tuyển tập 500 bài toán hình không gian chọn lọc nguyễn đức đồng

De phuc vu cho cac do'i tUcfng t\i hoc : Cac bai giai luon chi tiet va ddy du, phan nho tCrng loai toan va dua vao do cac phucfng phap hop l i.. Mac du chiing toi da co g^ng het siic tr

• BOI Dl/dNG HQC SINH GIOI

BOG

BAN GIAO V I E N NANG K H I E U TRl/CfNG THI

TUYEN TAP 500 BAITOAN

•

HDIH imm GIAN

C H O N LOG

• •

• PHAN LOAI VA PHUdNG PHAP GIAI THEO 2 3 CHUYEN

• B o i difdng hoc sinh g i o i

• C h u a n b i t h i T i i t a i , D a i hoc va Cao d a n g

(Tdi ban idn 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

n ta

p Che ban : (04 ) 3971489

-6

Hanln chinli : (04

) 39714899

; Ton

g Bie

n tap : (04 ) 3971501

1

• Fax : (04 ) 3971489

9

* *

*

Chiu trdch nhiem xuat

ban:

Gidm doc

- Tong bien tap:

T

S

PHAM THI TRA

M

Bien tap:

THU

Y HO

A

Saa bdi:

THA

I VA

N

Che ban:

Nha sach

THAI

AN

SACH L

L 195OH2014

-In 1.00

0 cuon, kh

o 1

7 x

24cm tai Con

4 2014/CXB/01-127/OHQGHN

ngay 10/03/2014

Quyet din

h xua

t ba

n so: 198L

K TN/ QO

- NXBOHQGH

N nga

y 15/04/2014

in xon

g v

a nop

NhSm phuc vu cho viec ren luyen va on t h i vao Dai hoc bkng phucrng phdp t i m

hieu cac de t h i dai hoc da ra, de tiT nang cao va chuan bi kien thiJc can thiet

De phuc vu cho cac do'i tUcfng t\i hoc : Cac bai giai luon chi tiet va ddy du, phan

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

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

tranh khoi nhiJng thieu sot Chiing toi mong don nhan moi gop y, phe binh tii quy

dong nghiep ciing doc gia de Ian xuat ban sau sach ducfc hoan thien hcfn

Cuoi Cling, chiing toi xin 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 giiip da chiing toi moi mat d l bo sach dUdc ra dcfi

NGUYEN DtfC DONG

BANG K

E CA

C KI HIE

U VA CHLf

VIET TAT TRONG SAC

H

CAC K

I HIE

U TOA

N HO

C v A

CAC Tl/

VIET TAT

(i)

•

< => : (i

ABC)

; (EFG)] :

va

(EFG)

• =

> : (i) ke

dilcfng

• C > :

g tru

e A

• = : don

g nha

t

• D

o : Phep

doi xiiTn g

true 0

• i : khon

g don

g nha

t

• Q (0; cp )

0; k) : Phep v

i t

u tam

C

• D

N : din

h nghl

a

• St p

: Die

n tic

h l

y

• Sx q

: Die

n tic

h xun

g quan

h

• H Q

h

• CMR : chiJn

g min

h ri

ng

• A' =

''7(ai

A : A' l

a

hi nh

xuong dtfcfn g

thftng (d)

• V

T :

ve tr

ai

• d[M

• BDT : ba

t d in g

thijfc

• d[M

; (ABC)

I : khoang eac

h tii

diem

M

• y cb t

: ye

u ca

u ba

i toa

• d pc m

: die

u pha

i chuCn g

m

in

phang (a) v

a (P)

• g t

: gi

a

th ie

t

• (S

; AB; D) = (AB) : n

hi die

n c an h

B

• K

L : ket lua

hai dUom

g t hi ng

d

• D

K : dieu kie

n

va d'

• P

B : phan ban

• [HTCABCT

d

• CPB : chiT

a pha

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 khong gian la mot mon hoc ve cac v $ t the t r o n g k h o n g g i a n ( h i n h h i n h hoc

trong khong gian) ma cac diem h i n h t h a n h nen vat the do thudng thiTcrng k h o n g ciing nftm trong mot mat 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 gian con xay di/ng t h e m mot doi tuong can nghien ciifu nCfa la

k h a i n i # m mgt 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 ta

lien tuang den mot mat ban bang phang, mot mat ho nildc yen lang, mot 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 ky hieu bdi cac chCf i n L a T i n h n h a : (P), (Q), (R), hoac cac chCf thudng H y Lap nhU (a), ((5), (y),

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

ph&ng CO nhutng t i n h chat cu the 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 diem d ngoai mot mat ph^ng tiiy y)

O T I E N DE 2: N e u mpt dtfdng th&ng v a mpt m a t p h ^ n g c6 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 3: N e u h a i m a t ph&ng c6 d i e m c h u n g t h i c h t i n g c 6 v 6 so' d i e m c h u n g :

n e n h a i m a t p h S n g do c S t n h a u theo m p t dUdng t h ^ n g d i q u a v 6 so' d i e m

c h u n g a y Di/cfng t h a n g ay goi la giao tuyen cua h a i mat ph^ng

O T I E N D E 4: C o m p t v a c h i m p t m $ t p h a n g d u y n h a t d i q u a b a d i e m p h a n bi#t

k h o n g t h ^ n g h a n g

O T I E N D E 5: T r e n m p t m § t p h a n g t u y y t r o n g k h o n g g i a n c a c d i n h ly h i n h h o c ph&ng scf c a p (da hoc tCr Idp 6 den Idp 10 va cac d i n h l y nang cao) d e u diing

O T I E N DE 6: Moi doan th&ng trong k h o n g gian d e u c6 dp d a i x a c d i n h : tien de neu

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

• TiT do chung ta c6 mot so each xac d i n h m a t ph4ng n h i / sau :

O H E Q U A 1: C o mpt v a c h i mpt mfit p h S n g d u y n h a t d i q u a m p t d U d n g t h S n g v a mpt d i e m n S m n g o a i dt^dng t h a n g do

O H E QUA 2: C o mpt v a c h i mpt m^t phdng duy nhat di q u a h a i di^cAig th^ng cSt n h a u

O H E Q U A 3: C o m p t v a c h i mpt m ^ t p h a n g d u y n h a t d i q u a h a i di^c/ng t h d n g song song

• Dong thdi ta phai hieu t h e m r k n g mot mat phang se rong khong bien gidi va dUcmg t h ^ n g c6

do dai v6 tan mac du ta se bieu dien no mpt each h i n h thiifc hflu han va k h i e m ton nhU sau:

• De thuc hien dirge phep ve c h i n h xdc m6t h i n h h i n h hoc trong k h o n g gian ngoai cac dudng thay ve lien net, ta can phai nam chac di/pc k h a i n i e m di/dng khuat ve bkng 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 mpt d o a n c u e bp n a o do k h i v a c h i

k h i ton t a i it n h a t m p t m a t p h S n g du'ng p h i a trvC6c h o ^ c p h i a t r e n c h e n o mpt

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

• Muo

n xa

c din

h nh

^n

h mo

t ma

t ph^

ng trong khong gian ta con chon thu thuat thU

c h

an

h :

