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

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BAN

T527T

GI AO VIEN N AN G

N J GU Y EN

K H I EU TRUCiNG T H I

eC/ C e O N G

( C hu b ie n)

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

D AN G

BOG
Ha

NOI

NHA XUATBAN OAI HO C Q UO C G IA HA NOI

BAN GI AO V I E N NANG K H I E U TRl/CfNG T H I
NGUY EN DLfC D 6 N G {Ch u hien)

TUYEN TAP 500

BAITOAN

•

HDIH i mm GI AN
C H O N LOG
•

•

•

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

• B oi difdng hoc s inh gioi
• C h u a n b i t h i Tii ta i, D a i hoc va Cao da ng
(Tdi

ban idn thvt ba, c6 svCa chUa bo

sung)

THir ViEN TiiVH BiKH liik^m

NHA X UA T BA N DA I H O C QUOC GI A 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
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• Fax: (04) 39714899

*

Chiu
Gidm

Bien
Saa

t r d ch

**

n hiem

xu a t

ban:

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

tap:

TH UY HOA

bdi:

TH AI

Che ban :
Trinh

VAN

N h a s a ch H O N G A N

bay bia:

TH AI V A N

SACH LI E N 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

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

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

mp

( A B C ) va ( E F G )

(il

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

->

• C > : Phep t i n h tien vectcf v
V

• D A : Phep doi xOmg true A
• Do : Phep doi xiiTng true 0

: 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

• 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.

• V s AHc = V ( S . A B C ) : the t i c h h i n h chop

• D N : dinh nghla

S.ABC

• H Q : he qua

• Sxq : D i e n t i c h xung quanh

• D L : dinh ly

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

• B i : budc i

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

• C M R : chiJng m i n h r i n g

• V

: The t i c h

xuong m a t p h i n g (a)

• A ' = ''Vfd) A : A ' l a h i n h chieu cua

• T H i : t r u d n g hop i
A

• V T : ve t r a i

xuong dtfcfng thftng (d)
• 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

• 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

den mat phang ( A B C )
• d p c m : dieu p h a i chuCng m i n h
• (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
•

(3r 3^

: goc

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

• 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 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

•

Mu on xac d in h n h ^ n h mot ma t p h ^ n g tr on g kh on g gian ta con chon thu thu at thUc h a n h :
M p t h i n h t a m g i a c , t ii" g i a c h o a c d a g i a c p h &n g ( 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 g p i c a c m& t p h ^ n g do l a m ^i t
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 t hd'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 d o k h o n g l a b i e n c u a m a t p h d n g bi k h u a t d o ,
t h i d i ^d n g t h &n g d o c u n g tii'oTng vlng k h u a t c u e bp h a y t o a n bp.

No i h a i d i e m m a i t n h a t c 6 m p t d i e m k h u a t t h i dUpc m pt dUcfng k h u a t c ue bp h a y

•

M p t d i e m nhm t r o n g m p t m $ t ph&ng h i n h thuTc bi k h u a t t h i g o i l a d i e m k h ua t .

•

t o a n bp : n e u h a i di i c t ag do k h o n g l a b i e n c u a c a c m ^t phA ng 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 kh u a t cue bo, do (d) c6 1 doan ve

net dijft doan n k m du di (a).

S

•

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

CO mpt doan ve duft doan n k m sau (SAC)

(hien

n h ien (d) cu ng d sau cac ma t (SAB ), (SBC)).
•

C a nh AC b i h a i ma t pha ng (SBC) v£l (SBC) che

kh u a t toan bo, do ca doan AC xem n h u hoan toa n d
sau dong th d i h a i ma t p h ^ n g (SAB ), (SBC).
-A A .

c./—1 —^VF J L ^

•

•

A ] H b i che toa n bo do ca doan A ] H n k m sau ma t

p h i n g ( AiAD D i) , mSc dij no d trU( AB B jAi) va d tr en ma t p h a n g (AB CD ).
• (d) b i che kh u a t cue bo vi c6 doan E F ve net duTt
doan n a m sau h a i ma t pha ng ( AD D iAj) ; (C D D jC j),
mac dij doan E F d phia t r Uc J e ha i ma t phang
( AB B jAi) ; (B C C iB i); va d tr en ma t pha ng (AB CD ).

