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

516.23076

BAN GIAO VIEN NANG

T527T

N J G U Y E N

KHIEU TRUCiNG THI

eC/C e O N G

( C h u bien)

PHAN LOAI
VA PHl/OfNG PHAP
GIAITHEO
CHUYEN DE
• BOI Dl/dNG HQC SINH GIOI
• CHUAN B! THI TU TAI, DAI HOC VA CAO
BOG
Ha

NOI

DANG

NHA XUAT BAN OAI HOC QUOC GIA HA NOI


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

NGUYEN DLfC D 6 N G {Chu hien)

TUYEN TAP 500

BAITOAN

•

HDIH imm GIAN
C H O N LOG
•

•

•

PHAN LOAI VA PHUdNG PHAP G I A I THEO 2 3 CHUYEN

• B o i difdng hoc s i n h g i o i
• C h u a n b i t h i T i i t a i , D a i hoc v a Cao d a n g
(Tdi ban idn thvt ba, c6 svCa chUa bo

sung)

THir ViEN TiiVH BiKH liik^m

NHA XUAT BAN DAI HOC QUOC GIA H A NOI


NHA XUAT BAN DAI HOC QUOC GIA HA NQI

16 Hang Chuoi - Hai Ba Trcfng - Ha Npi
Dien thoai: Bien tap - Che ban: (04) 39714896
Hanln chinli: (04) 39714899; Tong Bien tap: (04) 39715011
• Fax: (04) 39714899

*

Chiu
Gidm

Bien
Saa

trdch

**

nhiem

xuat ban:

doc - Tong bien tap: T S . P H A M T H I T R A M

tap:

THUY

bdi:

THAI

Che ban:
Trinh

HOA
VAN

N h a sach H O N G A N

bay bia:

THAI V A N

SACH LIEN K E T

TUYEN TAP 500 BAI TOAN HJNH KHONG GIAN CHON LOG
Ma so: 1L - 195OH2014
In 1.000 cuon, kho 17 x 24cm tai Cong ti Co phan V3n hoa VSn Lang - TP. Ho Chi IVlinh.
So xuat ban: 664 - 2014/CXB/01-127/OHQGHN ngay 10/03/2014.
Quyet dinh xuat ban so: 198LK - TN/QO - NXBOHQGHN ngay 15/04/2014.
in xong va nop IIAJ chieu quy il nSm 2014.


LCilNOIDAU
Chung t o i x i n g i d i t h i $ u den doc gia bp sdch: Tuyen t a p cdc b ^ i toan d k n h cho
hoc sinh Idp 12, chuan b i t h i vao cac trucrng D a i hoc & Cao d i n g .
Bo sach gom 7 quyen :
.

T U Y E N T A P 546 B A I T O A N T I C H P H A N

.

T U Y E N T A P 540 B A I T O A N K H A O S A T H A M SO

.

T U Y E N T A P 500 B A I T O A N H I N H G I A I T I C H

.

T U Y E N T A P 500 B A I T O A N H I N H K H O N G G I A N

.

T U Y E N T A P 696 B A I T O A N D A I SO

•

T U Y E N T A P 599 B A I T O A N L U O N G G I A C

.

T U Y E N T A P 6 7 0 B A I T O A N RCJI R A C V A C l / C

TRI

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 n a n g 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 m o n g don n h a n m o i 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 N I l 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 da
giiip da chiing t o i m o i m a t d l bo sach dUdc r a dcfi.
NGUYEN

DtfC

DONG

3


•

(i)

B A N G K E CAC K I H I E U V A CHLf V I E T T A T T R O N G

• [ ( A B C ) ; ( E F G ) ] : goc tao bori 2

mp

( A B C ) va ( E F G )
->

• C > : Phep t i n h tien vectcf v
V

• D A : Phep doi xOmg true A
• Do : Phep doi xiiTng true 0
• Q(0; cp) : Phep quay t a m O, goc quay
(p.

• V T ( 0 ; k ) : Phep v i t u t a m 0, t i so k.
• D N : dinh nghla
• D L : dinh ly

• Stp : D i e n t i c h t o a n p h a n
: The t i c h

• C M R : chiJng m i n h r i n g

A

: goc

• B i : budc i
• T H i : t r u d n g hop i
• V T : ve t r a i

xuong dtfcfng thftng (d)

(3r3^

SACH

CAC K I H I E U T O A N HOC v A CAC T l / V I E T T A T

<=> : (i) tUcfng dUcfng
(il

• => : (i) keo theo
• <!> : k h o n g tUdng dilcfng
• d> : k h o n g keo theo
• = : dong n h a t
: k h o n g dong n h a t

• i

• Sv\nc = S ( A B C ) = d t ( A B C ) : d i e n . t i c h
AABC
• V s A H c = V ( S . A B C ) : the t i c h h i n h chop
S.ABC

• H Q : he qua

• Sxq : D i e n t i c h xung quanh
• V

• A ' = ''7(ai A : A ' la h i n h chieu ciia A

xuong m a t p h i n g (a)
• A ' = ''Vfd) A : A ' l a h i n h chieu cua

• d [ M ; (D)l : k h o a n g each tiT d i e m M d e n
ducfng t h i n g (D)
• d [ M ; ( A B C ) I : k h o a n g each tii diem M
den mat phang ( A B C )

• (a; P ) : goc n h i d i e n tao bcfi 2 mfa m a t
phang (a) va ( P )
• ( S ; A B ; D) = ( A B ) : n h i dien c a n h A B
•

tao bdi h a i dUomg t h i n g d

• V P : ve p h a i
• B D T : bat d i n g thijfc
• y c b t : yeu cau b a i toan
• d p c m : dieu p h a i chuCng m i n h
• gt : gia

thiet

• K L : ket luan
• D K : dieu k i e n
• P B : phan ban

va d'
• [ H T C A B C T I : goc tao bdi du&ng t h i n g d
va

• C P B : chiTa p h a n ban

mp(ABC)

4

Chuyen

de 1 :

TONG QUAN V E C A C KHAI NIEM
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.

•

N h i f 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 i t 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 3: N e u h a i m a t p h & n g c 6 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 t h e o m p t d U d n g 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 m a t 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 b i # t
khong th^ng hang.

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 l y 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 n a n g cao) d e u d i i n g .
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 :
O

H E Q U A 1: C o m p t v a c h i m p t 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
m p t d i e m n S m n g o a i dt^dng t h a n g do.

O
O

H E Q U A 2: C o mpt v a c h i mpt m^t p h d n g duy n h a t d i q u a h a i di^cAig t h ^ n g cSt n h a u .
H E Q U A 3: 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 h a i di^c/ng t h d n g
song song.

