Chuyên đề trọng điểm bồi dưỡng học sinh giỏi hình học không gian phần 1

516.23076

NeuyiNmMcsini

CH5270

#

*

m

m

B6 DubnG HQC smH Gidi

HinH HOC KHOnC GIHH
Danh cho hoc sinh khoi 12 chifong trinh chuan va nang cao
On tap va nang cao kl nang lam bai
Bien scan theo noi dung va cau true de thi cua Bo GD&OT

«

M
MNci(SBC)
PQC(SAD)

K€(SBC)

DVL.013499

'K

A I _ AJ _ 1
I C ~ JK ~ 2

AI ^ 1
AC' ~ 3

NHA XUAT BAN DAI HOC QUOC GIA HA NO!


DU0nc3 Higc sin

HiRH HOC KHOnC GIHH
Danh cho hpc sinh khd'i 12 chaong trlnh chuan va nang cao
On tap va nang cao kl nang l^m b^i
Bien soan theo noi dung cau true de thi cua Bp GD&OT

TKUViiN ifeHEiNHTflOAN

i i>-

.

J .i^'Kj n i l ;

•ithr • tin

I* Mij'.i i(;iy.'ii':i.>.

nr.? v f i c , * n .


Cni/ONG I .

D A I CUONG

nlNH H Q C KHONG GIAN

Buoc dau tien lam quen voi H m h hpc khong gian, cac b^n cac b^n phki nho
ky cac khai ni^m va nhung tinh chat sau sau:
L KHAINI^MMdDAU

Cuon sach H I N H H Q C K H 6 N G G I A N , du(?c bien soan n h ^ mvc dfch

1. M S t p h i n g :

H I N H K H 6 N G G I A N K H 6 N G C 6 N L A M6'I L O L A N G C H O C A C B A N

M|t bang, m^t ban, mat nuoc ho yen l?uig, m§t san nha,... cho ta hiiJti knh

HOC SINH PHO THONG .

Cuon H I N H K H 6 N G G I A N duban den nang cao H I N H K H 6 N G G I A N L 6 P 11, H I N H K H 6 N G G I A N

mpt phan cixa m$t phang. M$t ph5ng khong c6 be day va khong c6 gioi h^n.
De bieu dien m^t phJing ta thuong diing hinh binh hanh hay mpt mien goc
va ghi ten cua m^t phJing do vao mpt goc ciia hinh bieu dien (nhu hinh 1).

L O P 12 va dac bi?t L U Y $ N T H I D A I H O C .

De k i hi?u mat phang , ta thuong dung chu cai in hoa hole chu cai H i L^p

Noi dung cuon sach gom :

dat trong dau ( ) . V i d\i: mat phSng (P), mat phSng (Q), m|t phSng ( a ) , m^t

Chuong I : Dai cuong hinh hpc khong gian.

phang (p) hoSc viet tat la mp(P), mp(Q), m p ( a ) , m p ( p ) , ho|c (P), (Q),

Chuong I I : Quan h^ song song trong khong gian.
Chuang I I I : Quan h$ vuong goc trong khong gian.

{«)'(P)-

Chuong I V : The tich khol tru, the tich khoi chop.
Chuong V : Matcau, mat tru, mat non.

'

Chuong V I : Bai tap tong hgp lop 12 hoc ky I.

^

„ ,

^

•

'

,

Hit ,p,rmrt;r

Chuong V I I : Bai tap tong hop luyen thi Dai Hoc.
Cuon sach duoc trinh bay ngan gon, ro rang, voi mot luong bai tap rat ?
Ion va rat day du, cac bai tap duoc giai chi tiet, chat che, de hieu. Nham
giiip cac em c6 mot djnh huong de giai quyet van de ciia bai tap do.
Cuon sach ditgc phan bo 6 chuang, moi chuang dugc tom tat ly thuye't day
du. Bai tap trong moi chuang duoc phan dang ro r^ng, moi dang c6 tom tat