Mpt hi nh t am g ia

c, tii

" g ia

c ho ac d

a gia

c ph&n

g (k ho ng genh), dUcfn

h m

pt m^

t ph Sn

g tr on

g kh on

g gia

n

Ta gpi ca

c m&t

p h^

g hi nh

thvCc v

di ca

c k

y h ip

u (A BC ), (A BC D) , (C ),

txictng

vtng

• Ma

t ph dn

g hi nh thu^c

hi kh ua

t n eu c

6 mp

t h ay n hi eu m^t ph&n

^m t ro ng m

^t ph&n

g h in

h thd'

c m

a ma

t d

o

hi kh

a kh

i dUcTn

g t h^

ng do k ho ng la b ie

n cu

a ma

t p hd ng

tii'oTng vlng

k hu

at cu

e b

p ha

y to an

bp

• Mp

t die

m nhm

t ro ng mpt m$

t ph&n

g h in

h thuT

c b

i k hu

at th

i go

i la d ie

m khuat

• No

i ha

i d ie

m m

a i

t n ha

t c

6 mp

t die

m k hu

at th

i dUp

c mp

t dUcfn

g kh ua

ai diicta

g d

o k ho ng la b ie

n c ua c ac m^t phAn

g h in

bi (a) ch

e khua

t cu

e bo, d

o (d) c

6 1 doan

ve

net dijf

t doa

n nkm dudi (a)

S

• (d)

bi ma

t ph

^n

g (SAC) ch

e khua

t cu

e bo, d

m sa

u (SAC) (hie

n

nhie

n (d) cun

g d sau cac ma

t (SAB), (SBC))

• Canh AC b

i h

ai ma

t phan

g (SBC) v£

l (SBC) ch

e

khua

t toa

n bo, d

i ha

i m

at ph

^n

g (SAB), (SBC)

-A

A

c /—

1—

^V FJ L^

• A]

phin

g (AiADDi), mSc di

j n

o d trU<Sc

ma

t phan

g

(ABBjAi) v

a d tre

n m

at phan

g (ABCD)

• (d)

bi ch

e khua

ve net

duTt

doan nam sau h

ai ma

t phan

g (ADDiAj);

(CDDjCj),

mac di

j doa

n EF

t phan

g

(ABBjAi);

(BCCiBi);

va

d tre

n m

at phan

g (ABCD)

• CA

C K

Y HI EU

AN ^fHd

Thiir tr

f Ky hie

u

Y nght

a Gh

m A thuoc ducfn

g thing (d) ha

y dadn

g

thin

g (d) chur

a A Hay

vie

2

A i

(d) Die

m A or ngoai difdn

g thin

g (d) ha

y

dUcfng thin

g (d) khon

g chtif

a A Ha

y vie

d)

3 (d) c (a) DU&ng

thin

g (d) nk

m tron

g ma

t phin

g (a)

hay (a) qua

y quan

h (a) ne

u (a) lu

u dong Hay

viet nh

am

la

:

(d) e (a)

4 (d) / / (a) Difcrng

thin

g (d) son

g son

g \6\

t ph in

g

(a)

Cach viet kha

thin

g (d) ei

t ma

t phing (a) ta

i A each vie

t khd

c :

(d) n

(a) = {A}

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

Cling chijfa 3 diem A, B, C phan biet k h o n g

t h i n g hang

Cach viet khac : (a) = (p)

C phan biet va k h o n g t h i n g hang

(ABC) : la m a t p h I n g

h i n h thijfc vdi ba

dildng bi&n A B , BC,

AC

ducfng t h i n g (d) khong qua A

Co so cua phiiong phap t i m giao tuyen cua

hai mat p h l n g (a) va (P) can thUc hien 2 budc

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

n Ba : Difdng t h i n g A B la 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 diem chung S

• Luc nay ta c6 h a i trifcfng hap :

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

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

=> SI la giao tuyen can t i m

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

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

S _

Difng xSy song song vdi (dj) hay (da)

=> xSy la giao tuyen can t i m

m CA

C B AI T OA

a (SBC)

a (SBD), t

a c

6 :

• S

la die

g

t uT giac lo

i ABC

va (2 )

suy r

a :

(SAC) o

(SBD) = SO (ycbt)

a (SDC) cun

m chung

• Ha

l gia

c AB

CD

theo gi

a thie

t khong song song

^ AB ^ CD = E : l

a die

m chun

g thu

t hai

Do d

o :

(SAB) n

(SDC) = SE (ycbt)

Tucfng tif: (SAD

D ^ BC; d

o AD/

/ BC

Bai 2

Cho ti

l die

n ABC

D Go

i Gj,

Ga l

a trpn

g tar

n ha

i tam giac BC

D v

a ACD La

g die

m cii

a B

D, AD, CD.Tim cac gia

B) hi

(G1G2B) n

(ACD) c/ (ABK

J

hi

(GiGaB) n

(ACD) = GgK hoSc

AK

d

(ABK) ^ (CIJ) =

G,G2

Bai 3

Cho hin

la hin

h bin

h han

h ta

m O

aJ

Tim giao tuyen cua ha

i m

at phSn

g (SAD) v

a (SBC)

hi

Tim giao tuyen cua ha

i mS

t phin

g (SAB) v

a (SCD)

c/ Ti

t ph^

ng (SAC) v

a (SBD)

a (SBC), t

• D

e y A

D c (SAD); B

C c (SBC) m

a AD // BC

Ta dun

g xS

y // A

D hoa

c BC

[(SAD) = (xSy; AD)

^ |(SBC) =

(xSy; BC)

=^ (SAD) n (SBC) =

xSy (ycbt)

hi

Tifang tir, difn

g uS

v // A

B hoft

c

CD

8

=> (SAB) r^ (SCD) = uSv (ycbt)

c/ Goi O = A C n B D , tiTcrng t a bai 1

=> (SAC) n (SBD) = SO (ycbt)

B a i 4

Cho h i n h chop S.ABCD c6 day la h i n h t h a n g A B C D vdi A B la day Idtn Gpi M la mot diem bat ky t r e n SD va E F la difang t r u n g b i n h cua h i n h thang

a/ T i m giao tuyen ciia h a i mSt p h i n g (SAB) va (SCD)

b/ T i m giao tuyen cua h a i m a t phSng (SAD) va (SBC),

c/ T i m giao tuyen cua h a i mSt p h a n g ( M E F ) va ( M A B )

Doc gia t u giai tUcfng t u n h u cac bai t r e n

Goi I , J , E, F thur t a Ik t r u n g d i e m cac doan t h i n g A D ,

BC, SA, SB theo thur tvt d6 Thifc h i e n cac lap luan nhtf cac

bai toan t r e n ;

a/ (SG1G2) n (ABCD) = I J (ycbt)

b/ (CDGiGa) n (SAB) = E F (ycbt)

c/ (ADG2) ^ (SBC) = xG2y (ycbt)

T r o n g do xGay // A D hoSc BC

L o a i 2 : T l M G I A O D I £ M C U A D U d N G T H A N G 1fA M A T P H A N G

Ca sd cua phaang phap t i m giao d i e m O cua dudng t h a n g

(a) va mat phSng (a) la xet 2 h a i k h a nSng xay r a :

n Trirdng hop (a) chiJa dudng t h S n g (b) va (b) l a i c&t diicrng

Giai

• D

e y den KB > K

D =

>

KN khong son

o C

D = I

Ma KN

c (MNK) C

D (MNK) =

I (ycbt)

g (ADC)