C A C KY H I E U C A N ^f Hd

Thiir trf

Y nghta

Ky h ieu

(d) n (a) = A

5

(d) // (a)

4

(d) c (a)

3

A i

2

A e (d)

1

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

(d)

G h i ch u
Ha y viet n h a m la :
Ac(d)
(d)

hay

H a y viet n h a m la :

Ha y viet n h a m la :

DU&ng t h i n g (d) n km trong mat p h i n g (a)

A cz( d )

D iem A or ngoai difdng t h i n g
dUcfng t h i n g (d) kh on g chtifa A.

Ca c h v i e t k h a e :

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.

ea ch viet khdc :

D ifdng t h i n g (d) e i t ma t p h i n g (a) ta 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

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

(«)^(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

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

(di)//(d2)

(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
Ch o tiif gia c l o i A BC D c6 cac c a n h d o i k h o n g so n g so n g va d i e m S d n g o a i ( ABCD ) . T i m
giac t u ye n ciia :
a/

( SAC) va ( SBD ) .

hi

( SAB) va ( SD C) ; ( SAD ) va ( SBC) .

Giai
a/

Xe t h a i m a t p h a n g ( SAC) va ( SBD ) , t a c6 :
•
•

S l a d i e m c h u n g th uf n h a t .

(1)

T r o n g tuT gia c l o i ABC D , h a i d u cm g ch eo AC

n BD = O : d i e m c h u n g t h ijf n h i (2).

^

Ti/ (1) va (2) su y r a :
( SAC) o ( SBD ) = SO ( yc b t )
hi

Xe t h a i m a t p h a n g ( SAB) va ( SD C) c u n g c6 :

H a i c a n h b e n A B va CD cu a t i l gia c ABC D

•

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

•

t h eo gia t h i e t k h o n g so n g so n g.
^

AB ^ C D = E : l a d i e m c h u n g th ut h a i .

Do do : ( SAB) n ( SD C) = SE ( yc b t )
Tu cfn g t i f : ( SAD ) n ( SBC) = SF ( yc b t ) ; vd i F = AD ^ BC; do A D / / BC.

Bai 2
Ch o t i l d i e n ABC D . Go i G j , Ga l a t r p n g t a r n h a i t a m giac BCD va AC D . La y t h e o thuT t i i I ,
J, K l a t r u n g d i e m ciia BD , A D , C D . T i m cac gia c t u ye n :

( ABK) ^ ( CI J) = G,G2

d

( GiGa B) n ( ACD ) = GgK h oSc A K

hi

(G1G2C) n ( ABD ) = I J

a/

(G1G2C) o ( AD B)

aJ

hi

(G1G2B) n ( ACD )

c/

( ABK) o (CIJ>.

Bai 3
Ch o h i n h ch o p S. ABCD c6 d a y ABC D l a h i n h b i n h h a n h t a m O .
T i m gia o t u ye n cu a h a i m St p h i n g ( SAB) va ( SCD ) .

hi

T i m gia o t u ye n cu a h a i m a t p h Sn g ( SAD ) va ( SBC) .

aJ
c/

T i m gia o t u ye n ciia h a i m a t p h ^ n g ( SAC) va ( SBD ) .

Giai
aJ

Xe t h a i m a t p h Sn g ( SAD ) va ( SBC) , t a c6 :

De y A D c ( SAD ) ; BC c ( SBC) m a A D // BC.

•

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

•

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

| (SBC) = (xSy; BC)

=^ ( SAD ) n ( SBC) = xSy ( yc b t ) .
hi

Tifa n g t i r , d ifn g u Sv // A B h oft c C D

8


=> ( SAB) r ^ ( SCD ) = u Sv ( ycb t )
c/

Go i O = AC n B D , tiTcrng t a b a i 1
=> ( SAC) n ( SBD ) = SO ( ycb t ) .

Bai 4
Ch o h i n h ch o p S . ABCD c6 d a y la h i n h t h a n g AB C D v d i A B l a d a y Idtn . G p i M la m o t d i e m

b a t k y t r e n SD va E F l a d ifa n g t r u n g b i n h cu a h i n h t h a n g .
a/

T i m gia o t u ye n ciia h a i m St p h i n g ( SAB) va ( SCD) .

b/

T i m gia o t u ye n cu a h a i m a t p h S n g ( SAD ) va ( SBC) ,

c/

T i m gia o t u y e n cu a h a i m St p h a n g ( M E F ) va ( M AB ) .