Hlnhl

2. Diem thupc mat phang:

n'^iri

Cho diem A va mat p h i n g ( a ) .
Khi diem A thupc m^t phSng ( a ) , ta

phuong phap giai bai tap. Bai tap phan bo trong moi chuang hoac moi

noi A nam tren (a)

phan la t u de den kho, trong moi dang deu c6 long vao nhung bai tap ciia

(a) chua A, hay mat phSng (a) d i qua

dang thi Dai Hoc. Nhiing bai tap de nham muc dich giiip cac ban nam ro ly

diem A va k i hi^u A G ( a ) , dupe bieu

thuye't va phuong phap de chung minh hoac giai quyet mot van de cu the,
va t u do cac ban c6 ky nang de giai quyet nhung bai tap kho hon. Tac gia
rat hy vpng quyen sach nay la nguoi ban dong hanh tot cho cac em trong
qua trinh hpc va trong nhung ky thi.

i

Tuy c6' gang nhieu trong qua trinh bien soan, nhung cuon sach khong
the tranh khoi nhirng thieu sot. Tac gia rat mong nhan dugc y kien dong
gop chan thanh qui bau t u phia ban doc, cac em hoc sinh va cac ban dong
nghiep gan xa de nhOng tai ban Ian sau sach se hoan thien hon.
Xin chan thanh cam on !
Tac gia NGUYEN Q U A N G SON

hay m|it phSng
Hinh 2

dien 6 hinh 2 .
Khi diem A khong thupc mat phSng
(a) ta noi diem A nam ngoai m^t
phSng (a) hay m|t phSng (a) khong
chiia diem A va k i hi?u la

Ag(a),

dupe bieu dien d hinh 3 .
I I . CAC T I N H CHAT DLTOCTHl/A N H A N
T i n h chat 1: Co mpt va chi mpt duong
thang di qua hai diem phan bi?t.

Hinh 3


T i n h c h a t 2: C o mot v a chi m o t m a t

-

.A

p h a n g di q u a ba d i e m idiong thang hang.

diem

phan

bi^t

thuoc

mpt

duong
thuoc

duong

thSng

thang
d.

d

va

K h i do

d

xac

diem

diem

dinh

mpt

A

A
va

mat

p h S n g , k i h i e u la m p ( A , d ) , h o a c m p

T i n h c h a t 3: Ne'u m o t d u o n g t h a n g c 6
hai

Cho

khong

(d, A ) h a y ( d . A ) .

mat

M a t p h S n g d u p e hoan toan xac djnh khi biet no c h i i a hai d u o n g thSng c3t nhau:

p h a n g thi m p i d i e m c u a d u o n g t h S n g

C h o h a i d u o n g t h a n g cat n h a u a v a b .

d e u thuQC m a t p h a n g do.

K h i d o h a i d u o n g t h a n g a v a b xac d j n h

• D

m p t m a t p h a n g v a ki h i e u la m p ( a , b)

T i n h c h a t 4: T o n tai b o n d i e m k h o n g

h a y (a , b ) , h o a c m p (b , a) hay (b, a ) .

cung thuoc mpt mat ph5ng .

T i n h c h a t 5: N e u hai m a t p h a n g p h a n
bit^t C O m p t d i e m c h u n g thi c h u n g c o n
CO m p t d i e m c h u n g k h a c n u a .

H I N H CHOP VA Tir DIEN

T u tinh c h a t n a y s u y ra: Ne'u hai m a t
p h a n g p h a n biet c 6 m p t d i e m c h u n g thi

1.

c h u n g se c 6 mot d u u n g t h a n g c h u n g di
qua

diem

chung

ay.

Duong

c h u n g la d u y nhiit c h u a tat ca cac d i e m

t h u o c m a t p h ^ n g ( a ) . L a n l u p t noi d i e m S v o i cac d i n h A ^ , A j , A 3 , . . . , A,., ta

c h u n g cua hai mat phang do . D u o n g

dupe

t h a n g c h u n g d o d u p e goi la giao t u y e n

V i d u : D u o n g t h a n g c h u n g d c u a hai m a t p h a n g p h a n biet ( a ) v a ( p ) d u p e
v a ki hieu la d = ( a ) n ( p ) .