•

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 n e t , 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 trvC6c 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 xac d i n h n h ^ n h m o t m a t p h ^ n g t r o n g k h o n g gian t a con chon t h u thuat thUc h a n h :
M p t h i n h t a m g i a c , tii" g i a c h o a c d a g i a c ph&ng ( k h o n g g e n h ) , dUcfng i r o n ,
l u d n x a c d i n h m p t m ^ t p h S n g t r o n g k h o n g g i a n . T a gpi c a c m&t p h ^ n g do l a m^it
p h S n g h i n h thvCc v d i c a c k y h i p u ( A B C ) , ( A B C D ) , ( C ) , ... txictng vtng.

M p t dvictng t h d n g n ^ m t r o n g m ^ t p h & n g h i n h thd'c m a m a t do h i k h u a t c u e bp

•

M a t p h d n g h i n h thu^c h i k h u a t n e u c 6 m p t h a y n h i e u m ^ t ph&ng n a o do c h e n o .

•

h a y t o a n bp v a k h i dUcTng t h ^ n g do k h o n g l a b i e n c u a m a t p h d n g b i k h u a t do,
t h i di^dng th&ng do c u n g tii'oTng vlng k h u a t c u e bp h a y t o a n bp.

Noi h a i d i e m m a it n h a t c 6 mpt d i e m k h u a t t h i dUpc mpt dUcfng k h u a t cue bp h a y

•

Mpt d i e m nhm t r o n g m p t m $ t ph&ng h i n h thuTc b i k h u a t t h i goi l a d i e m k h u a t .

•

t o a n bp : n e u h a i diictag do k h o n g l a b i e n c u a c a c m^t phAng h i n h thufc c h e no.
•

C A C H I N H A N H M I N H HQA
\(d)

•

(d) b i (a) che k h u a t cue bo, do (d) c6 1 doan ve

net dijft doan n k m dudi (a).

S

•

(d) b i m a t p h ^ n g (SAC) che k h u a t cue bo, do (d)

CO m p t doan ve duft doan n k m sau (SAC)

(hien

n h i e n (d) cung d sau cac m a t (SAB), (SBC)).
•

C a n h AC b i h a i m a t p h a n g (SBC) v£l (SBC)

che

k h u a t toan bo, do ca doan AC x e m n h u hoan t o a n d
sau dong t h d i h a i m a t p h ^ n g (SAB), (SBC).
-AA.

c./—1—^VFJL^

•

• A ] H b i che t o a n bo do ca doan A ] H n k m sau m a t
p h i n g ( A i A D D i ) , mSc dij no d trU( A B B j A i ) va d t r e n m a t p h a n g (ABCD).
• (d) b i che k h u a t cue bo v i c6 doan E F ve net duTt
doan n a m sau h a i m a t p h a n g ( A D D i A j ) ; ( C D D j C j ) ,
mac dij doan E F d p h i a trUcJe h a i m a t phang
( A B B j A i ) ; ( B C C i B i ) ; va d t r e n m a t p h a n g (ABCD).

C A C KY H I E U C A N ^fHd

Thiir trf

Y nghta

Ky hieu

(d) n (a) = A

5

(d) // (a)

4

(d) c

3

A i

2

A e (d)

1

D i e m A thuoc ducfng t h i n g (d) hay dadng
t h i n g (d) chura A.

(d)

(a)

Ghi chu
H a y v i e t n h a m la :

Ac(d)
(d)

hay

H a y v i e t n h a m la :

H a y v i e t n h a m la :

DU&ng t h i n g (d) n k m t r o n g mat p h i n g (a)

Acz(d)

D i e m A or ngoai difdng t h i n g
dUcfng t h i n g (d) k h o n g chtifa A.

Cach v i e t khae :

Difcrng t h i n g (d) song song \6\t p h i n g

(d) e (a)

hay (a) quay quanh (a) neu (a) luu dong.

e a c h v i e t khdc :

Difdng t h i n g (d) e i t m a t p h i n g (a) t a i A.

(d) n (a) = 0

(a).

(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 :
(d,) n (da) = {A}

7
8

9

(di)//(d2)

H a i difdng t h f t n g ( d i ) , (da) song song n h a u
neu chiing.

H a y v i e t n h a m la :

(«)^(P)

H a i m a t p h a n g (a) va ([5) t r i i n g nhau k h i
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 :

M a t p h a n g (a) xac d i n h bdi ba d i e m A , B,

(ABC) : l a m a t p h I n g
hinh
thijfc
vdi
ba
dildng bi&n A B , BC,
AC.

(a) = (ABC)

C p h a n b i e t va k h o n g t h i n g hang.

10

(a) = (A; d)

M a t p h i n g (a) xac d i n h b d i d i e m A

va

ducfng t h i n g (d) k h o n g qua A.
11

(a) s ( d i ; dz)

M a t p h I n g (a) xac

d i n h bdi h a i

( d i ) o (da) = 0
(a) = (p)

(A; d) ; la m a t p h I n g
h i n h thufc

dudng

t h i n g d i , da.

• d i , da CO the song
song hoac dong quy.
•

( d i ; da) la m a t

p h l n g h i n h thuTc.
Loai 1: T t M G I A O T U Y I N C U A H A I M A T P H A N G
I. p m r o N G P H 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
CO

•
n

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

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) h a y (da)
=> xSy la giao t u y e n can t i m .

7

m. C A C B A I T O A N C O B A M

Bai 1
Cho tiif giac l o i A B C D c6 cac canh doi k h o n g song song va d i e m S d ngoai (ABCD). T i m
giac tuyen ciia :
a/ (SAC) va (SBD).

hi

(SAB) va (SDC); (SAD) va (SBC).

Giai
a/ Xet h a i m a t p h a n g (SAC) va (SBD), t a c6 :
T r o n g tuT giac l o i A B C D , h a i ducmg cheo A C

•

S la d i e m c h u n g thuf n h a t .

•

(1)

n B D = O : d i e m c h u n g thijf n h i (2).

^

Ti/(1) va (2) suy r a :

(SAC) o (SBD) = SO (ycbt)
hi

Xet hai m a t p h a n g (SAB) va (SDC) cung c6 :
H a i canh ben A B va C D cua t i l giac A B C D

•

S la m o t d i e m chung.

•

theo gia t h i e t k h o n g song song.
^

A B ^ C D = E : la d i e m c h u n g thut h a i .

Do do : (SAB) n (SDC) = SE (ycbt)
Tucfng t i f : (SAD) n (SBC) = SF (ycbt); v d i F = A D ^ BC; do A D / / BC.