Doc gia t u g i a i tUcfn g t u n h u cac b a i t r e n .
Bai 5
Ch o h i n h ch o p S . ABCD c6 AB C D l a h i n h b i n h h a n h . Go i G, , G2 l a t r o n g t a m cac t a m gia c
SAD ; SBC. T i m gia o t u y e n cu a cac cSp m St p h a n g :
a/

(SGiG^ ) va ( AB CD )

b/

( CD Gi Gz) va ( S AB)
Uv Cd n g

0/

( AD G2 ) va ( SBC) .

d§Ln

Go i I , J , E , F thur t a Ik t r u n g d i e m cac d o a n t h i n g AD ,
BC, SA, SB t h e o thur tvt d 6 . Th ifc h i e n cac l a p l u a n n h t f cac
bai toan t r e n ;
a/

(SG1G2) n ( ABCD ) = I J ( ycb t )

b/

( CD GiGa ) n ( SAB) = E F ( ycb t )

c/

( ADG2 ) ^ ( SBC) = xG2 y ( ycb t )
T r o n g do xGay // A D h oSc 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

P H i r ON G

PH ANG

P H AP

Ca sd cua p h a a n g p h a p t i m gia o d i e m O cu a d u d n g t h a n g
(a ) va m a t p h Sn g ( a ) l a x e t 2 h a i k h a n S n g xa y r a :
n

T r i r d n g h o p ( a ) ch iJ a d u d n g t h S n g ( b ) va (b ) l a i c&t d iicr n g
t h d n g (a ) t a i O.
T i m O = (a ) n ( b )
=> O la d i e m ca n t i m .

n

T r t fd n g h a p ( a ) k h o n g chiifa dUcm g t h i n g n a o ca t (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 ca n t i m .

n . CAC

BAI TOAM G O B A N

Bai 6
Ch o tuf d i e n AB C D . Go i M , N I a n l u g t la t r u n g d i e m cua AC va BC. L a y d i e m K e B D sao
ch o K B > K D . T i m gia o d i e m ciia h a i d u d n g 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/ C D

( M N K ) = P (ycbt)

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

Ba i 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)

Ba i 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.
H i ^d n g 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 : 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 pha i chufng min h ba diem
trong yeu cau b ^ i toa n 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 kh on g 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
B a i 10
Xet ba diem A, B , C kh on g thuoc ma t p h i n g (u). Goi D, E, F Ia n lu ot la giao diem ciia AB ,
EC, CA va (g). ChCifng m i n h D , E, F th a n g hang.
Gi ai
De y tha y D, E, F viTa a tr on g mp(AB C ) vifa d tr on g mp(a).
Do A, B, C g (a), nen (a) va (AB C) pha n b iet nhau .
=> ( a ) n (AB C) = A (A chuTa D , E, F)
D, E, F t h i n g h a n g tr en A (dpcm).
B a i 11
Ha i ta m giac AB C , A B C khong dong p h i n g c6 AB n A B ' = I , AC n A C = J , B C n B C = K
ChiJfng min h I , J , K t h i n g ha ng.
Gi ai
De y I , J , K Ia n lu ot d tr en h a i ma t p h i n g pha n
(P) ^ (AB C) va (Q) = (A'B 'C ).
N en no 1^ diim chu ng cua h a i ma t p h i n g do

I , J , K e (A ) = (AB C) n ( A ' B C )
=> 1, J , K t h i n g h a n g (dpcm).
B a i 12
Cho A, B la h a i diem d h a i p h ia khac nha u doi vdi ma t p h i n g a va A B c i t a ta i O. D itog
hai dUdng t h i n g x'Ax, y'B y song song nha u theo thuf tiT c i t a ta i M va N . ChuTng m i n h M , N ,
O t h i n g hang.
Kvldng
T i f Ong

^

t\l:

•

fM, O,

dSn

NG(S)

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

=> M , N , O t h i n g h a n g tr en (8)

11


t o a l 4 : Cm iUG

M W fl M Q T D t f Ci N G T H A N G T R O N G K H O N G G I A N
Q U A M O T D I £ M C O D IN H

I.