T i n h c h a t 6: 7 ren m o i m a t p h a n g , cac ket q u a d a biet trong h i n h hpc p h a n g

tam

giac

chop,

S A j A j , SAjA-,,..., SA,^A,.

n t a m giac

S A j A j , SA2A3

ki h i e u la S . A , A 2 A 3 . . . A , , . T a gpi S la dinh

A , A 2 A 3 . . . A | , la mat
SAj^Ajdupc

gpi

day
la

cm

cac

hinh
mat

III. C A C H X A C D I N H M O T M A T

PHANG

hen

cm

hinh

gom

da

S A ^ A , d u p e gpi
cm

hinh

cac t a m g i a c
chop,

giac

la

hinh

chop, c o n d a giac

SAjAj, SA2A3,
cac

doan

thang

phang

dupe

hoan

toan

T a gpi h i n h c h o p c 6 d a y la tam giac, t u giac, ngij giac,..., l a n l u p t la hinh
tam i^idc, hinh chop ti'f giac, hinh chop ngu giiic,

C o ba each x a c d j n h m p t m a t p h a n g :

xac

C h o b o n d i e m A , B , C , D k h o n g d o n g p h S n g . H i n h g o m b o n t a m giac

d j n h k h i biet n o d i q u a ba d i e m k h o n g

A B C , A C D , A B D v a B C D gpi la hinh tie dien (hay ngMn g p n gpi la tu dien)
d u p e k i h i ^ u la A B C D . C a c d i e m A , B, C , D gpi la c a c dinh

'

:j

M a t p h a n g d u p e h o a n toan xac d j n h

chop

....

thang hang.
-

d o a n t h S n g A B , B C , C D , D A , C A , B D gpi la c a c canh cm

cm

tu dien

tie dicn.

va
Cac

. H a i canh

khi biet n o d i q u a m p t d i e m v a c h u a

k h o n g d i q u a m p t d i n h gpi la hai canh doi dien cua tie dien. C a c t a m giac A B C ,

mpt d u o n g thang k h o n g di q u a d i e m do.

A C D , A B D , B C D gpi la cac mat ciia tie dien . D i n h k h o n g n S m tren m a t gpi la
dinh doi dien cm

mat do .
,;V... ••

4

chop,

Hinh

S A , , S A 2 , S A 3 , . . . , S A , , d u p e gpi la cac canh ben ciia hinh chop .

deu dung.

Mat

n

AjA2A3...A„ va

ciia hai m a t p h a n g .

-

Khai niem:
T r o n g m a t p h ^ n g ( a ) c h o d a giac loi A , A 2 A 3 . . . A ^ . L a y m p t d i e m S k h o n g

thang

gpi la G I A O T U Y E N c u a hai mat p h a n g ( a ) v a

,!'; '

%'


H i n h tu d i # n c6 b o n m 3 t la cdc tarn gidc deu g p i la h i n h tii diftt deu .
A

S

BAI TAP G I A I CHI T I E T
Co b o n d g n g toan chi'nh la: T i m giao t u y e n cua hai m a t p h 3 n g . T i m giao
d i e m ciia d u o n g th3ng va mat phang. T i m thiet di?n ciia h i n h chop. C h u n g
m i n h ba d i e m thJing hang, ba d u o n g thMng d o n g q u i ... Ta Ian l u p t xet t u n g
d ^ n g m p t n h u sau:

D A N G 1: Tim giao tuyen cua hai m|t phang.

,1.