Ta

CO

: AD

n (MNK) =

E (ycbt)

Bai 7

Cho ti

J die

n ABCD La

y die

m M tren A

HtfdTng di

n

Doc gi

a t

u giai, xe

m hin

h ben

a/ C

D (MNK) =

P (ycbt)

b/ AD

n (MNK) =

Q (ycbt

)

Bai 8

Cho hin

h cho

p tu

f gia

c S.ABCD La

y tre

n SA, S

ii t\i

sao

cho MP khong th

e c&t

A

B ha

y CD Tim giao die

n (MNP) =

K (ycbt)

Trong mp(SAC) M

K o AC = H

1

ma M

K c (MNP) |

=> A

C r

> (MNP) =

H (ycbt)

Bai 9

Cho mo

t tam giac AB

g chil

a ta

m giac Tre

va tron

g ma

t phin

g (ABC) t

a la

y mo

t die

m O Din

B, BC, A

C v

a SC

Hi^dn

g di

n

Tuang tu, do

h ben)

AB

n (MNO) =

E (ycbt)

BC

o (MNO) =

F (ycbt)

AC

n (MNO) =

G (ycbt)

SC

n (MNO) =

H (ycbt)

10

Loal 3 : Cfll/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 pmroNG P H A P

Co so cua phiiong phap can phai chufng m i n h ba diem

trong yeu cau b ^ i toan la d i e m chung cua 2 mSt phSng nao

do (chfing ban A, B, C nSm t r e n giao tuyen (d) cua h a i m a t

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

O day k h o n g loai triJ k h a n&ng chiJng m i n h difoc difdng

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

n CAC B A I T O A N C O BAM

B a i 10

Xet ba diem A, B, C k h o n g thuoc m a t p h i n g (u) Goi D, E, F Ian luot la giao diem ciia A B ,

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

G i a i

De y thay D, E, F viTa a t r o n g mp(ABC) vifa d t r o n g mp(a)

Do A, B, C g (a), nen (a) va (ABC) p h a n biet nhau

=> (a) n (ABC) = A (A chuTa D, E, F)

to al

4 :

CmiUG

MWfl MQ

T DtfCiN G

T HA NG

T RO NG KHONG

GIAN

QU

A MO

T DI

£M C

h diXcrn

g thing (d)

qua mo

t die

m c

o dinh :

Ta can tim tre

n (d) h

ai die

m tu

y y A; B va chuTng mi

g vd

i mo

t die

m I

co dinh c6 sS

n tron

g

khon

g gian

=> (d) qu

a I CO din

h (dpcm)

IL PHtfONG

PHAP,

Co sd cua phiTcfn

g pha

p ca

n thu

c hie

thin

g a

co dinh

a (a) chil

a d (liOi dong)

• B

2 : Tim giao die

m I = a ^ d

=> I la die

m c

o dinh ma

d d

i qu

a

m CA

C B AI

OA

N CO

ai die

m c

o dinh trong khong gian

t die

m M luu dong trong khong gian sao ch

o MA

n a = I va M

m c

o din

h

Giai

Goi O = A

B n (a) =

> O

co dinh (

vi

A, B

co dinh

vk

a 2

phia cua (a))

Ta CO : mp

(P) =

(MA

; MB) n (a) = I

J

De y tha

y :

O e I

J =

> O,

I, J thing hang

Nghia la dacfng thin

Bai 1

4

Cho hin

h thang ABC

D (A

B // C

D v

a AB > CD) Xe

t die

m S

e

(ABCD) v

a ma

B = M, a

n S

D = N ChuTn

g minh difdng thing M

m c

o din

h

De thay

dxiac nga

y MN

c (SBD) v

a

AC

c (SAC) v

a MN

o AC = O t

hi O

e

BD = (SBD) n (SAC)

=> MN qua O

co dinh (dpcm)

12

Bai 15

C h 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 (xOy) 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 mot d i e m co d i n h

Co so cua p h i f a n g p h a p l a t a c a n chiifng m i n h d U d n g thiif n h a t

qua giao d i e m ciia 2 d i f d n g c o n l a i b a n g 2 budrc co b a n :

O Ca ch k ha

c

Doc g ia churng m in

h r Sn

g J

F q ua

O = I

G n C

D =

> C D;

i t am g ia

c AB C, A B

C sa

o c ho

B c

at A'

B' a

E , A

C cd

t A

C d F

a d ie

m E,

, G t hS ng h an

g

b/ Ch iJ ng m in

h difcfn

g th an

g AA ', BB ', CC d on

g quy

Gidi

a/ De

y th ay E , F , G l

a b

a d ie

m ch un

g cu

a h

ai ma

t p h^

ng p ha

n bie

t

(a ) ^ ( AB C) v

a ( P) = (A B' C)

Do do : E , F , G

e (A ) = ( a)

n

(P)

Va

y E,

, G t h^

ng h an

g (d pc m)

b/ Nh an x

et nh

u s au :

: AA' , BB

' cr

(EAA'); AA ' o B B' #

0

^ BB' , CC

c (GBB')

; BB

; CC

n AA

' #

0

^ AA ', BB ', CC d on

g qu

y t

ai O ( dp cm )

Chuyen de

2 :

QUAN HE SONG

SONG

to ai 1: CHtJN

G MWf

l HA

I DLfCJN G

T HA NG

S ON

G S ON

g ph ap c an thiic h ie

n h

ai hxidc

CO b an cho d in

h n gh ia

a / / b

• Bi : K ie

m tr

a h

ai difdn

g th an

g a

c un

g tr on

g mo

t m

at

ph an

g ha

y hife

u n ga

m ra ng h ie

n nh ie

n die

u d

o x ay

a

ne

u ch un

g tr on

g 1 h in

h ph an

g na

o do

(1 )

• B2 : Du ng d in

h ly T ha le

s, ta

m gia

c don

g da ng , ti nh c ha

t ba

c ca

u ( ti nh c

\6i difdn

g thi

J ba ) l

a h

ai ca nh c ua h in

h th an

g, ha

y h

ai ca nh doi cu

a h in

h ha

i difcfn

g th

^n

g d

o k ho ng c6 d ie

m ch un

g

(2 )

n CAC BA

p S.