P H i r ON G

PH AP,

Ca sd cu a p h u cfn g p h a p chuTng m i n h diXcrng t h i n g ( d )
q u a m o t d i e m co d i n h :
T a ca n t i m t r e n ( d ) h a i d i e m t u y y A; B va chuTng m i n h
2 d i e m d o 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
k h o n g gia n .
=> ( d ) q u a I CO d i n h ( d p cm ) .

I L P H tfON G P H AP ,
Co sd cu a phiTcfn g p h a p ca n t h u c h i e n b a bifd c ccf b a n :
n

B i : T i m dUctn g t h i n g a co d i n h d n go a i m St p h 5 n g co
d i n h ( a ) m a ( a ) ch ila d (liOi d o n g) .

•

B2 : T i m gia o d i e m I = a ^ d
=> I l a d i e m co d i n h m a d d i q u a

m. C A C B AI TO A N C O B A N

Bai 13
Ch o 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 k h a c n h a u cu a m &t p h i n g co
d i n h a. Xe t d i e m M lu u d o n g t r o n g k h o n g g i a n sao ch o M A n a = I va M B n a = J . ChuTng
m i n h d ifd n g t h i n g I J l u o n d i q u a m o t d i e m co d i n h .

Giai
Go i 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 cu a ( a ) )
T a CO : m p ( 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 h a n g .
N g h i a l a d acfn g t h i n g I J d i q u a O co d i n h ( d p cm )

Bai 14
Ch o h i n h t h a n g A B C D ( AB / / CD va A B > CD ) . Xe t d i e m S e ( AB CD ) va m a t p h i n g a lu u
d o n g q u a n h AC v d i a '-^ SB = M , a n SD = N . ChuTng m i n h d i fd n g t h i n g M N l u o n lu o n d i q u a
m o t d i e m co d i n h .

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

va

AC c ( SAC) va M N o AC = O t h i O e
B D = ( S BD ) n

( SAC)

=> M N q u a O co d i n h ( d p cm ) .

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 Sm

trong mat

phing

( 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

qua

m o t 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 [(Ox; 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 gh i a la d u d n g t h a n g M N = A lo u d Qn g n h On g
v a n q u a I co d i n h . ( d p c m )

t o a l S : C H O N G M W fl B A O U d N G T H A « G T R O N G K H O N G G I A N D O N G Q U Y
L PH U ON G

PHAP,

Co s o c u a 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 t h iif n h a t
q u a 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 c o b a n :

•
•

Bi : T i m (d,) o

(d^)

= O

B2 : Ch u f n g m i n h ( d ; j ) q u a O .
=> ( d i ) , ( d 2 ) , ( 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,

Co s d c u a p h a a n g p h a p l a t a c a n c h i j fn g m i n h

ch u n g

doi m o t ca t n h a u va 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 b i fd c c a b a n :
d],
•

B i : Xa c d i n h <

c: a ; d j

da = I i

0

id3

\

d a , d ;j cz P; d a ^ d 3 = I2
A

di

d g, d , e Y; d g n d j = I 3

\

\

a , p, y p h a n b i §t

•

B2 : K e t l u a n ( d , ) ; ( d a ) ; ( 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
Ch 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 , AC , B D sao ch o E F n B C = I ,
E G o AD = J (vd i I ^

C wk J

^

B).

Ch 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 d i fc fn g t h A n g C D ; I G v a J F , t a t h a ' y :
CD , I G e ( BD C) v a CD
• I G, J F c ( E F G)

IG /

0

va I G n J F * 0

J F , CD e ( ACD ) v a J F r> CD * 0
Va b a 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 c h u r n g 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 q u y .

B a i 17
Ch o h a i t a m gia c A B C , A B C s a o c h o A B c a t A 'B ' a E , A C cd t A C d F ; B C c a t B C d G .