,

Phuong phap:
M u o n tim giao tuyen ciia hai mat phMng? Ta t i m hai d i e m chung thuQc ca
hai mat p h a n g . N61 hai d i e m chung do duVedang nay diem chung thu nhat thuang de tim. Diem chung con lai cac ban phai
Hinh chop tir giac

tim hai dieang thang Ian lugt thuoc hai mat phhng, dong thai chung lai thuoc mat
phang thie ba va chiing khong song song. Giao diem cua hai ditong thUng do la
diem chung thi'e hai.

s

Cac ban p h a i nho ky: Giao tuyen la duong thang chung cua hai m a t phJing,
CO nghla la giao tuyen la duong th5ng vua thupc mat p h i i n g nay vua thuQC
mat p h a n g kia.
D g n g toan t i m giao tuyen, thuong giao tuyen ciia nhOng cau h o i dau hay
dupe s u dyng de t i m giao d i e m de l a m bai t?ip 0 nhung cau sau. Ta xet cy
thenhij'ng bai toan sau:
Cho t u giac ABCD sao cho cac canh d o i k h o n g song song v o i n h a u . Lay
m p t d i e m S k h o n g thupc mat phang (ABCD). Xac d i n h giao t u y e n cua :
a) M p (SAC) va m p (SBD).
b) M p (SAB) va m p (SCD).

(Sj tj

c) M p (SAD) va m p (SBC).
. Hinh chop tu- giac c6 day la hinh binh hanh

LOI GIAI
a) N h i n h i n h ta de d a n g tha'y S la
d i e m c h u n g t h u nhat, noi A C va
B D lai c h u n g cat nhau tai O, t h i
O la d i e m c h u n g t h u hai, A C va
B D cat nhau la v i c h u n g c i i n g
thupc m a t phSng day ( A B C D ) .

.1 n,>-> QDa .'JBA ,i:r:>/,

Cach t r i n h bay giao tuyen ciia
(SAC) va (SBD) n h u sau:
Ta CO S e ( S A C ) n ( S B D )

(1)

Trong m p ( A B C D ) gpi O = A C n B D .


V\\

fOe AC,ACc:(SAC)
,
/=^Oe{SAC)n(SBD)
OeBD,BDcz(SBD)
V

^
(2)
/

,

.

V

;

b) T i m giao t u y e n cua m p (IBC) va m p ( D M N ) .
T r o n g m p ( A B D ) goi E = BI n D M , v i :

v^

E€BI,BIc(lBC)

Tir (1) va (2) suy ra (SAC) n ( S B D ) = S O .

EeDM,DMc(DMN)

b) Cau b c u n g t u o n g t u (SAB) va (SCD) c6 d i e m c h u n g t h i i nhat la S, d i e m
c h u n g t h i i hai la E, v o i E la giao d i e m ciia A B va C D v i hai d u a n g thSng nay

=>Ec:(lBC)n(DMN)

c u n g thuQC m a t p h a n g ( A B C D ) va chung k h o n g song song v o i n h a u . Cach

T r o n g m p ( A C D ) goi F = C I n D N , v i :

t r i n h bay c u n g n h u cau a):
Ta CO S € ( S A B ) n ( S C D )

(3)

,

jEe

AB,ABc(SAB)

5

|EGCD,CDC{SCD)

'FGDN,DNC(DMN)

Ee(SAB)n(SCD)

(4)

Cho t u d i e n A B C D . Lay cac d i e m M thuoc canh A B , N thuoc canh A C sao

c) Cau nay c u n g vay S la d i e m chung t h u nhat, d i e m c h u n g thiV hai la giao
d i e m ciia A D va BC, v i hai d u o n g thSng nay cung thuoc m p ( A B C D ) va
c h i i n g k h o n g song song, ta t r i n h bay n h u sau:

FeBC,BCc(SBC)

cho M N cat BC. Goi I la d i e m ben trong tam giac BCD. T i m giao tuyen ciia:
a) M p ( M N I ) va m p (BCD).
b) M p ( M N I ) va m p ( A B D ) .
c)

(5)

( A B , C D e ( A B C D ) ) Goi F

M p ( M N I ) va m p ( A C D ) .
LOI GIAI

A D n BC, v i :

a) M p ( M N I ) va m p (BCD).