AB CD c6 G

j, G2 , G3

,

G, I an lucft la t ro ng t am c ac t am giac S

A

Ch um

g mi nh

tiJf

gi ac G iG aG gG , l

a h in

h bi nh h an

h

14

Cho diem S d ngoai m a t phSng h i n h b i n h h a n h A B C D X e t m S t p h d n g 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 i n h t h a n g

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

Cho tuT dien A B C D Goi M , N I a n li^gt l a t r u n g diem cua BC va B D Ggi P l a diem

tren canh A B sao cho P ?t A va P # B X e t 1 = P D A N va J = PC o A M

Co so ciia phuong phap m o t l a sii dung d i n h l y phuong giao tuyen song song

De chiing m i n h d // a t a can thUc h i e n h a i bade CO ban chufng m i n h : d

• E l : Chufng m i n h d = y o p m a

• B2 : K e t luan t i f t r e n d // a

y r- a = a

p n a = b a//b

g die

u kie

g (d) son

bang h

ai btfdr

c :

•

Bi : Quan sat v

a qua

n l

y gi

a thie

t tim dudng thing

• B

2 : K

et lua

n (d) / / (a) the

o die

u kie

n ca

n v

a dii

m cA

c BA

I T OA

N CO B

AM

Ba i2

1

Tron

g tu

f die

n ABCD, chufn

g minh rSng dean no'i ha

i tron

g ta

m Gi,

G2

cua ha

i A

ABC

va

AABD t

, A

2 l

a trun

g die

m B

C v

a BD theo thut ti

T do, t

Theo din

h l

y Thales, t

a c

6 :

'0,02/

/A, A

2 B

'm

a A,A2 //CD (tinh chat dUcrn

g trun

g binh)

Theo tin

G1G

2 /

/ (ACD) (dpcm)

Ba

i 2

2

Cho hinh chop S.ABCD

day la hin

h bin

h hanh ABC

D Go

i

M, N la trung diem SA

va

SB

Chijfng minh : M

N / / (SCD) v

a AB //

NCD)

Gi

ai

Theo tin

h cha

t dudn

g trun

g bin

h trong tam giac

=>

MN //

A

B, m

a AB //

CD

=>

MN //

CD

cz

(SCD)

Theo dieu kie

O Ca ch k ha

c

De y M

N = (MNCD) n (SAB) v

a tron

g ha

i ma

B D

MN //

B v

a CD =>

MN //

(SCD) 3 CD (ycbt)

TifOng tyl

:

AB //

N c (CDMN) =

>

AB //

CDMN) (dpcm)

Ba

i 2

3

Xet ha

i hinh bin

h hanh ABC

D v

a ABEF khong dong phlng Go

i

M, N la h

g minh rin

g M

N / / (DEF)

3 3

Gi

ai

De y thay M, N la trong tam cua ba

i ta

m gia

c ABD

Keo da

i th

i D

M o E

N =

P : la trung diem A

B

PM PX

h l

y Thale

s

^ MN //

D c (EFDC) ^ (DEF) (dpcm

) D

16

n ckc

B AI

OA

N CO

a t ia c un

g ch ie

u, so ng s on

g va Ichong d on

g ph

^n

g Ax , By , C

B' = C

C c

6 d

o d

ai kh ac k ho ng ChOfn

g mi nh ( AB C) //

C // AC

c

(ABC)

Ne

n ta c6

ai dUcrng t h^

ng d on

g qu

y A B' , A

C

tr on

g mp (A 'B 'C ') th oa d ie

u ki en ( I)

=> (A B' C) //

(A BC ) (d pc m)

Bai 2

7

Ch

o hi nh b in

h ha nh A BC

e A

x c

a Cy

on

g so ng c un

g ch ie

g ma

t p hS ng ( AB CD ) Chiifn

g mi nh ( B;

x) //

(D

; C y)

Gi&i

Ti ra ng t

u x et

ai m at

hi ng ( B;

A x) v

a

(D

; Cy ), thu

T ta chuTa ca

c c ap d ud ng t hi

ng

do ng quy

fAB//CD IAx//Cy

=> (B

; A x) //

(D

; C y) ( dp cm )

Bai 2

8

Ch

o ha

i h in

h bi nh h

^n

h A BC

D va A BE

F d

ro ng h ai m at p h^

ng k ha

c nh au

Giai

Ha

i ma

t ph li ng ( AD F) v

a ( BC E) thiif ti

T chuT

a ca

c c ap dirdng

th dn

g do ng quy / A

;

iAF//BE

AD//BC

(A DF ) / / ( BC E) ( dp cm )

D an

g 2 : C

H UfN G MINH CA

C D

l /d

N G TH

A NG D6N G PHANG

LP Bi rO NG PB

AP

Ccf sd

ua p hu an

g ph ap chiifng m in

h ca

c d ud ng t hi ng d

i, d2 , dg

d on

g ph in

^n h ai bi/<Jc ca

an :

• Bi : C hv in

g mi nh d ], dg , ds ,

d

oi mo

t c

at nh au v

a c un

g so ng s on

g vd

i m

pt ma

• B2 : Ket luan d], d2, ds, c (a) // (P) => di, d2, d^j, dong phing trong (a); (a) phai

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

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

Do cac tam gidc can tai dinh A nen cac

phan giac ngoai song song vdi canh day, nen :

At, //BC c (BCD)

AtaZ/CDe (BCD)

;At3//BDc(BCD)

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

Htfdng d i n

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

DE SH L i"

HA IJ ES T RO NG K H6 NG G IA

N

•k

Di nh l yi (thu|ln) :

Hai ditang thing tuy

y d,, d2 trong

khong gian chdn tren

cdc in^t phdng song

song nhau (a)

II (P)

II (y) tao ra cdc doan thang tUcmg

• TrUd

c kh

i x

et di nh l

y dao , t

a qu an tarn d en h

ai kh

ai

ni em s au

hi xe

t d en

ac d ay t

y so , c hi ng h an :

A, A

B1B9

• (A i;

B]

) l

a c ap goc c ua d ay t

y s

o (

*)

• (A2

; B2 ) v

a (A3

; B3 ) l

a c ac cStp ngo

n cu

a d ay t

y s

o (

*)

• Do an n

oi ca

n la (doan) ba

c t ha ng c ua d ay ty so (*)

• Di nh l

y :

Neu c6 day ty

so trong khong gian

:

A,A

(*) da, xdy ra tren hai

dudng

thdng (d,), (d2) thi

cdc bac thang AiB,,

A2B2, A3B3 cung song

song vai mot mg,t

phdng c6

dinh

O Gh

i c hu : Ta c6 p ha

t bie

u kh ac c ua d in

h

ly

Th al

es da

o nhi

f s au :

Vai dieu kien c6

day ty

so (*)

da xdy

ra tren hai

dudng thdng (dj), fd^

) thi

mot trong

3 bac

thang A,B,, A2B2, A^B^

se song song vdi

mot

mat phdng chda hai

bac thang con lai

"A ,B

i //(a ) = (A2B2 ; A3B3)

A2 B2 //

(P )- (A 3B 3;

Ai Bi ) (A )

A3 B3 //

(Y ) S ( A,

Bi ; A2B2)

(di) (goc)A

i

(ngpn tren) A

2

(ngon difdi) A3

L PHirON

G PH AP ,

Co s

d c ua p hu on

g ph ap chufng mi nh dUdng t hi ng s on

g so ng v

di mS

t p hi ng b

h l

y T ha le

s da

o t ro ng k ho ng g ia

n go

m h

ai bud

c c

a b an s au

ay :

• Bi : X ac d in

h t re

n h

ai du dn

g t hi ng tiiy y c ha ng h an

di) , (d2 ) d

e t im

Aj As B,B3

Xd

c d in

h c Sp ( Ai

; B j)

a c Sp goc, cd

c c Sp ( Aj

; B2 ) v

a (A3 B3 ) l

a h

ai cS

p ngon

n B2 : L uc do c ac doan b ac t ha ng A iB

i, A2B2 , A3B

3 dira

c ke

t 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 a m t r o n g m a t 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 dien A B C D c6 A B = CD Goi M va N la hai diem lUu dong t r e n A B va CD sao cho