Bai 2
Cho t i l d i e n A B C D . Goi G j , Ga la t r p n g t a r n h a i t a m giac B C D va A C D . L a y theo thuT t i i I ,
J , K la t r u n g d i e m ciia B D , A D , C D . T i m cac giac tuyen :
aJ

(G1G2C) o ( A D B )

hi

(G1G2B) n ( A C D )

c/

( A B K ) o (CIJ>.

a/ (G1G2C) n ( A B D ) = I J
(ABK) ^ (CIJ) =

d

(GiGaB) n ( A C D ) = GgK hoSc A K

hi

G,G2

Bai 3
Cho h i n h chop S . A B C D c6 day A B C D la h i n h b i n h h a n h t a m O.
T i m giao t u y e n cua h a i mSt p h i n g (SAB) va (SCD).

hi

T i m giao t u y e n cua h a i m a t phSng (SAD) va (SBC).

aJ

c/ T i m giao t u y e n ciia h a i m a t p h ^ n g (SAC) va (SBD).

Giai
aJ

Xet h a i m a t phSng (SAD) va (SBC), t a c6 :
De y A D c ( S A D ) ; BC c (SBC) m a A D // BC.

•

S la d i e m c h u n g thur n h a t .

•

Ta d u n g xSy // A D hoac BC.
[(SAD) = (xSy; AD)
^

|(SBC) = (xSy; BC)

=^ (SAD) n (SBC) = xSy (ycbt).
hi

Tifang t i r , difng uSv // A B hoftc C D

8


=> (SAB) r^ (SCD) = uSv (ycbt)
c/ Goi O = A C n B D , tiTcrng t a b a i 1
=> (SAC) n (SBD) = SO (ycbt).
Bai 4
Cho h i n h chop S . A B C D c6 day la h i n h t h a n g A B C D v d i A B l a day Idtn. Gpi M la m o t d i e m
bat ky t r e n SD va E F l a difang t r u n g b i n h cua h i n h t h a n g .

a/ T i m giao t u y e n ciia h a i mSt p h i n g (SAB) va (SCD).
b/ T i m giao t u y e n cua h a i m a t phSng (SAD) va (SBC),
c/ T i m giao t u y e n cua h a i mSt p h a n g ( M E F ) va ( M A B ) .

Doc gia t u g i a i tUcfng t u n h u cac b a i t r e n .
Bai 5
Cho h i n h chop S . A B C D c6 A B C D l a h i n h b i n h h a n h . Goi G,, G2 l a t r o n g t a m cac t a m giac
SAD; SBC. T i m giao t u y e n cua cac cSp mSt p h a n g :
a/ (SGiG^) va ( A B C D )

b/ (CDGiGz) va (SAB)
UvCdng

0/

(ADG2) va (SBC).

d§Ln

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 l a p l u a n nhtf cac
bai toan t r e n ;
a/ (SG1G2) n ( A B C D ) = 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
L

PHirONG

PHANG

PHAP

Ca sd cua phaang phap t i m giao d i e m O cua dudng t h a n g
(a) va m a t phSng (a) l a xet 2 h a i k h a nSng xay r a :
n

T r i r d n g hop (a) chiJa dudng t h S n g (b) va (b) l a i c&t diicrng
t h d n g (a) t a i O.
T i m O = (a) n (b)
=> O la d i e m can t i m .

n

Trtfdng hap (a) k h o n g chiifa dUcmg t h i n g nao cat (a).
T i m ( P ) ^ ( a ) v a ( a ) n ( P ) = (d)
>

T i m O = (a) o (d)

=> O la d i e m can t i m .

n. CAC

BAI TOAM G O B A N

Bai 6

Cho tuf d i e n A B C D . Goi M , N I a n lugt la t r u n g d i e m cua A C va BC. L a y d i e m K e B D sao
cho K B > K D . T i m giao d i e m ciia h a i dudng t h i n g CD va A D v d i ( M N K ) .
9


•

De y den K B > K D
Do do t r o n g (BCD)
Ma K N c ( M N K )

•

Giai
=> K N k h o n g song song C D
K N o CD = I .
CD

( M N K ) = I (ycbt)

Taong t a xet I M c ( M N K ) , t r o n g ( A D C )
Ta CO :

AD n IM = E

=> A D n ( M N K ) = E (ycbt)

Bai 7
Cho tiJ dien A B C D . L a y d i e m M t r e n A C va h a i d i e m N va K thuf tiT nSm t r o n g cac t a m
giac B C D va A C D . D u n g giao d i e m cua CD va A D \di ( M N K ) .

HtfdTng d i n
Doc gia t u g i a i , x e m h i n h ben.
a/

CD

( M N K ) = P (ycbt)

b/ A D n ( M N K ) = Q (ycbt)

Bai 8
Cho h i n h chop tuf giac S.ABCD. L a y t r e n SA, SB va BC ba d i e m M , N , P theo t h i i t\i sao
cho M P k h o n g t h e c&t A B hay C D . T i m giao d i e m cua SC va A C v d i ( M N P ) .

Giai
ThUdng t h u d n g do ycbt t i m giao d i e m
NP o SC = K
ma NP c (MNP)
Trong mp(SAC)
ma M K c (MNP)

SC n ( M N P ) = K (ycbt)
M K o AC = H1
|

=> A C r> ( M N P ) = H (ycbt)

Bai 9
Cho m o t t a m giac A B C va m o t d i e m S d ngoai m a t p h i n g chila t a m giac. T r e n SA va

SB

ta lay hai d i e m M , N v a t r o n g m a t p h i n g (ABC) ta lay mot d i e m O. D i n h ro giao diem cua
( M N O ) v d i cac dudng t h i n g A B , B C , A C va SC.
Hi^dng d i n
Tuang t u , doc gia t u g i a i (xem h i n h ben)
A B n ( M N O ) = E (ycbt)
BC o ( M N O ) = F (ycbt)
A C n ( M N O ) = G (ycbt)
SC n ( M N O ) = 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 p h i i o n g phap can p h a i chufng m i n h ba d i e m
trong yeu cau b ^ i t o a n l a d i e m chung cua 2 mSt phSng nao
do (chfing b a n A, B, C nSm t r e n giao t u y e n (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 l o a i 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. C A C B A I T O A N C O BAM
B a i 10
Xet ba d i e m A, B, C k h o n g thuoc m a t p h i n g (u). Goi D, E, F I a n l u o t l a giao d i e m ciia A B ,
EC, CA va (g). ChCifng m i n h D, E, F t h a n g hang.
Giai
De y t h a y D, E, F viTa a t r o n g m p ( A B C ) vifa d t r o n g mp(a).
Do A, B, C g (a), nen (a) va (ABC) p h a n b i e t nhau.
=> ( a ) n (ABC) = A (A chuTa D , E, F)