Fe(SAD)n(SBC)

Goi H = M N n B C ( M N , B C c ( A B C ) )

(6)

Taco: I e ( l M N ) n ( B C D )

Tir (5) (6) suy ra ( S A D ) n (SBC) = S F .

a) T i m giao t u y e n ciia 2 m p (IBC) va m p (JAD).

, <

b) Lay d i e m M thuoc canh A B , N thuoc canh A C sao cho M , N k h o n g la
LOI GIAI

I e (IBC)

, :^Ie(lBC)nfjAD)
I e A D , A D c (JAD)
/
'
'

T u (1) va (2) ^

( I B C ) n ( J A D ) = IJ

'

^

=^He(lMN)n(BCD)

(2)

T r o n g m p (BCD) goi E va F Ian l u g t la giao d i e m cua H I v o i B D va C D .

(1)

Je(jAD).,,

^

HeBC,BCc(BCD)

b) M p ( M N I ) va m p ( A B D ) .

a) T i m giao t u y e n ciia 2 m p (IBC) va m p (]AD).

j6BC,BCc(lBC)

//

Tu(1)va(2): (IMN)n(BCD) =HI

t r u n g d i e m . T i m giao tuyen ciia m p (IBC) va m p ( D M N ) .

Va

(l)

HeMN,MNc(lMN)
va <

Cho t u dien A B C D . Goi I , J Ian l u g t la t r u n g d i e m cac canh A D , BC.

Co

(2)

Tu'(l)va (1): ( i B C ) n ( D M N ) = E F

Tir (3) va (4) suy ra ( S A B ) n ( S C D ) = SE

FeAD,ADc(SAD)

•'" •^- • • ' i ' ;

' ^''^
=^Fc(lBC)n(DMN)

Taco SG(SAD)n(SBC)

g

FeCI,CIc(lBC)

'''' • >

T r o n g m p ( A B C D ) goi E = A B n C D , v i :

(l)

'

(2) ^

'..fin th

fM

Co

Va

M e (MNI)
Me ABc(ABD)
EeHIc(MNl)
EGBD(=(ABD)~

r>Me(MNl)n(ABD)

(3)

••;

EG(MNl)n(ABD)

T u (3) va (4) suy ra ( M N I ) n ( A B D ) = M E .

(4)'
^,

i^'^^'''


c) M p ( M N I ) va m p ( A C D ) .
Co

N e ( M N I ) n ( A C D ) (5).

N e AC c (ACD)
FeHIc(MNl)
a
\) ( 6 ) .
FeCDc(ACD)

V

Cho h i n h chop S.ABCD day la h i n h binh hanh t a m O. G p i M , N , P Ian l u p t la

Ne(MNl)

t r u n g d i e m cac canh BC, C D , SA. T i m giao tuyen cua :

''" • ' ^ i
,,•4.

,

.

, V, „

^

,
/,

,

c) M p ( M N P ) va m p (SBC).

(vi M N ,

Cho h i n h chop S.ABCD c6 day A B C D la h i n h thang c6 A B song song C D .
cua:

•

Co

-i

iymf^rxlOm}^

b) M p (SAD) va m p (SBC).

Co

a) T i m giao t u y e n cua m a t phcing (SAC) va (SBD).
(l)

P£SA,SAC(SAB)

FeMN,MNc(MNP)
F€AB,ABc(SAB)
(2)

^

b) M p ( M N P ) va m p (SAD).