C h i J n g m i n h d i fc fn g t h a n g A A ' , B B ' , C C d o n g q u y .

b/

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

a/

Gid i
a/

D e y th ay E , F , G l a b a d ie m ch u n g cu a h a i m a t ph ^ n g p h a n biet
(a) ^ ( AB C) va (P) = ( AB 'C) .
D o do : E , F , G e ( A) = ( a ) n ( P ) .
Vay E , F , G th ^ n g h a n g (dpcm ).

b/

N h an xet n h u sau :
: 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 CC) ; C C n AA' # 0

A A ', B B ' , C C d o n g q u y t a i O ( d p c m ) .

Chuyen

de 2 :

QU AN H E SONG SONG

t o a i 1: CH t J N G M W fl H A I D LfCJN G

THANG S O N G

SON G

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 d ifd n g 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 h ife u n g a m r a n g h i e n n h i e n d i e u d o 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 d o . ( 1)

•

B 2 : D u n g d in h ly Th a le s , t a m giac dong dan g, tin h ch at bac cau ( tin h ch at cun g song
s o n g \ 6i

d ifd n g th iJ 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 d o i c u a h i n h b i n h h a n h ,

... d e k h a n g d i n h h a i d ifc fn g t h ^ n g d o k h o n g c 6 d i e m c h u n g . ( 2 )
T i f ( 1) v a ( 2 ) => ( y c b t )

n . CAC BAI TOAN CO BAN
B a i 18
C h o h i n h c h o p S . A B C D c 6 G j , G 2 , G3 , G , I a n lu cft 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


Gi ai
SG,
SE
Theo tin h chat tr on g tarn, ta c6 : - i,
SG3
t
[ SH

SG2
SF

2

SG4

2

SK

3

3

Dinh ly Thales va tin h chat diTcfng tr u n g b in h
G,G2/ / = - E F ; E F 7 / = i A C
'

^

3

2

/ / = G; j G4

• G1G2

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

^

'

3

2

G1G2G3G4 la h in h b in h h a n h (dpcm).
B a i 19
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.
S

Gi ai
D6 y thay ha i mS t phSng (a) va (P) c6 2 diem M vfl N 1^ d i^ m chu ng.
=> M N = (a) n (SB C)

'(a) 3 AD
ma ^ ( S B C ) 3 B C
iAD//BC

N

va theo each dimg M N // A D (hoftc B C)
=> A D N M la h i n h th a n g day lorn A D . (dpcm)
B a i 20
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 tu y y
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 rS ng : I J // C D .
Gi ai
Xet h a i ma t pha ng ( A M N ) va (PCD ) c6 h a i diem chu ng la I va J .
IJ = ( A M N )

r-.

(PCD )

'CD c (PCD)
N himg < MX CT (AMN)
• va MN // CD
^

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

t o ai Z : Cf lt Jf JG M W H DiidfiG T H A N G S O N G S O N G T f Cl M A T F H A N G
L PHtrOWG P H AP ,
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 :
•

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

•

B2 : Ket lu a n tif tr en d // a.

d

y r- a = a
pna = b.
a//b

15


n . PHOONG PHAP^
Ca sd ciia phifcng phap la stf du ng dieu kien can va du
chijfng m i n h di/dng t h i n g (d) song song vcJi ma t pha ng

(a)

b a ng h a i btfdrc :
•

B i : Qu an sat va qu an ly gia th iet t i m du dng t h i n g ou
viet (A) cz (a) va chiJng m i n h (d) // (A).

•

B2 : Ket lu a n (d) // (a) theo dieu kien can va dii.

m. c A c

B AI TOAN C O BAM

Bai 21

Tr on g tuf dien A B C D , chufng min h rSng dean no'i h a i tr on g ta m G i, G 2 cua ha i A AB C

va

AAB D th i song song v6[ (AC D ).
A

Gi ai
Goi A i , A2 la tr u n g d iem B C va B D theo thut tiT do, ta c6 :
3

AA, ' AAg

2

AG)

AG2

Theo d in h ly Tha les , ta c6 :
' 0 , 0 2 / / A , A2

B

'ma A, A2 //CD (tinh chat dUcrng trung binh)
Theo ti n h bSc cau

=>

G 1G 2 // CD c: (ACD )

=j.