AM = CN Chutng m i n h M N luon song somg vdi mSt p h I n g co dinh^

Giai

Neu dat A B = CD = a; A M = C N = x De y thay t r e n A B va C D ta co day t y

AM ^ CN

AB CD

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

thang AC, M N va B D ciing song song v<Ji mot mSt phIng (a)

due nay (a) chUa co d i n h v i day t y so — chUa la h k n g so)

a

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

DC, CB theo thuf t i l do t h i (a) = (EFG) va (a) thoa yeu cau la

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

Bai33

Cho hai h i n h binh h a n h A B C D va A B E F k h o n g dong phIng; tren cAc dUOng cheo A C va

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

Cho h i n h vuong A B C D va A B E F d t r o n g hai m a t p h I n g khac nhau T r e n cac difdng cheo

AC va BF, ta Ian lugt lay cac d i e m M , N sao cho A M = B N Chutng m i n h r a n g M N // (CEF)

Ap dung d i n h ly Thales cho cac doan bac t h a n g :

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

^ M N // (CEF) (dpcm)

Bai 3

5

Tr en h

ai ti

a Ax v

a By c h6

o nh au , ta I an

luat

la

y ha

i di em

M N sao ch

o AM

Chufng m in

h rS ng M

N lu on lu on s on

g so ng v

di mp

t mS

t ph Sn

g c

o di

nh

Tr ad

e he t:

HUoTng

dim

By la'

y d ie

m N , din

h bd

i : B

N, =

1

Ax la

y die

m

Mj din

h bd

i : A M; =

k (v

i k > 0 , ch

o tn/dc )

Hi en n hi en 1^

h

ai di em M

j \k

N ] c

o di nh

Th eo g ia t hi

et va

tii

ea ch d an

g tr en h in

h ta

co :

AM , AM , AM

i BN

i

BN ,

BN AM

B

N

Ne

n th eo d in

h ly dao cu

a di nh l

y Th al

^s M

N

lu on lu on s on

g so ng v

di mS

t ph in

g c

o di nh ((5) =

(A

; Bd ) ch aa A

B va d ad ng t hi ng

d qu

a B s on

g

so ng v

di N, M]

(d pc m)

Bai 3

6

Ch

o ha

i da dn

g th in

g ch eo n ha

u

dj va

d2 M

la m

ot di em c hu ye

n do ng t re

n do ng t re

n d2

T

im q

uy ti eh t ru ng d ie

m I cua do an M

N

Hu&ng dan

Go

i AB la d oa

n vu on

g go

e ch un

g cu

a

di va

d2

(

A e d

j, B

d2); O

la t ru ng d ie

m eii

a AB

Ta eo : = = i

di

d'2

O M'

N'

OB I

X

Th eo d in

h

ly

T ha le

s da

o th

i 01

nkm

t ro ng m

at ph ln

g

(P )

q ua

O so ng s on

g vd

i

di va

da, tiif

c la m

at ph an

g xa

c

di nh b

ai ha

i da dn

g th in

O Ia

n lacr

t so

ng

so ng v

di d ] va

d-^

Giai han

; M v

a N c ha

y tr en d ] va k ho ng

co ra

ng

huge ne

n I c ha

y tii

y y t re

n (

P)

• Dao :

L ay m

ot di em

a dO ng dacfng t hi ng s on

g

so ng v

di d

'2,

da dn

g th in

g na

y ci

t

d'l

t

ai E La

y di em M ' e d

'l

do'i xO ng vdri O qua

E

MI c

lt d

'2

d N'.

D in

h

ly

d ad ng t ru ng b in

h ch

o th ay

I

la

tr un

g di em cua M'N ' TC

f M' va N ' dif ng cac da dn

g th in

g so

ng

so ng v

di AB Ch un

g Ia

n la

gt el

t d] d

M va

c OM 'M

A va O N' NB d eu la n hf ln

g hi nh c hS n ha

t :

=> , ^ ^ ^ _

^ MN 'N M' la m

ot hi nh b in

h ha nh

do d

o I la t ru ng d ie

y ti eh t ru ng d ie

m I cua do an M

N

la

m

at ph in

g

(P )

i qu

a O s on

g so ng

Chuyen de 3 : PHlTONG PHAP T I E N D E

Ta da t h a y di/gc k h i giai toan h i n h hoc t r o n g k h o n g gian tif h a i chuyen de trUdc mot each chi/a tiidng m i n h iSm viec sCf dung h a i t i e n de 5 va t i e n de 6 the nao ?

Den day, de" khftc phuc viec do Chung toi diTa vac mot chuyen dfe PHUCfNG PHAP TIEN' DE vdi mot mong muon la doc gia se thiTc sa thay difoc mot each c h i n h xac han, tUdng m i n h hon :

si^ c a n thiet c u a t i e n de 5 v a t i e n de 6 Hien n h i e n viec gidi thieu rong r a i nhif the doi hoi

doc gia can chuan bi mot i t kien thiife ve sir vuong goc va nhOmg k h a i n i e m ve cac h i n h khoi Sau nhOrng suy n g h i va t r a n t r d t r o n g suot qua day hoc va viet sach chiing toi hy vpng duge doc gia dong cam vcii vi$e dat chuyen de 3 a v i t r i nay t r o n g quyfin sach c h i mot each

lidc le cung la du

Co so ciia phifong phap la sii dung sii c^n t h i e t cua h a i t i e n de 5 va t i e n de 6 d4 xay diTng

va chufng m i n h mot so bai toan co ban t r o n g k h o n g gian k h i h i n h t h a n h nen cac v a t the (hien

nhien 4 t i e n de d trirdc da duoc ngam hieu la luon luon di/gc sOf dung)

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

B a i 37

Cho a, b, c la ba difdng t h i n g k h o n g ciing nkm t r o n g mpt m a t phAng va doi m o t cSt nhau

Chufng m i n h rSng : a, b, c dong guy

G i a i

T h a t vay : gia sijf a, b, c k h o n g dong quy, t h i cac giao diem ciia chiing lap t h a n h ba diem

khong t h a n g hang va ba difcfng thftng cung nam t r o n g mot m a t phang T r a i v d i gia t h i e t

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

B a i 38

Cho 3 t i a Ox, Oy, Oz doi mot vuong goc

a/ Chufng m i n h r k n g ba t i a do k h o n g cung n k m t r o n g mot m a t phang

b/ Ijay t r e n ba t i a Ox, Oy, Oz Ian lifgt cac diem A, B, C (khac goc O) Chijfng m i n h r a n g :

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

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

va chufng to r a n g a, [3, y n h o n

G i a i

a/ T h a t vay : gia sCf ba t i a cijng thuge mot m a t

phang, v i Ox va Oy ciing vuong goc v6i Oz, nen Ox va

Oy cung n a m t r e n mot du'dng t h a n g Dieu do t r a i vdi

gia t h i e t

Do do ycbt di/gc chufng m i n h b a n g phep chufng

m i n h phan chufng

b/ Ap dung bat d a n g thufc Bunhiacovky, ta eo :