^HeBDc(SBD)
(2)

Ta c6: •

^

T u ( l ) v a (2) suy ra ( S A C ) n ( S B D ) = SH .
Va CO

b) T i m giao t u y e n ciia m a t ph^ng (SAD) va (SBC).
(3)

I e A D c (SAD)
UlE(SAD)n(SBC)
I € BC c (SBC)

=:>PG(MNP)n(SAD)

EeMN,MNc(MNP)
Ee A D , A D c ( S A D )

Ee(MNP)n(SAD)

(4)
. it':

T r o n g mp(SAB) g p i K = P F n SB , c6:

' Id

,1

1 Ui

^
'
^ =>Me(ADM)n(SBC)
MeSC,SCe(SBC)
^
> y
f

fKGPF,PFc(MNP)
'K
B , S B ec ( S
BC)
K €eSSB,SB
(SBC)

H i, /:

[ M € ( A D M )

" ^ ^^^^^

^-^^^9

[ M e (MNP)

Vac6<|
'
'
, \>M6(MNP)n(SBC)
MeBCBCc(SBC)
^
/

(5)

Tir (5) va (6) suy ra ( M N P ) n (SBC) - M K
•l€(ADM)n(SBC)

Tir (5) va (6) suy ra ( A D M ) n ( S B C ) = M I

(3)

PeSA,SAc(SAD)

c) M p ( M N P ) va m p (SBC).
(4)

Tir (1) va (2) suy ra ( S A D ) n (SBC) = S I .
c) T i m giao t u y e n m p ( A D M ) va m p (SBC).

PG(MNP)

Tir (3) va (4) suy ra ( M N P ) n ( S A D ) = PE

T r o n g m p ( A B C D ) gpi I = A D n B C , c6:

IeBC,BCc(SBC)

(l)

T u (1) va (2) suy ra ( M N P ) n ( S A B ) = P F

e A C c (SAC)

jle AD,ADc(ADM)

T i d v

P6(MNP)

=>Fe(MNP)n(SAB)

T r o n g m p ( A B C D ) g p i H = A C n B D , c6:

Taco: S € ( S A D ) n ( S B C )

d ) M p ( M N P ) va m ^ (SCD).

(ABCD))

P€(MNP)n(SAB)
LOIGIAI

=>H€(SAC)n(SBD)

AB, A D c

a) M p ( M N P ) va m p ( S A B ) .

c) M p ( A D M ) va m p (SBC).

' J H

•

Gpi F = M N n A B , E = M N n A D

Gpi I la giao d i e m cua A D va BC. Lay M thupc c^nh SC. T i m giao tuyen

Taco S € ( S A C ) n ( S B D )

b) M p ( M N P ) va m p (SAD).
LOI GIAI S

T u (5) va (6) suy ra ( M N I ) n ( A C D ) = N F

a) M p (SAC) va m p (SBD).

a) M p ( M N P ) va m p (SAB).

(6)

d) M p ( M N P ) va m p (SCD).
-; {I)

U7

i;T

G p i H - P E n SD ( P E , SD c ( S A D ) ) , c6:

v

/

(6)


HGPE,PECI(MNP)

a). Tim giao tuyen cua (AMN) va (BCD)
He{MNP)n(SCD)

(7)

HGSD,SDCI(SCD)

Trong (ABD ) goi E = A M n BD, c6:

Ee A M , A M c ( A M N )
EGBD,BDC(BCD)

Ne(MNP)

Va

CO ^

, ,

Tir

Ne(MNP)n(SCD)

/
[ N € C D , C D C ( S C D )

va (8) suy ra:

(MNP)

n

(SCD)

\

(8)

=^ E G

( A M N ) n (BCD)

(l)

Trong (ACD ) gpi F = A N n CD, c6:

= NH

FG

A N , A N C ( A M N )

Cho t u dien S.ABC. Lay M e S B , N e AC, l e S C sao ciio M I khong song
' F G C D , C D C ( B C D )

song vai BC, N I khong song song voi SA. Tim giao tuyen ciia mat phang
^FG(AMN)n(BCD)

(MNI) voi cac mat (ABC) va (SAB).
LOI GIAI

Tu(1) v a ( 2 ) : ( A M N ) n ( B C D ) = F.F

a) Tim giao tuyen cua mat phang ( M N I ) va (ABC).
NG(MNI)
Yi

^

b. Tim giao tuyen cua ( D M N ) va (ABC)

^

Trong (ABD ) goi P = D M n AB , c6:

\

P G D M , D M C ( D M N )

N G AC,ACc(ABC)
=>NG(MNl)n(ABC)

(2)

• PG(DMN)n(ABC)

(3)

P G AB,ABc(ABC)

(l)

Trong (ACD) goi Q = D N n AC , c6:

Trong mp(SBC) goi K = M I n BC, vi:

Q G D N , D N C ( D M N )

KGMIC=(MNI)

^
' : ^ Q e D M N n(ABC)
Q G AC,ACc(ABC)
^

' ^
'

(4)
^ '

•ml

K G B C , B C C ( A B C )

=:>Ke(MNl)n(ABC)

Tu(3)va(4) :

( D M N ) r , ( A H C ) - I'Q

T u (1) va (2) suy ra:

Cho tii dien ABCD. Lay I e AB, ] la diem trong tam giac BCD, K la diem

(MNl)n(ABC) = N K

trong tam giac ACD. Tim giao tuyen cua mat phang (IJK) voi cac mat

b) Tim giao tuyen cua mat phang

(MNI)

voi

cua t u dien

(SAB).

/
Goi

J= NI n SA(NI, SA c

LOI

GIAI

(SAC))

Goi M - DK n A C ( D K , AC c ( A C D ) ) ; N = DJ n B C ( D ] , BC cz (BCD))

Me

(MNI)

Ta co:

>M6(MNl)n(SAB)
MGSB,SBC(SAB)

Va: H = M N n K j ( M N , K J c ( D M N ) ) .

(3)

Vi H G M N , M N c (ABC) :=> l i G ( A B C )
'JGNIC(MNI)

Va

J G ( M N I ) n (SAB)

CO

JeSA,SAc(SAB)'
Tu (3) va (4) suy ra:

(MNI)

n (SAB) =

(4)

Goi P - H I n B C ( H I , BC c (ABC)),
Q - PJr,CD(PJ,CDc(BCD))

MJ.

Cho tu dien ABCD, M la mot diem ben trong tam giac ABD, N la mot
diem ben trong tam giac ACD. Tim giao tuyen cua cac cap mp sau :
a) ( A M N ) va (BCD)

"

M'Ti;^f7 •

b) (DMN) va (ABC)
LOI GIAI

T = QK n AD(QK, AD c (ACD))

Theo each dung diem a tren ta c(S:
'-

( I J K ) n ( A B C ) = IP; (IJK)n(BCD) = PQ
( l J K ) n ( A C D ) = Q T ; ( l J K ) n ( A B ) = TI

Wi • M"/
13


D A N G 2: Tim giao diem ciia duong thang va m|t phing.

Cho tu dien A B C D . Tren A B , A C , B D Ian luot lay 3 diem M , N , P sao cho
M N khong song song voi B C , M P khong song song voi A D . Xac djnh giao
diem cua cac duong thing B C , A D , C D voi mp (MNP).

Phuang phap: Muon tim giao diem cua duong thiing d va m|t phling (P),
CO hai each lam n h u sau:

LOIGIAI

Cach 1: N h u n g bai don gian, c6 san mot mat phaiTg (Q) chua duong thSng d
va m p t duong thSng a thuQC mat ph3ng (P).

> s

Tim giao diem cm BC va mp(MNP)

Giao diem ciia hai duong thang khong

Gpi H = M N n B C ( M N , B C c ( A B C ) ) .

song song d va a chinh la giao diem ciia
d va mgt phang (P).
Cach 2: T i m myt mat
chua duong

phang

Ta c6:

(Q)

thang d, sao cho

,

de

HeBC
H e M N , M N c: ( M N P )

r:>H = B C n ( M N P )

dang tim giao tuye'n voi mat phang
(P). Giao diem ciia duong thang d

Tim j^iao diem cua AD va mp(MNP)

va mat phang (P) chinh la giao diem

G
^

cua duong thang d va giao tuyen a
Cho tu dien A B C D . Goi M, N Ian lugt la trung diem cua A C va B C . K la dien-

leAD
Vi \
, =>I = A D n ( M N P )
[ltMP,MPc(MNP)

tren B P sao cho K D < K B . Tim giao diem cua C D va A D voi mp (MNK).