D, E, F t h i n g h a n g t r e n A (dpcm).
B a i 11
H a i t a m giac A B C , A B C k h o n g dong p h i n g c6 A B n A B ' = I , A C n A C = J , BC n B C = K
ChiJfng m i n h I , J , K t h i n g h a n g .
Giai
De y I , J , K I a n l u o t d t r e n h a i m a t p h i n g p h a n
(P) ^ (ABC) va (Q) = ( A ' B ' C ) .
N e n no 1^ diim c h u n g cua h a i m a t p h i n g do
I , J , K e (A) = (ABC) n ( A ' B C )
=> 1, J , K t h i n g h a n g (dpcm).
B a i 12
Cho A, B l a h a i d i e m d h a i p h i a khac n h a u doi v d i m a t p h i n g a va A B c i t a t a i O. D i t o g
hai dUdng t h i n g x'Ax, y ' B y song song n h a u theo thuf tiT c i t a t a i M va N . ChuTng m i n h M , N ,
O t h i n g hang.
Kvldng
TifOng

^

t\l:

•

fM, O,

dSn

NG(S)

[8 = (Ax; By) n (a)

=> M , N , O t h i n g h a n g t r e n (8)

11


t o a l 4 : CmiUG MWfl M Q T DtfCiNG T H A N G T R O N G KHONG G I A N
Q U A M O T D I £ M C O DINH
I.

PHirONG

PHAP,

Ca sd cua phucfng phap chuTng m i n h diXcrng t h i n g (d)
qua m o t d i e m co d i n h :
Ta can t i m t r e n (d) h a i d i e m tuy y A ; B va chuTng m i n h
2 d i e m do t h i n g h a n g v d i m o t d i e m I co d i n h c6 sSn t r o n g
khong gian.
=> (d) qua I CO d i n h (dpcm).

IL PHtfONG PHAP,
Co sd cua phiTcfng phap can thuc h i e n ba bifdc ccf ban :
n

B i : T i m dUctng t h i n g a co d i n h d ngoai mSt p h 5 n g co
d i n h (a) ma (a) chila d (liOi dong).

•

B2 : T i m giao d i e m I = a ^ d
=> I l a d i e m co d i n h ma d d i qua

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

Bai 13
Cho A , B l a h a i d i e m co d i n h t r o n g k h o n g g i a n d ve h a i p h i a khac n h a u cua m&t p h i n g co
d i n h a. X e t d i e m M luu dong t r o n g k h o n g g i a n sao cho M A n a = I va M B n a = J . ChuTng
m i n h difdng t h i n g I J luon d i qua m o t d i e m co d i n h .

Giai
Goi O = A B n (a) => O co d i n h ( v i A , B co d i n h vk
a 2 p h i a cua (a))
T a CO : mp(P) = ( M A ; M B ) n (a) = I J
De y t h a y : O e I J => O, I , J t h i n g hang.
N g h i a l a dacfng t h i n g I J d i qua O co d i n h (dpcm)

Bai 14
Cho h i n h t h a n g A B C D ( A B // C D va A B > CD). X e t d i e m S e ( A B C D ) va m a t p h i n g a luu
dong quanh A C v d i a '-^ SB = M , a n SD = N . ChuTng m i n h difdng t h i n g M N luon luon d i qua
mot d i e m co d i n h .

De t h a y dxiac n g a y M N c (SBD)

va

AC c (SAC) va M N o A C = O t h i O e
B D = (SBD) n

(SAC)

=> M N qua O co d i n h (dpcm).

12


Bai 15
Cho 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

phing

(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 .

Giai
D e y t h a ' y k h i ( a ) q u a y q u a n h A B co d i n h n h t f n g vAn co :
( a ) n [ ( O x ; O y ) ^ (P)] = A ( q u a M , N )
1 AB CO d i n h

Nhung -j[P

ABo(p) = I e A

CO d i n h

N g h i a la d u d n g t h a n g M N = A l o u dQng nhOng
v a n qua I co d i n h . ( d p c m )

toal S : C H O N G MWfl B A O U d N G T H A « G T R O N G KHONG G I A N D O N G Q U Y
L PHUONG PHAP,

Co so cua p h i f a n g p h a p l a t a c a n c h i i f n g m i n h d U d n g thiif n h a t
qua g i a o d i e m c i i a 2 d i f d n g c o n l a i b a n g 2 budrc co b a n :

•
•

Bi : T i m (d,) o

(d^)

= O

B2 : C h u f n g m i n h (d;j) q u a O .
=> ( d i ) , (d2), (d.i) d o n g q u y t a i O ( d p c m )

Q. P H U O N G

PHAP,

C o s d c u a p h a a n g p h a p l a t a c a n chijfng m i n h

chung

doi m o t cat n h a u v a d o i m o t d t r o n g 3 m a t p h a n g p h a n
b i e t q u a 2 bifdc ca b a n :
d],
•

B i : Xac d i n h <

c: a; d j

0

da = I i

id3

\

da, d;j cz P; da ^ d3 = I2
A

dg, d , e Y; dg n d j = I3

di

\

\

a, p, y p h a n bi§t
•

B2 : K e t l u a n ( d , ) ; (da); (d;,) d o n g q u y t a i 0 = I i = I2 = I3

m. c A c

BAI TOAN C O BAM

Bai 16
C h o t i l d i e n A B C D . G o i E , F , G l a b a d i e m t r e n b a c a n h A B , A C , B D sao c h o E F n B C = I ,
E G o A D = J ( v d i I ^ C wk J

^ B). C h i j f n g m i n h C D , I G v a J F d o n g q u y .

Giai
X e t b a difcfng t h A n g C D ; I G v a J F , t a tha'y :
CD, I G e ( B D C ) va C D
• IG, J F c (EFG)

IG / 0

va I G n J F * 0

J F , C D e ( A C D ) va J F r> C D * 0
Va ba m a t p h a n g ( B C D ) , ( E F G ) , ( A C D ) l u o n p h a n b i e t
( v i I ?t c v a J ?t D ) => C D , I G , J F d o n g q u y t a i O ( d p c m ) .

13


O Cach khac
D o c g i a churng m i n h r S n g J F q u a O = I G n C D => C D ; I G v a J F d o n g quy.

B a i 17
C h o h a i t a m giac A B C , A B C sao cho A B c a t A ' B ' a E , A C cdt A C d F ; B C c a t B C d G .
a/

Chufng m i n h b a d i e m E , F , G t h S n g h a n g .

b/ C h i J n g m i n h difcfng t h a n g A A ' , B B ' , C C d o n g quy.