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

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

c/ Ap dung d j n h ly h a m cos cho AABC, ta c6 :

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

cosa =

< =>

a nho

n (dpcm)

Va

^ + b

^V b^

g khong gian ba tia

Ox, Oy, O

Ox, Oy, O

z pha

i don

g phSng

Gi

ai

Gia siif Ox, Oy, O

z khon

B =

1 (dvcd)

Dong thd

i tren tia doi Oz' cu

a ti

a Oz, t

a cho

n die

m C sao ch

" A

C =

OC = OAcos60

" O

C = -

2

Din

h l

y ham cosin trong

C 2OB.OCcos60°

-« BC' =

1 +

i ~ 2.1.-.i =

C = (1)

Tifong tu : AB

^ = OA' + OB^

- 2OA.OBcosl20'' =

1 +

1 2.1.1(-1

-) =

3

« AB =

Va

(2)

Ttf (1) v

a (2) t

a difg

c : CA + C

B = A

B

< =>

C e A

n dau)

Vay Ox, Oy, O

z ph

ai don

g phang (dpcm)

Ox, Oy, O

a y(5z = 90" ChiJn

g min

Hi^cTng da

n

Gia sU

Ox, Oy, O

z khon

do the

o tht

f

tu cac die

m A, B, C sao ch

o : OA = a

; O

B = OC = a

Stf dun

g din

h l

y ham cosin

=> AB = A

C = VOC^ + OA^ - 2 0C O

A.

CO S4 5"

AB = A

C =

ha^

.a ' - 2.a>/2.a ^

= =

a

Ma : B

1/ Gia sd O x ; O y ; Oz 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 Oy hoac n a m t r o n g m i e n goc xOz ( x e m h l ) hoac 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) hoac xOy = 1 5 0 ° * 6 0 ° ( h l ) (v6 l y v d i g i a t h i e t xOy = 6 0 ° )

Luc do, (1) + (2) ch

o ta gia thi§'t:

2 co

s fliOt)

1 OA OB

OC

= 1 <=>

OD

V ' ) OE

' =

1 oO

E = 2cosf

(3)

Ha

i tia OA; O

C c

o dinh trong khong gian nen tia phan giac O

D cun

g c6' din

h

Hai tia OB; O

D c

o din

h tron

g khon

g gia

n ne

n tia phan giac O

E c

o din

J

= cons

t ch

o ta die

m E

co din

h trong khong gian

Vay kh

i A, B, C lou dong sao ch

o

nhung luon qua E

co din

i m

at ph

^n

g (ABC) lii

U don

g the

o

Cho hin

h bin

h han

h ABCD Tren cac can

h SA, SB

m A], B

i, C

j sa

o ch

o SA

SC : k (k > 0, k cho san) ChuTn

i C

i thay doi th

i mp(AiBiCi) ca

t S

O ta

i 1 die

m c

o din

h (vd

i O = A

C o BD)

Gi

ai

Goi AM la trung tuyen cua ZiAB

C tu

y y con

B', C tuy

y thij

f t

ii tren A

B v

a AC (h.2)

Kh

i AM

o B'C = M', t

— (*

) (Do

c gi

a tu chufng min

Do d6, ne

u go

i S

O ^ A,C, = O', th

O

SAi SC

i SO'

(1) (h.l)

Nen SA SC ,

SAi SC]

h (ycbt)

D nh

o nhat, ha

y ch

i r

a gi

a tr

i nh

o nh

at do

Gi

ai

Goi S la dien tic

h AMB

D S = i

BD.MO =

|

aV2

M0 (1)

a

Nhun

g minM

O = dlO; SA] = O

D =

O th

i S

O la dudng cao

=> ASO

A vuon

g ta

i 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 khi 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)

Giai

• Do M, N, I thuoc hai mat phing phan bi$t (P) va (ABC)

nen thing hang

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

ma thing hang thi 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^ 3 diem th^ng

h^ng

Goi d la diTcrng thSn

g d

i qu

a A, B va

C Dudn

g thing nay

cung vd

i die

m D xac din

h mo

t m

at phftn

g (a) T

a c

6 : / ^r

v ^ ^

D; A

; B

; C

e (a) (1)

De y tha'y (1

) tr

ai vo

l gi

a thie

t "khon

g c

6 4 diem nao /

tron

g n diem

da ch

o cun

g phang"

/"

\

Vay khon

g th

e c

6 3 diem nao tron

g n diem

ay thin

g han

g (dpcm)

i tin

h cha

t l

a 4 diem bat ky nao tron

• K

hi n = 4 t

hi ba

i toa

n difan

g nhie

n dung

• Gia

h ma

t phlng (a) X6

t die

m

Aj (v(J

i 3 < i < n) The

o gi

a thie

t 4 diem Aj;

a A

j e (a) vd

i mo

i i = 4

; 5

; n

Nhir va

y t

at c

a n diem

ay thup

c mp(a) (dpcm)

i I;

J Ia

n lug

t l

a tru

ng die

m cu

a ha

n can

h AC M

at phln

g (IJM) ca

t can

h BD t

ai

N Chufn

g min

t die

n IMJ

N than

Tron

g m

at phin

g (IJM)

h l

y Thale

s da

o th

i AC

; BD

; IJ nam trong

=> O la trun

K trong tam giac MI

dt (MIJ) =

dt (NIJ)

Vay IJ chia thie

t die

n IMJN thanh hai pha

n die

n tic

h

bang nhau (dpcm)

D

Ti

m die

m M trong khong gian sao ch

o I = MA

^ + MB^

a tru

ng die

m cii

a AB; C

ng tuye'

n ch

o :

MA^ + MB^ = 2MI'^

AB-'

2

28

2

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

=> L = 4MG^ + IJ'' + > + IJ'' = hang so

2 2 Dang thufc xay ra k h i va chi k h i M = G => E = MA^ + MB^ + MC^ + M D ^ dat gia t r j nho nhat k h i va chi k h i M d G, trong tarn cua tuT dien (ycbt)

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

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

OM " ON ~ OP " 3

Vay the tich tuT dien O M N P nho nhat b^ng - abc xay ra k h i mp(a) = m p ( M N P ) dugc d i n h nhu tren (ycbt)

Chuyen de

4 :

QUAN H

E V

U6

NG G

OC

Loai

1: DUCIN

G T HA NG W ON

G G OC fdJ^T

HA

NG

Dang

1 : C

HQ NG M IN

H Ol/dN

G THAN

G V

UO NG

GOC V

dl MA

T PHAN

G

BANG

DIEU KIEN

CAN VA

dla;bcu;

a^

b =

0 =>d_L

a (dpcm)

n CA

C B AI T OA

N CO B

' A

B 1 SA

CI

(SAD)

^^AB L DA

c (SAD)

AB 1

(SAD) (dpcm)

va S

B =

De y den ha

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)

TCr (1) v

a (2) ch

o : AC

1 (SBD

) (dpcm)

30

B a i 53

Cho h i n h ch6p S.ABCD c6 day la h i n h cha n h a t \k SA J_ (ABCD) Goi A E , A F 1^ dudng

cao cua cac ASAB va ASAD ChOfng m i n h r ^ n g SC 1 (AEF)

