i

m

at ph

^n

g hinh ch

C ? nh

at ABCD MQ

t Cx, Dy t

g min

h ran

g ABEF la hin

h ch

Q nh

^t

Gi

ai

Bi

y EC

1 (ABCD) EB

: I

k

udng xie

h chieu

Ma A

B 1 C

B

=>

A

B 1 E

B (1)

Theo din

h ly gia

o tuye

n song son

g th

i ABEF la hin

h binh

h^nh v

a n

o thoa (1) ne

n la hin

h ch

if nh

at (dpcm)

O Ca ch k ha

c

: D

l y tha

y DC

1 (EBC) ma AB/

/C

D

=> A

B X

(CDE

)

r

EB =>

A

B 1 E

B (1)

=>

(dpcm)

Bai 73

Trong h i n h chop S A B C D d a y la h i n h chC n h a t A B C D G o i S H la diTcfng cao h i n h chop va

SK; SL thijf t u la dirdng cao cac t a m giac S A B va SCD Chijfng m i n h r S n g H , K , L t h S n g h a n g

D e y v d i S H I ( A B C D )

G i a i

fSH : la dudng xien

^HK : ]k hinh chieu

MaAB 1 S K (each difng) A B i H K ( d i n h l y ba dif&ng vuong goc)

TucJng t u C D ± H L ( d i n h l y ba di/cing vuong goc)

Cho tiJ dien SABC c6 A B C l a t a m giac deu canh a, cac mftt (SAB); (SBC) v a (SCA) h a p v d i

(ABC) cac goc bftng nhau v a bftng a

1/ Chiitng m i n h rftng : h i n h chieu H cua S l e n (ABC) l a t a m du&ng t r o n n o i t i e p AABC

2/ Tinh tong dien t i c h 4 mftt cua tuT dien S.ABC

: 1

1/ Goi 1, J , K I a n lifot la h i n h chieu ciia H l e n BC; C A ; A B

Do d i n h l y ba dUcfng t h f t n g vuong goc

BC 1 S I ; C A 1 SJ; A B 1 S K

Do d6 goc p h i n g cua cac m f t t ben (SBC); (SAC) v ^ ( S A B )

tao vdi (ABC) I a n lugt la S ^, SJi?, gS^

g l i l =S3tl - SKfi = a

De y t h a y t a m giac vuong S H I , S H J , S H K bftng nhau n e n :

H I = H J = H K

Vay, H l a t a m dUdng t r o n HQI t i e p AABC ( H cung l a t r o n g t a m , true t a m , t a m difdng t r 6 n

ngoai tiep cua AABC) (dpcm)

2/ Theo d i n h l y dien t i c h v a h i n h chieu t a c6 : •JSHBC = ^sac cos a <=> -ISSBC