Gidi
a/

D e y t h a y E , F , G l a b a d i e m chung cua h a i m a t ph^ng p h a n biet
(a) ^ ( A B C ) v a (P) = ( A B ' C ) .
D o do : E , F , G e (A) = ( a ) n (P).
V a y E , F , G th^ng h a n g (dpcm).

b/ N h a n x e t n h u s a u :
: AA', B B ' cr ( E A A ' ) ; A A ' o B B ' # 0
^ B B ' , C C c ( G B B ' ) ; B B ' r^ C C * 0
Ice,
^

AA' c ( F C C ) ; C C n AA' # 0

A A ' , B B ' , C C d o n g quy t a i O (dpcm).

Chuyen

de 2 :

QUAN HE SONG SONG

t o a i 1: C H t J N G MWfl HAI DLfCJNG

THANG

SONG SONG

I. PHirONG PHAP
C o S0 c u a p h a o n g p h a p c a n t h i i c h i e n h a i hxidc CO b a n c h o d i n h n g h i a a // b
j a , b c: (a)
'a^b = 0
•

B i : K i e m t r a h a i difdng t h a n g a c u n g t r o n g m o t m a t
p h a n g h a y hifeu n g a m r a n g h i e n n h i e n d i e u do x a y r a
n e u c h u n g t r o n g 1 h i n h p h a n g n a o do. ( 1 )

•

B 2 : D u n g d i n h ly T h a l e s , t a m giac dong dang, t i n h c h a t bac cau ( t i n h c h a t cung song
s o n g \6i

difdng thiJ b a ) l a h a i c a n h c u a h i n h t h a n g , h a y h a i c a n h doi c u a h i n h b i n h h a n h ,

... de k h a n g d i n h h a i difcfng t h ^ n g do k h o n g c6 d i e m c h u n g . ( 2 )
T i f ( 1 ) v a ( 2 ) => ( y c b t )

n. C A C BAI TOAN CO BAN
B a i 18
C h o h i n h c h o p S . A B C D c6 G j , G 2 , G3, G , I a n lucft l a t r o n g t a m c a c t a m g i a c S A B , S B C ,
S C D , S D A . C h u m g m i n h tiJf g i a c G i G a G g G , l a h i n h b i n h h a n h .

14

Giai
SG,
SE
Theo t i n h chat t r o n g tarn, t a c6 : - i ,
SG3
t
[ SH

SG2
SF

2

SG4

2

SK

3

3

Dinh l y Thales va t i n h chat diTcfng t r u n g b i n h
G,G2// = - E F ; E F 7 / = i A C
'

^

3

2

• G1G2

// = G;jG4

G.G,, // = - H K ; HK// = - AC
^

'

3

2

G1G2G3G4 l a h i n h b i n h h a n h (dpcm).
B a i 19
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 l a h i n h t h a n g .
Giai

S

D6 y thay h a i m S t phSng (a) v a (P) c6 2 d i e m M vfl N 1^ d i ^ m chung.
=> M N = (a) n (SBC)

'(a) 3 AD
ma^(SBC)3BC

iAD//BC

N

va theo each d i m g M N // A D (hoftc BC)
=> A D N M l a h i n h t h a n g day lorn A D . (dpcm)
B a i 20
Cho tuT dien A B C D . Goi M , N I a n li^gt l a t r u n g d i e m cua B C va B D . G g i P l a d i e m t u y y
tren canh A B sao cho P ?t A v a P # B. X e t 1 = P D

A N va J = PC o A M .

ChiJng m i n h r S n g : I J // C D .
Giai
Xet h a i m a t p h a n g ( A M N ) v a (PCD) c6 h a i d i e m chung l a I va J .
IJ = ( A M N )

r-.

(PCD)

'CD c (PCD)
N h i m g < MX CT (AMN)
• va MN // CD
^

I J // M N hoac C D (dpcm).

toai Z : CfltJfJG M W H DiidfiG T H A N G S O N G S O N G TfCl M A T F H A N G
L PHtrOWG P H A P ,

Co so ciia phuong phap m o t l a sii dung d i n h l y phuong giao t u y e n song song.
De chiing m i n h d // a t a can thUc h i e n h a i bade CO b a n chufng m i n h :
•

E l : Chufng m i n h d = y o p m a

•

B2 : K e t l u a n t i f t r e n d // a.

d

y r- a = a
p n a = b.
a//b

15


n . PHOONG PHAP^
Ca sd ciia phifcng phap la stf dung dieu k i e n can va du
chijfng m i n h di/dng t h i n g (d) song song vcJi m a t p h a n g

(a)

b a n g h a i btfdrc :
•

B i : Quan sat va quan l y gia t h i e t t i m dudng t h i n g ou
v i e t (A) cz (a) va chiJng m i n h (d) // (A).

•

B2 : K e t l u a n (d) // (a) theo dieu k i e n can va dii.

m. cAc

BAI T O A N C O BAM

Bai21
T r o n g tuf dien A B C D , chufng m i n h rSng dean no'i h a i t r o n g t a m G i , G2 cua h a i A A B C

va

A A B D t h i song song v6[ ( A C D ) .
Giai

A

Goi A i , A2 l a t r u n g d i e m BC va B D theo thut tiT do, t a c6 :
AG2

3

AA, ' AAg

2

AG)

Theo d i n h l y T h a l e s , t a c6 :
' 0 , 0 2 / / A , A2

B

' m a A,A2 //CD (tinh chat dUcrng trung binh)
Theo t i n h bSc cau

=>

G1G2 // CD c: (ACD)

=j.

G1G2 // (ACD)

(dpcm)

B a i 22
Cho h i n h chop S.ABCD day l a h i n h b i n h h a n h A B C D . G o i M , N l a t r u n g d i e m SA va SB.
Chijfng m i n h : M N // (SCD) v a A B // ( M N C D ) .
Giai
Theo t i n h c h a t dudng t r u n g b i n h t r o n g t a m giac
=> M N // A B , ma A B // CD
=> M N // C D
Theo dieu k i e n can va du
O

cz

(SCD)

=> M N // (SCD)

(ycbt).

Cach khac
De y M N = ( M N C D ) n (SAB) va t r o n g h a i m a t p h a n g do

chiJa theo thijf tiT cac doan t h i n g C D // A B

D

=> M N // (SCD) 3 CD (ycbt)

M N // A B va C D
TifOng tyl :

A B // M N c ( C D M N )

=> A B // ( C D M N ) (dpcm).

B a i 23
Xet h a i h i n h b i n h h a n h A B C D va A B E F k h o n g dong p h l n g . Goi M , N l a h a i d i e m thoa
AM - i AC va BN = - BF . Chufng m i n h r i n g M N // ( D E F ) .
3
3
Giai
De y t h a y M , N l a t r o n g t a m cua b a i t a m giac A B D va
A B E theo thijf t u do.