G i a i

D 6 y A D ± ( S A B ) v a B C / / A D

(Sau nay ta c6 the chufng m i n h (1) bSng d i n h l y 3 dadng

vuong goc se n h a n h hdn hoac bdng t i n h chat giao tuyen

cua hai mftt p h i n g vuong goc)

Hon nOa E A 1 SB (c^ch dung) (2)

Mot cAch tuang tif S C I A F (4)

Tif (3) va (4) => SC 1 (AEF) (dpcm)

B a i 54

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

gia sii SA = SB ChOfng m i n h rSng CD 1 (SIJ)

Cho tijf dien A B C D c6 H , K la true tarn cdc tarn gidc A B C va DBC Gia 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

B a i 56

Cho tiJ dien SABC c6 SA 1 (ABC) Goi H va K I a n luot l a trifc tarn ckc tarn gidc ABC v^

SBC Chufng m i n h ;

a/ A H ; SK; BC dong quy b/ SC 1 ( B H K ) c/ H K 1 (SBC)

C = A' D

C

BC

^A A' 'B CI SA (do SA

X (A BO )

Ta

CO :

C 1 (SAA' ) =

>

BC

1 SA '

Vay

CO

the no

i : A H;

SK

va BC dong quy (dpcm)

C 1 B

K (1 )

Ma tk ha ct ac

o : \

^^

^^

^ JBHXSA (doSAKABO)

nen : BH

1 (SAC ) ^ B

H 1

SC

Tif (1 ) v

a (2 ) t

a su

y r

a : S

C ± ( BH K) (dpcm)

d

Theo cau a/ t

a c

6 : B

C ± (SAA') =

>

BC ±

HK

Theo cau b/ t

a c

6 : S

C _L

BH K) =

>

SC J_

HK

Tif (3 ) v

a (4 ) t

a su

y r

a : H

K ± (SBC) (dpcm)

Cho hinh vuong AB CD nam trong mat phAn

g (P) Qu

a A ditag nijf

c vd

i mp(MCB ) cS

t (P

)

tai R difdn

g thSn

g qu

a M vuong

goc \6i

mp(MCD) ca

t (P ) ta

i S

1/ Chufn

g min

h : A; B

; R thang hang

2/ Ti

n R

S kh

i M lifu don

g tre

n nijf

a dudn

g t hi ng

R 1 (MBC ) MR

1

BC

Ma d

a CO : A

D 1 AM ^ AD

1 (MAR ) =

a AR

va

AB

cung vuong go

TiJong ti

f tre

n

= > A , D , S thang hang

2/ D

o M

R ± (MBC) ^

d M c6 difcrn

// BC

)

MA ' = A B.A

R

Tifcrng tif : M A' = A D.A

S

AB A

R = A D.A

S

^ AR = A

S

=j> I thuoc difdn

g t hi ng A

C

Do R chay tren tia A

a S chay tren tia A

v (ti

a do

i ti

a AD ) ne

Vay, I chay tren tia do

Vay sau khi la

32

Dang 2 : C H J N G M I N H Dl/dNG T H A N G V U O N G G O C Vdl M A T P H A N G

B A N G T R U C Dl/CiNG T R O N

L PHirONG P H A P

Ca stf ciia phi/ang phap chiirng m i n h difdng t h i n g d vuong

goc v6i mat phang a bftng van dung d i n h nghia t r u e dxidng

tron: 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 chii'a

dUdng t r o n t a i t a m c u a n o bang h a i btfdp ca ban nhU sau :

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

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

T i m diem O d day each deu cac d i n h da giac day ABC

OA = OB = OC =

• Ba : No'i h a i d i e m S, O do t h a n h true d cua di/crng t r 6 n

No la dudng thftng vuong goc v d i moi mat phang ehiJa

duac di/cfng t r o n (ABC)

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

B a i 58

Cho h i n h vuong A B C D canh a Ve cung ve mot phia (ABCD); cac doan AA'' C C vuong goc

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

a/ Dat : SA = SB = SC = a > 0 (cho san)

BC = 2 C I = 2 ^ = a V s (ASIC 1^ niJra A deu)

Ta CO : •iCA = a (.\ASC deu)

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

< <^ B C ' = CA^ + A B 2

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

Theo d i n h l y Pythagore dao => ACAB vuong t a i A (dpcm)

T he

o ti nh c ha

t dUcfn

g

t ru ng t uy en

Cifng

v

di ca nh h uy en cua t am

g'lAc A

BC v uo ng

B = S

I la

true ducrng tr on

n go

ai ti ep zVABC =>

SI 1 ( AB C) (d

pc

m)

Ba i6

0

Ch

o hi nh c ho

p S.

AB CD d ay A BC

D la h in

h th

oi c

6 IJ AC = 60

" v

a SA = SB = SC

g SG

x (A BC D) Vd

i G l

a t ro ng t am t am g ia

c AB

C

Gi

ai

De y A AB

h thoi ) AA BC d eu

=> G l

a ta

m d ud ng t ro

n AB

C ng oa

i ti ep t am g ia

c de

u AB

C

Do

do ; G

A = G

B = G

C

^ G

6

(d ) tr ue eiia d ud ng t ro

n (A BC )

Gia t hi

et c

6 : S

A = S

B = SC

S e (d )

To

m la

i S

G c (d ) ha

y SG

1 (A BC ) ha

y SG

1 (A BC D) (d pc m)

p S.

AB CD c6 SA = SC = SD v

a AD

t = 9 0"

Go

i 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 g ia

c AD

C (6 = 9 0°

)

=> lA = IC =

ID

Kc

t hcr

p g ia t hi

et SA = SC = SD

=i>

SI la t ru

e du dn

g tr on ( AC D) ng oa

i ti ep \

AC

D

=^

SI 1 ( AC D) ^ ( AB CD )

« SI

l (A BC D) (d pc m)

p S.

AB CD c6 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 ijf ng m in

h rft ng

0 1

1 (B CD ) va S

A 1 ( AB CD )

Gi

ai

De y d en t in

h ch

at cii

a du dn

g tr un

g tu ye

n ufn

g \'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

c de

u AB CD ( la

m cii

a lu

c gia

c de

u la O )

=>

OB = OC = OD

ue di /d ng t ro

n (B CD ) ng oa

i ti ep t am g ia

c BC

D

=>

IO

1 (B CD ) (d pc m)

IB = IC =

ID

(1)

Ma S

A / / - O

I 2 SA

1 (A BC D) = ( BC D) (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 ta sit dung d i n h nghia : d 1 a => d 1 a (tuy y trong a),

qua 2 bade ca ban :

• B i : Quan sat va quan l y gia thiet t i m mp(a) chijfa dudng thdng

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

Goi I la trung diem canh C D va de y h a i trung tuyen cung la

dudng cao trong h a i tarn giac can cung ddy CD la : AADC va ABCD

Cho h i n h chop S.ABCD c6 A B C D la niJfa h i n h luc giac deu va SA 1 (ABCD) M o t m a t

phang qua A vuong goc vdi SD t a i D ' cat SB; SC t a i B', C ChuTng m i n h t i l giac AB'C'D' noi tiep difoc