G 1G 2 // (ACD )

(dpcm)

B a i 22
Cho h i n h chop S.AB CD day la h i n h b in h h a n h AB C D . G oi M , N la tr u n g diem SA va SB.
Chijfng m i n h : M N // (SCD) va AB // ( M N C D ) .
Gi ai
Theo ti n h cha t du dng tr u n g b in h tr on g ta m giac
=> M N // AB , ma AB // CD
=> M N // C D
Theo dieu kien can va du
O

cz

(SCD)

=> M N // (SCD)

(ycbt).

Cac h khac
De y M N = ( M N C D ) n (SAB ) va tr on g ha i ma t pha ng do

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

TifOng tyl :

D

=> M N // (SCD) 3 CD (ycb t)

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

=> AB // ( C D M N ) (dpcm).

B a i 23
Xet ha i h i n h b in h h a n h A B C D va A B E F kh on g dong p h l n g . Goi M , N la h a i diem thoa
AM - i AC va B N = - BF . Chufng min h r i n g M N // (D E F).
3
3
Gi ai
De y tha y M , N la tr on g ta m cua b a i ta m giac AB D va
AB E theo thijf tu do.
Keo da i t h i D M o E N = P : la tr u n g diem AB .
^

PE

PD

PX

PM

1

3

Theo d in h ly Tha les
^

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

D

16


Bai 24
H i n h ch op 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 thiJc vecto : 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 // AB ; 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 diTdng t h S n g ch eo 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 cho A M = B N . C hijfng m i n h r S n g dU cfng 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 gia 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 '
D e y A A M N ' c a n d A n e n t i a p h a n gia c n g o a i A t cu a S TA J T 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 xa 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 on g s on g v6i m&t p h a n g co d i n h

( d p cm) .

t o al 3 : H A I M A T P H A N G S O N G S O N G

Dang

1 : CHQNG MINH HAI MAT PHANG SONG SONG

L P m r OH G PHAP
Co

s d cu a p h u o n g

phap

ch iJ n g m i n h

ha i ma t

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 : C hu fng m i n h

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

t h a n g a , b d o n g q u y thijf t i f s on g s o n g v d i h a i
dU oing 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".
•

B 2 : K e t l u a n ( a ) // (P) th e o d i e u k i e n c a n v a d u .

THL; VJENTifJHglNHTHUAN

17


n . ckc

B AITOAN 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 I c h o n g d o n g p h ^ n g A x , B y , C z M n lifo t 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 d o d a i k h a c k h o n g . Ch Of n g m i n h ( A B C ) / / ( A B C ) .

Giai
AA' =3 BB'

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

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

Nen

t a c 6 h a i dUcrn g t h ^ n g

c

(ABC)

dong

quyAB ', A C

t r o n g m p ( A'B 'C') t h o a d i e u k i e n ( I ) .
=> ( A B ' C ) / / ( A B C ) ( d p c m )

Bai 27
Ch o h i n h b i n h h a n h A B C D . Ti r A v a C k e A x c a C y s o n g s o n g c u n g c h i e u v a k h o n g n k m
t r o n g m a t p h S n g ( A B C D ) . Ch i i f n g m i n h ( B ; A x ) / / ( D ; C y ) .

Gi&i
Ti r a n g t u x e t h a i m a t p h i n g ( B ; A x ) v a
( D ; C y ) , thuT t a ch u Ta c a c c a p d u d n g

thing

d o n g qu y.

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

Bai 28
Ch o h a i h i n h b i n h h ^ n h A B C D v a A B E F d tr o n g h a i m a t p h ^ n g k h a c n h a u . Ch i l n g 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 ) th iif tiT ch u Ta c a c c a p d i r d n g
th d n g d o n g qu y.

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
LP B i r ON GP B AP
Cc f s d c u a p h u a n g p h a p c h i i f n g m i n h c a c d u d n g t h i n g d i , d 2 , d g... d o n g p h i n g l a c a n p h a i
th i Tc h i ^ n h a i bi/ •

B i : Ch v i n g m i n h d ] , dg, d s , ... 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)
nao 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 BA 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 De (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, F A

=> 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 L i " T H A I JE S T R O N G K H 6 N G
• k D i n h l y i (thu|ln) : Hai dit ang t hing
khong gian chdn t ren cdc in^t phdng

GIAN

t uy y d,, d2 t rong
song song nhau (a)

II (P) II (y) t ao ra cdc doan t hang t Ucm g ling t y le :
A,A.