Keo d a i t h i D M o E N = P : l a t r u n g d i e m A B .
^

PE

PD

PX

PM

1
3

Theo d i n h l y T h a l e s
^

M N // E D c ( E F D C ) ^ ( D E F ) (dpcm)

D

16


Bai 24
H i n h c h o p S . A B C D c6 d a y l a h i n h b i n h h a n h A B C D , t a r n O . G o i M , N I a n \\iqt l a t r u n g
d i e m S A , S B v a x e t h e t h i J c v e c t o : 3 S I - 2 S M = 3 SJ - 2 S N = 0*. ChuTng m i n h r S n g :
a/ I J / / ( S C D )

b/ S C / / ( M N O ) .

Hvfdrng d i n
a/ i

I J // M N , M N // A B ; A B // C D

M N // C D

CD c (SCD)

=> I J / / ( C S D ) ( d p c m )
b/

AM

AO

AS

AC

S C // M O c ( O M N )
S C // ( O M N ) ( d p c m )

Bai 25
C h o A x , B y l a h a i nijfa d i T d n g t h S n g c h e o n h a u . T r e n A x l a y d i e m M , t r e n B y l a y d i e m N
sao c h o A M = B N . C h i j f n g m i n h r S n g dUcfng t h i n g chufa d o a n M N l u o n l u o n s o n g s o n g w6i m a t
p h a n g CO d i n h .

Q u a A d u n g A x ' // B y ; q u a N d i f n g N N ' // B A ; v6i N ' e A x ' . L u c d o tii g i a c A N N B l a h i n h

b i n h h a n h n e n : A N ' = B N => A M = A N '
De y A A M N ' c a n d A n e n t i a p h a n giac n g o a i A t cua STAJT

se s o n g s o n g v6i M N ' v a t i a A t

n a y co d i n h h a y A B v a A t x a c d i n h m a t p h S n g co d i n h ( P ) .

Ta

lMN'//At

CO : <
[

( M N N ' ) // ( P )

N N ' // A B

V a y : M N // ( P ) tiifc l a M N l u o n l u o n s o n g s o n g v6i m&t p h a n g co d i n h

(dpcm).

toal 3 : HAI M A T P H A N G S O N G S O N G

Dang

1 : C H Q N G MINH HAI MAT P H A N G S O N G S O N G

L PmrOHG PHAP
Co

sd cua phuong

phap

chiJng m i n h

hai mat

p h a n g fx v a P s o n g s o n g n h a u t a c a n thiTc h i e n h a i
bUdc CO b a n t r o n g k h i siJf d u n g d i e u k i e n c a n v a d u
nhu sau:
•

B i : Chufng m i n h

" m a t p h a n g ( a ) c h i i a h a i dUcJng

t h a n g a, b d o n g

q u y thijf t i f s o n g

song v d i h a i

dUoing t h a n g a', b ' d o n g q u y t r o n g m a t p h a n g P".
•

B2 : K e t l u a n ( a ) // (P) t h e o d i e u k i e n c a n v a d u .

THL; VJENTifJHglNHTHUAN

17


n . ckc

BAITOAN C OBAM

Bai 26
T r e n b a t i a c u n g c h i e u , s o n g s o n g v a Ichong d o n g p h ^ n g A x , B y , C z M n lifot l a y c a c d i e m
A ' , B ' , C s a o c h o : A A ' = B B ' = C C c 6 do d a i k h a c k h o n g . ChOfng m i n h ( A B C ) // ( A B C ) .

Giai
AA' =3 BB'

D e y :
A'B' IIAB c (ABC)

; AA' = CC' =j. A'C // AC

c

(I)

(ABC)

N e n t a c 6 h a i dUcrng t h ^ n g d o n g q u y A B ' , A C
trong mp(A'B'C') thoa dieu k i e n (I).
=> ( A B ' C ) // ( A B C ) ( d p c m )

Bai 27
C h o h i n h b i n h h a n h A B C D . Tir A v a C k e A x c a C y song song cung chieu v a khong n k m
t r o n g m a t p h S n g ( A B C D ) . Chiifng m i n h ( B ; A x ) // ( D ; C y ) .

Gi&i
Tirang t u xet h a i m a t p h i n g ( B ; A x ) v a
( D ; C y ) , thuT t a chuTa c a c c a p d u d n g

thing

d o n g quy.

fAB//CD
IAx//Cy
=> ( B ; A x ) // ( D ; C y ) ( d p c m )

Bai 28
C h o h a i h i n h binh h ^ n h A B C D v a A B E F d trong h a i m a t ph^ng khac nhau. Chilng m i n h
( A D F ) // ( B C E ) .

Giai
H a i m a t p h l i n g ( A D F ) v a ( B C E ) thiif tiT chuTa c a c c a p dirdng
t h d n g d o n g quy.

iAF//BE
AD//BC

/ A;
( A D F ) // ( B C E ) ( d p c m )

Dang 2 : CHUfNG MINH CAC Dl/dNG THANG D6NG

PHANG

LPBirONGPBAP
Ccf s d c u a p h u a n g p h a p chiifng m i n h c a c d u d n g t h i n g d i , d2, dg... d o n g p h i n g l a c a n p h a i
thiTc h i ^ n h a i bi/•

B i : C h v i n g m i n h d ] , dg, ds, ... d o i m o t c a t n h a u v a c u n g s o n g s o n g v d i m p t m a t p h i n g ((i)
n a o do.

18


•

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

Bai 29
Cho tiJ dien ABCD c6 AB = AC = AD. Chufng minh rSng ba diTcfng phan giac ngo^i cdc goc
SAC. CAI), I5AB cung nSm trong mot mat phlng.
Giai
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)
;At3//BDc(BCD)

At,, At2, At3 // (BCD)
=> A t , , At2, Ata dong ph^ng (trong (P) // (BCD) \k (P) qua A) (dpcm).
Bai 30
Cho hinh chop day la luc gidc deu. Chufng minh rang giao tuyen cua mat ben doi nhau thi
dong phlng.
Giai
De y thay :
(SAB) n (SED) = t, // AB, E D
(SBC) o (SEE) = ta // BC, E E
^(SCD) n (SEA) =

=>

// CD, FA

t , , ta, tg//(ABCDEF)

Vay t , , t2, tg dong ph^ng trong (a) // (ABCDEF)
va (a) qua S. (dpcm)
Bai 31
Tren bon tia phan biet Ax, By, Cz va Dt song song cung chieu, lay cac diem A', B', C , D'
sao cho AA' = BB' = CC = DD'. Chutng minh r i n g A B , B'C, CD', D A ' , A C , B'D" cung song
song vdi mat ph^ng ABCD.

Htfdng d i n
Doc gia t u giSi iMng t\l hai bai toan tren.
19


D E S H Li" THAIJES T R O N G K H 6 N G
•k D i n h l y i (thu|ln) : Hai ditang thing
khong gian chdn tren cdc in^t phdng

GIAN

tuy y d,, d2 trong
song song nhau (a)

II (P) II (y) tao ra cdc doan thang tUcmg ling ty le :
A,A.