Theo each difng (a) ^ (AB'C'D') =

> AB' 1 SD (5)

Tif (4) v

a (5) =

>

AB' 1 (SBD); m

a (SBD) 3 B'D'

=>

AB ' ± B'D' « A BT

T = 9 0"

(6 )

Cung tif (5) v

a (6) =

> Tu

T gia

c AB'C'D' no

i tie

p dua

c (dpcm)

g minh

D IE

U KI EN D AI

h die

u kie

n

(=>) : Gia sij

f A

B 1 CD CD ± (ABH)

D ± A

H

Ap dung

he thiJ

c luan

g trong ta

m gia

c vdr

i M

la trun

g die

m CD

f

AC'

- AD

' -_

2 CD

.M

H

=> <i

iB C' -B D^

-

2C D

.M

H

=> AC' - AD' = BC' - BD' (dpcm)

• Di

^u k i$

n

(<=) : Gia AC' - AD' = BC' - BD' (1)

Gpi M

la trun

g die

m CD, AH

j,

BH 2

2Ci5.MH

7 (2

)

tuong tfng, t

a C

O : \

;B C' -

B D' = 2CD.MH2

(3)

Sijf dun

g (1) ch

o (2) v

a (3) :

2C D

.MH, =

2CD.MH2

=>

MH, =

la \.\S

(4) t

a c

6 CD

1 (ABHi) ^

( AB H2 ),

ca ha

i ma

t phSn

g (ABHi)

o d

o C

D 1 A

B (dpcm)

O Ke

t lugi n

: Die

u kie

e ha

i can

h do

i AB

Ba

i 6

7

Chiifng min

h rkn

Gi

ai

Goi E

la trun

g die

m can

h AB The

o tin

h cha

t cu

a ta

'A

B ^ C E

c

( CD E)

|A B

1

D E

C

( CD E)

=> AB

1 (CDE)

; m

a CD

c (CDE)

AB

1 C

D (dpcm)

Tuang tir t

a chiifn

g min

h difa

c B

C 1 A

D v

a A

C L

BD (dpcm)

O Gh

i ch

u :

Boc gid xem each

chiing minh khdc a

phdn goc cua hai

diictng thang trong

khong gian.

Viec nha tinh chat

nay, chung toi xin

nhdc : rdt ti^n ich trong

qua trinh

tinh

todn tren cdc til

dien dSu

Doc gid cUng c6

the dung diSu kien

dai so cila Bai

66

36

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

doi thuT ba cung vuong goc nhau

• Ghi chii : Co thi chvCng minh bdi todn bdng phuang phdp vecta

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 van dung d i n h ly ba dudng vuong goc n h u sau :

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

;HM la hinh chieu (ciia AM xuong (a))

t h i difcrng (d) n'km trong (a) thoa :

(d) 1 A M (ducrng xien) c* (d) J H M (hinh chieu)

Do do philcfng phdp gom 2 budc thuc hanh :

do t i m ra dudng xien ® va h i n h chieu ®

n B2 : Dirdng t h i n g thut @ la (d) nam trong mat p h l n g (c

• Neu : (D 1 ® (D J_ ® => (ycbt)

• Neu : (D ± ® (D 1 ® => (ycbt)

n cA c B

AI TO AN C OB

De y thay : S

A 1 (ABCD)

Ma B

C 1 A

B

Cho hin

a ABC

D l

a hin

h ch

a nhat Chuto

, (SCD), (SAD) de

u Ik

nhflng tarn giac vuong

=> B

C 1 SB (din

=> ASB

C vuon

g a

B (ycbt)

Tirang ti

i D

C 1 SD (din

h l

y 3 dudng vuon

g goc) =

> ASD

C vuon

g d

D (ycbt)

Tom la

i hin

a nhifn

g tam giac vuon

g (dpcm)

D c

6 AB _L CD

vk

AC

1 BD Go

i H

la hin

h chie

u cu

a A xuong (BCD)

o B

H r>

CD = B,

CH saochoCHnBD

= C

i

DH sao ch

o D

H n BC = D

j Ta

CO :

CD ± A

B (gi

a thiet)

AB

H) 3B

Ti/ong

ta BD ± (ACH) z C

H BD ± C

H t

ai Cj

=> CH

la dudn

g ca

o ABC

D (2)

Tii

(1) v

a (2) ch

Di (v

i H

la trif

c tam ABCD) =

> B

C J_ A

Dj

=> B

C 1 (AHDi) ^ (ADD

i) 3

A

D B

C ± A

D (dpcm)

Ba

i 7

2

Cho Cx, D

h chC

? nha

t ABCD MQ

t mp(P

) qu

a

AB cat Cx, D

y t

ai E

va F Chufn

g min

h ran

g ABE

F l

a hin

h ch

Q nh^t

Ma AB

Theo din

h ly giao tuye

n son

g son

g th

i ABE

F l

a hin

h bin

n l

a hin

h chi

f nha

t (dpcm)

O Ca ch k ha

c : D

l y thay DC

1 (EBC) m

a AB//

CD

=> AB

X (CDE) r EB

=>

AB

1 E

B (1 ) => (dpcm)

Bai 73

Trong hinh chop S.ABCD day la h i n h chC nhat ABCD Goi S H la diTcfng cao h i n h chop va

SK; SL thijf t u la dirdng cao cac t a m giac SAB va SCD Chijfng m i n h rSng H , K , L thSng hang

D e y v d i S H I (ABCD)

G i a i

fSH : la dudng xien

^HK : ]k hinh chieu

MaAB 1 SK (each difng) A B i H K (dinh ly ba dif&ng vuong goc)

TucJng t u CD ± H L (dinh l y ba di/cing vuong goc)

' A B x HK

Tom t a i : <CD ± H L => H , K, L thftng hang (dpcm)

B HK// H L

Bai 74

Cho tiJ dien SABC c6 A B C la t a m giac deu canh a, cac mftt (SAB); (SBC) va (SCA) hap vdi

(ABC) cac goc bftng nhau va bftng a

1/ Chiitng m i n h rftng : h i n h chieu H cua S len (ABC) la t a m du&ng tron noi tiep AABC

2/ Tinh tong dien tich 4 mftt cua tuT dien S.ABC

: 1

G i a i S

1/ Goi 1, J, K Ian lifot la hinh chieu ciia H len BC; CA; A B

Do dinh l y ba dUcfng thftng vuong goc

BC 1 SI; CA 1 SJ; A B 1 SK

Do d6 goc p h i n g cua cac mftt ben (SBC); (SAC) v ^ (SAB)

tao vdi (ABC) Ian lugt la S ^ , SJi?, gS^

g l i l =S3tl - SKfi = a

De y thay t a m giac vuong SHI, S H J , S H K bftng nhau nen :

H I = H J = H K

Vay, H la t a m dUdng tron HQI tiep AABC ( H cung la trong t a m , true t a m , t a m difdng tr6n

ngoai tiep cua AABC) (dpcm)

[SHAB = SsAB COS a I

2/ Theo dinh l y dien tich va h i n h chieu ta c6 : •JSHBC = ^sac cos a <=> -ISSBC