-k D i n h l y 2 ( d a o ) :
•

Tr U d c k h i x e t d i n h l y d a o , t a q u a n

t a rn d e n h a i k h a i

n ie m sau k h i xe t de n cac d ay 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 g o c 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 s o ( *) .

•

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

•

( A i ; B ] ) l a c a p g o c c u a d a y t y s o ( *) .

•

D i n h l y : Neu c6 day t y so t rong khong gian

:

A, A.

(*) da, xdy ra t ren hai

(di)
( g o c ) 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

t hdng
A,B,,

m at phdng

(dj),

(n gpn t re n ) A2

fd^)

t hi

m ot

A2B2, A^B^ se song

dudng

A2B2, A3B3 cung song song vai m ot m g,t phdng

t hdng (d,), (d2) t hi cdc bac t hang AiB,,
dinh.
O

Th a l e s d ao n h i f s a u :
Vai dieu kien c6 day t y so (*) da xdy ra t ren hai
dudng
t hang

t rong
song

3

bac

(n go n difdi) A 3

c6

(dz)
^ B, ( g o c )
B2 (n go n t re n )
B 3 (n go n dU(Ji)

vdi m ot

chda hai bac t hang con lai.

"A, B i / / ( a ) = ( A2B2 ; A3B3)
A2B 2/ / ( P ) - ( A3B 3;Ai B i )
A 3 B 3 / / ( Y) S ( A, B i

( A)

; A2 B 2 )

( m a t p h a n g co d i n h ) \

Dang 3 : CHUfNG MINH DJCiNG THANG SONG SONG MAT PHANG
BANG DINH LY THALES
L P H i rO N G P H A P ,
C o s d c u a p h u o n g p h a p ch u fn g m i n h d U d n g 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 Th a l e s d ao t ro 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 t i i y y c h a n g h a n ( d i ) , ( d 2) d e t i m t r e n d o d a y t y s o :

B, B3

AjAs

^ B1B2

^1^2

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

B 2 : L u c d o c a c d o a n b a c t h a n g A i B i , A2B2, A3B3 d i rac 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 am tro ng m at p h i n g (a) / / (()) => (d ) / / (p ).
BA I T O A N C O B A N

m. cAc

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

i(A; C) la cap go'c

AB

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

CD

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 vdue 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 : g o 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)
mat phIng

CO

d i n h v a cung so ng so ng

v a (a) tho a y eu cau la

vdri

A C , BD v a M N .

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

AM
AC

cho

BN
. Chutng m i n h rSng ta luo n co : M N / /
BF

(DEF).

Giai
Tir gia th i e t

AM

BN

AC

BF

(*)

A p d ung d i n h ly Thales dao cho cac

d o an

bac

thang : A B, M N , CF.
=> M N / / ( CDF) ; v i A B / / C D c

(CDF)

=> M N / / ( D EF) = (CDF) (d p cm)

Bai 34
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) .

Giai
Do hai h i n h v uo ng A BC D , A BEF b ang canh
bang nhau
„ .,,,..-^
Gia th i e t =>

AM

BN
=

AC

BF

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
^

nen

=> A C = BF.

M N / / (CEF)

(CEF)

(d p cm)
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 cheo nhau d j va d 2 . M la m o t d i e m c h u y e n d o n g t r e n d i va N la

mot

d i e m c h u y e n d o n g t r e n d 2 . 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

dinh

d'2

t h i 0 1 nkm

O song song vdi di va

bai

hai

di

AB.
i

IX

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

M'

O
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

l y 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

song vdi A B . C h u n g Ian lagt e l t d] d M va 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

P H l T ONG P H A P T I E N D E

de 3 :