-k D i n h l y 2 ( d a o ) :
•

TrUdc k h i xet d i n h ly dao, t a quan tarn d e n h a i k h a i
n i e m s a u k h i xet d e n cac d a y ty so, c h i n g h a n :

•

B2B3

A2A3

B1B9

A, A

(*)

D o a n n o i c a p goc v a c d c cftp n g o n l a ( d o a n ) b a c t h a n g c u a d a y t y so ( * ) .

•

(A2; B2) v a (A3; B3) l a c a c cStp n g o n c u a d a y t y so ( * ) .

•

( A i ; B ] ) l a c a p goc c u a d a y t y so ( * ) .

•

D i n h l y : Neu c6 day ty so trong khong gian :
thdng (d,), (d2) thi cdc bac thang AiB,,

A,A.

(*) da, xdy ra tren hai

(di)
(goc)Ai

G h i c h u : T a c6 p h a t b i e u k h a c c u a d i n h l y

thdng

thang A,B,,

(dj),

(ngpn tren) A2

fd^) thi

mot

dudng

A2B2, A3B3 cung song song vai mot mg,t phdng

dinh.
O

T h a l e s dao nhif s a u :
Vai dieu kien c6 day ty so (*) da xdy ra tren hai
dudng

trong

A2B2, A^B^ se song song

3

bac

(ngon difdi) A 3

c6

(dz)
^ B,(goc)
B2 (ngon tren)
B 3 (ngon dU(Ji)

vdi mot

mat phdng chda hai bac thang con lai.
" A , B i / / ( a ) = (A2B2 ; A 3 B 3 )
A2B2//(P)-(A3B3;AiBi)
A3B3//(Y) S (A,Bi

(A)

; A2B2)

(mat phang co dinh)\

Dang 3 : CHUfNG MINH DJCiNG THANG SONG SONG MAT PHANG
BANG DINH LY THALES
L PHirONG P H A P ,
C o s d c u a p h u o n g p h a p chufng m i n h dUdng t h i n g s o n g s o n g v d i m S t p h i n g b a n g e a c h suf
d u n g d i n h l y T h a l e s d a o t r o n g k h o n g g i a n g o m h a i budc c a b a n s a u d a y :
•

B i : X a c d i n h t r e n h a i d u d n g t h i n g tiiy y c h a n g h a n ( d i ) , (d2) de t i m t r e n do d a y t y so :

^1^2

^ B1B2

AjAs

B,B3

X d c d i n h c S p ( A i ; B j ) l a c S p goc, c d c c S p ( A j ; B2) v a (A3. B3) l a h a i c S p n g o n .
n

B 2 : L u c do c a c d o a n b a c t h a n g A i B i , A2B2, A3B3 dirac k e t l u a n c u n g s o n g s o n g v d i m S t
p h a n g ( P ) ( x e 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. c A c

BAI TOAN C O B A N

Bai 32
Cho tut dien A B C D c6 A B = C D . Goi M va N l a h a i d i e m lUu d o n g t r e n A B va C D sao cho
AM = C N . Chutng m i n h M N luon song somg vdi mSt p h I n g co dinh^

Giai
Neu dat A B = C D = a; A M = C N = x. De y t h a y t r e n A B va C D ta co day t y
AM ^ CN

i(A; C) la cap go'c

AB

|(M; N) v a (B; D) la hai cap ngon tUcJng ufng.

CD

Ap dung d i n h l y Thales dao t r o n g k h o n g g i a n t h i ba
thang A C , M N va B D ciing song song v
bac

mSt phIng (a)

due nay (a) chUa co d i n h v i day t y so — chUa l a 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)
mat phIng

CO

d i n h va cung song song

va (a) t h o a yeu cau l a

vdri

A C , B D va M N .

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

Bai33
Cho h a i 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
AM
AC

BF Ian lucft lay cac d i e m tuy y M , N sao cho

BN
. Chutng m i n h rSng t a luon co : M N //
BF

(DEF).

Giai
Tir gia t h i e t

AM

BN

AC

BF

(*)

A p dung d i n h l y T h a l e s dao cho cac

doan

bac

t h a n g : A B , M N , CF.
=> M N // ( C D F ) ; v i A B // C D c

(CDF)

=> M N // ( D E F ) = (CDF) (dpcm)

Bai 34
Cho h i n h vuong A B C D va A B E F d t r o n g h a i m a t p h I n g khac n h a u . T r e n cac difdng cheo
AC va BF, ta I a n l u g t 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 // ( C E F ) .

Giai
Do h a i h i n h v u o n g A B C D , A B E F b a n g canh
bang nhau
„.,,,..-^
Gia t h i e t =>

AM

BN
=

AC

BF

A p dung d i n h l y Thales cho cac

doan bac t h a n g :

A B , M N , C F voti de y E F cz ( C E F ) ; A B // E F c
^

nen

=> A C = B F .

M N // (CEF)

(CEF)

(dpcm)
21


Bai 35
T r e n h a i t i a A x v a B y c h 6 o n h a u , t a I a n luat l a y h a i d i e m M

N sao c h o A M = k . B N ( k >

0 cho t r a d e ) . C h u f n g m i n h r S n g M N l u o n l u o n s o n g s o n g v d i m p t m S t p h S n g co
HUoTng
Trade het:

dinh.

dim

B y la'y d i e m N , d i n h b d i : B N , = 1
A x l a y d i e m M j d i n h b d i : A M ; = k ( v i k > 0, cho t n / d c )

H i e n n h i e n 1^ h a i d i e m M j \k N ] co

dinh.