Ta da tha y di/gc k h i gia i toa n h in h hoc tr on g kh on g gian tif ha i chu yen de trUdc mot each
chi/a tiidng m i n h iSm viec sCf du ng h a i tien de 5 va tien de 6 the nao ?
Den day, de" khftc phuc viec do. Chu ng toi diTa vac mot chuyen dfe PHUCfNG PHAP TIEN' DE
vdi mot mong mu on la doc gia se thiTc sa thay difoc mot each chinh xac ha n, tU dng min h hon :
si^ c a n t h i e t c u a t i e n d e 5 v a t i e n d e 6. Hien n h ien viec gidi thieu rong r a i nhif the doi hoi
doc gia can chuan b i mot i t kien thiife ve sir vuong goc va nhOmg kh a i niem ve cac h in h khoi.
Sau nhOrng suy n gh i va tr a n tr d tr on g suot qua day hoc va viet sach chiing toi hy vpng
duge doc gia dong cam vcii vi$e da t chu yen de 3 a vi t r i nay tr on g quyfin sach chi mot each
lidc le cung la du.
L

nnxova

PHAP

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 m i n h mot so b a i toa n co b a n tr on g kh on g 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).
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 kh on g ciing nkm tr on g mpt ma t phAng va doi mot cSt nha u .
Chufng min h rSng : a, b, c dong guy.
Gi ai
Th a 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 cu ng nam tr on g mot ma t pha ng. Tr a i vd i 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 cu ng n k m tr on g mot ma t pha ng.
b/ Ijay tr en ba tia Ox, Oy, Oz Ia n 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 AB C , a, b, c la do da i OA, OB , OC. Ti n h cosa, cosp, cosy
va chufng to r a n g a, [3, y nhon.
Gi ai
a/ Th a t vay

: gia

sCf ba tia cijng thuge mot

ma t

phang, vi Ox va Oy ciing vu ong 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 ycb t di/gc chufng m i n h b a ng phep

chufng

min h phan chufng.
b/ Ap du ng bat da ng thufc B u nhiacovky, ta eo :
(AB +BC +CA)^ < 3( AB ' + BC^ + CA^) = 3(0A^ + 0B ^+ OB^ + OC^ + OC^
(AB + BC + CAf

+OA^)


c/ Ap du ng d jn h ly h a m cos cho AAB C, ta c6 :
BC^ = AC^ + A B ' - 2AC.AB .cosa
23


cosa =

>0

<=> a nhon (dpcm)
Tirang tif ta c6 : cos(i =

> 0; cosy =
Vc2 +

Va^ + b ^ V b ^ + c ^

> 0
b2.Va^+c 2

Do do : P, Y cung nhon (dpcm)
B a i 39
Cho tr on g kh on g gian ba tia Ox, Oy, Oz doi mot tao vdri nha u mot goc 120". Ch\jtng min h
rSng ba tia Ox, Oy, Oz pha i dong phSng.
Gi ai
G ia siif Ox, Oy, Oz kh on g dong phSng va ta chon sAn tr en Ox; Oy cac diem A, B theo thuT
tif do sao cho : OA = OB = 1 (dvcd)
Dong thdi tr en tia doi Oz' cua tia Oz, ta chon diem C sao cho OC 1 AC. Luc do AABC cho ta:
AC = OAsin60"

AC =

i

—
2

OC = OAcos60"

OC = 2

D in h ly h a m cosin tr on g A B OC cho ta :
BC^ = O B ' + O C - 2OB.OCcos60°

BC =

«

BC' = 1 + i

«

Do do :

~ 2.1.-.i =
2

AC = B C =

Tifong t u :
«

2

-

(1)

AB^ = O A' + OB^ - 2OA.OB cosl20'' = 1 + 1 - 2.1.1(-1) = 3

AB =

Va

(2)

Ttf (1) va (2) ta difgc : CA + CB = AB <=> C e AB <=> Ox, Oy; Oz dong p h ^ n g (v6 ly vdi dieu
gia stf b an dau)
Vay Ox, Oy, Oz pha i dong phang. (dpcm)
Ba i 40
Cho ba tia Ox, Oy, Oz sao cho xOy ^ x &

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

cung thuoc mot ma t phang.
Hi^cTng d a n
G ia sU Ox, Oy, Oz kh on g dong phSng va chon tr en do theo thtf
tu cac diem A, B, C sao cho : OA = a; OB = OC = a V 2 .
Do gia stf => A e

(OBC)

Stf du ng d in h ly h a m cosin
=>

AB

= AC

AB

Ma

= AC

: BC

=

VOC^

= ha^

- 20C . O A. C O S 4 5 "

+ OA^

- 2.a>/2.a. ^

. a '

= a ^/2.^/2 =

=

=

a

2a

24