T h e o g i a t h i e t v a tii e a c h d a n g t r e n h i n h t a co :

Nen

BN

BN,

AM

AM,

,

AMi
AM

t h e o d i n h l y dao

BNi
BN

cua d i n h l y T h a l ^ s

MN

l u o n l u o n s o n g s o n g v d i m S t p h i n g co d i n h ((5) =
(A;

Bd)

chaa A B

va

d a d n g t h i n g d qua

B

song

song vdi N , M ] . (dpcm)

B a i 36
Cho

h a i d a d n g t h i n g c h e o n h a u d j v a d2. M l a m o t d i e m c h u y e n d o n g t r e n d i v a N l a

mot

d i e m c h u y e n d o n g t r e n d2. T i m q u y t i e h t r u n g d i e m I cua d o a n M N .
Hu&ng

dan

G o i A B l a d o a n v u o n g goe c h u n g c u a d i v a d 2 ( A e d j , B
d 2 ) ; O l a t r u n g d i e m eiia
Taeo:

OB

=

IX

=

dinh

t h i 0 1 nkm

O song song vdi di va

bai

hai

di

AB.
i
d'2

T h e o d i n h l y T h a l e s dao
( P ) qua

trong mat

da, tiifc l a m a t

d a d n g t h i n g d'l va

d'2 q u a

phlng

phang

O I a n lacrt

O

M'
N'

xac
song

s o n g v d i d ] v a d-^.
Giai

han

; M

va

N c h a y t r e n d] va

k h o n g co

rang

huge n e n I c h a y t i i y y t r e n ( P ) .
•

: L a y m o t d i e m I G ( P ) . Q u a I t a d O n g dacfng t h i n g s o n g

Dao

s o n g v d i d'2, d a d n g t h i n g n a y c i t d ' l t a i E . L a y d i e m M ' e
do'i x O n g vdri O q u a
M I

clt

d'l

E.

d'2 d N ' . D i n h

t r u n g d i e m cua

ly dadng trung binh

M ' N ' . TCf M ' v a N ' d i f n g cac

cho

thay

dadng t h i n g

I

la

song

s o n g v d i A B . C h u n g I a n l a g t e l t d ] d M v a d2 d N .
H a i t a giac O M ' M A va O N ' N B deu la n h f l n g h i n h chS n h a t :
=>

,^ ^
^
M M = N N

_^

M N ' N M ' l a m o t h i n h b i n h h a n h do d o I l a t r u n g d i e m c u a

MN

V a y quy t i e h t r u n g d i e m I cua d o a n M N l a m a t p h i n g ( P ) d i qua O s o n g s o n g v d i d j v a

22


Chuyen de 3 :

PHlTONG P H A P T I E N D E

Ta da t h a y di/gc k h i g i a i t o a n h i n h hoc t r o n g k h o n g gian t i f h a i chuyen de trUdc mot each
chi/a t i i d n g m i n h iSm viec sCf d u n g h a i t i e n de 5 va t i e n de 6 t h e nao ?
Den day, de" khftc phuc viec do. Chung t o i 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 h a n , tUdng m i n h h o n :
si^ c a n t h i e t c u a t i e n de 5 v a t i e n de 6. H i e n n h i e n viec gidi thieu r o n g r a i n h i f t h e doi hoi
doc gia can chuan b i mot i t k i e n thiife ve sir vuong goc va nhOmg k h a i n i e m ve cac h i n h k h o i .
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 v i e t sach c h i i n g t o i h y v p n g
duge doc gia dong cam vcii vi$e d a t chuyen de 3 a v i t r i nay t r o n g quyfin sach c h i m o t each
lidc le cung l a du.
L

nnxova

PHAP

Co so ciia phifong phap l a sii d u n g 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 m o t so b a i t o a n co b a n t r o n g k h o n g gian k h i h i n h t h a n h n e n cac v a t the ( h i e n
nhien 4 t i e n de d trirdc da duoc n g a m 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 m p t m a t phAng va d o i m o t cSt n h a u .
Chufng m i n h rSng : a, b, c dong guy.
Giai
T h a t vay : gia sijf a, b, c k h o n g dong quy, t h i cac giao d i e m ciia c h i i n g lap t h a n h ba d i e m
khong t h a n g h a n g va ba difcfng t h f t n g cung nam t r o n g m o t m a t p h a n g . 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 m o t 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 p h a n g .
b/ Ijay t r e n ba t i a Ox, Oy, Oz I a n lifgt cac d i e m A, B, C (khac goc O). Chijfng m i n h r a n g :

(AB + BC + CAf ^ eiOA' + OB^ + OC^)
c/ K y hieu a, p, y la ba goc tarn giac A B C , a, b, c la do d a i OA, OB, OC. T i n h cosa, cosp, cosy
va chufng to r a n g a, [3, y n h o n .
Giai
a/ T h a t

vay

: gia

sCf ba t i a cijng thuge

mot

mat

phang, v i Ox va Oy ciing vuong goc v6i Oz, nen Ox va
Oy cung n a m t r e n m o t du'dng t h a n g . Dieu do t r a i v d i
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^
(AB + BC + CAf

+OA^)


c/ Ap dung d j n h ly h a m cos cho AABC, t a c6 :
BC^ = AC^ + A B ' - 2AC.AB.cosa
23


cosa =

>0

<=> a n h o n (dpcm)
Tirang tif t a c6 : cos(i =

> 0; cosy =
Vc2 +

Va^ + b ^ V b ^ + c ^

> 0
b2.Va^+c2

Do do : P, Y cung n h o n (dpcm)
B a i 39
Cho t r o n g k h o n g gian ba t i a Ox, Oy, Oz doi m o t tao vdri n h a u mot goc 120". Ch\jtng m i n h
rSng ba t i a Ox, Oy, Oz p h a i dong phSng.
Giai
Gia siif Ox, Oy, Oz k h o n g dong phSng va t a chon sAn t r e n Ox; Oy cac d i e m A, B theo thuT
tif do sao cho : OA = OB = 1 (dvcd)
Dong t h d i t r e n t i a doi Oz' cua t i a Oz, t a chon diem C sao cho OC 1 AC. Luc do AABC cho ta:
AC = OAsin60"

AC =

i

—
2

OC = OAcos60"

OC = 2

D i n h l y h a m cosin t r o n g A BOC cho t a :
BC^ = O B ' + O C - 2OB.OCcos60°

BC =

«

BC' = 1 + i

«

Do do :

~ 2.1.-.i =
2

A C = BC =

Tifong t u :
«

2

-

(1)

AB^ = O A ' + OB^ - 2OA.OBcosl20'' = 1 + 1 - 2.1.1(-1) = 3

AB =

Va

(2)

Ttf (1) va (2) t a difgc : CA + CB = A B <=> C e A B <=> Ox, Oy; Oz dong p h ^ n g (v6 l y v d i dieu
gia stf ban dau)
Vay Ox, Oy, Oz p h a i dong phang. (dpcm)
B a i 40
Cho ba t i a Ox, Oy, Oz sao cho xOy ^ x &

= 45" va y(5z = 90". ChiJng m i n h r i n g ba t i a do

cung thuoc m o t m a t phang.
Hi^cTng d a n
Gia sU Ox, Oy, Oz k h o n g dong phSng va chon t r e n do theo thtf
t u cac d i e m A, B, C sao cho : OA = a; OB = OC = a V 2 .
Do gia stf => A e

(OBC)

Stf dung d i n h l y h a m cosin
=>

AB

= AC

AB

Ma

= AC

: BC

=

VOC^

- 20C.OA.COS45"

+ OA^

. a '

= ha^

= a ^/2.^/2 =

- 2.a>/2.a. ^

=

=

a

2a

24