Tim j^iao diem cua CD vii iiip(MNP)

vua tim.

LOI GIAI

Tim giio diem ciia C D voi mp ( M N K ) .
Cac ban J e y N K

Ta CO

C D cung thuoc mat ph^ng (BCD) va chiing khong song

thuoc mat phang ( M N K ) suy ra I thuoc mat phSng ( M N K ) .
Ta CO

Vay I chinh la giao diem cua C D va (MNK).
Ta ccS the trinh bay nhu sau :

•••

IN c ( A C D )

NeAC,ACc=(ACD)

JeCD

•J = C D n ( M N P ) .

JeNI,NIc(MNP)

a) T i m giao tuye'n cua ( O M N ) va (BCD )

le NK, N K c ( M N K )
=>I = C D n ( M N K )

b) T i m giao diem cua B C voi (OMN)
-

c) Tim giao diem cua BD voi (OMN)

Tim giao diem ciia A D voi mp ( M N K ) .

a). Tim giao tuye'n ciia ( O M N ) va (BCD ):

Sau do tim giao tuye'n ciia ( A C D ) va (MNK), ta trinh bay n h u sau :
,\

MeAC,ACc(ACD)
IeNK,NKc(MNK)
IeCD,CDc(ACD)

.Me(MNK)n(ACD)

.Ie(MNK)n(ACD)

Vay ( N M K ) n ( A C D ) = M I . Goi H = M I n A D

LOI GIAI

fV

Chon mat phang ( A C D ) chiia A D .

Ta CO

'

song song voi C D . Goi O la diem ben trong tam giac B C D .

leCD

M e (MNK)

^

Cho tu di^n A B C D . Tren A C va A D lay hai diem M, N sao cho M N khong

Trong mp (BCD) goi I = C D n N K .

Ta c6: <

,

Goi J = N I n C D (Voi N I , C D c ( A C D ) ) .

song ne 1 hai duong thang nay se cat nhau tai mot diem I, nhung N K lai

Vi

le AD,ADc:(ACD)

,

(l)

I

Tac6:Oe(OMN)n(BCD)

(l)

Trong ( A C D ) , goi I = M N n C D .
|leMNc(MNO)
|leCDe(BCD)

.y'^, Vr
;!

(2)

=>Ie(MNO)n(nCD)(2)'
=^ H = A D n ( N M K ) .

T u (1) va (2): OI = ( O M N ) n (BCD )

,.:

i •,,

;(^•l:>A)^:aA,,>:^KIAr'

^


b) . Tim giao diem cua BC voi (OMN):
Gpi P = B C n O l ( B C , O I c : ( B C D ) ) .

Cho t u dien ABCD. Gpi M , N Ian iupt la trung diem cac canh AC, BC.

Tren canh BD lay diem P sao cho BP = 2PD. Lay Q thupc AB sao cho Q M

Vay : P = BC o ( O M N )

cat BC. Tim:
a) . Giao diem ciia CD va mp (MNP).

c) . Tim giao diem cua BD voi (OMN);

b) . Giao diem ciia A D va mp (MNP).

Trong (BCD), goi Q = BD n OI.

c) . Giao tuyen ciia mp (MPQ) va mp (BCD).

Vay : Q = BD n (OMN).
Cho t u di^n ABCD, lay M thupc AB, N thupc AC, I la diem thupc mien trong
cua tam giac BCD. Xac dinh giao diem cua cac duong thMng BC, BD, AD,
CD voi mp (IMN).
LOl GIAl

MNOBC(MN,BCC(ABC)),C6:

HeBC

MNC(MNI)

Va l e ( M N l ) n ( A C D )

mp(MNl)

Co\ = HI o DD ( H I , BD c (BCD)), c6:
^
"
• = nDn(MNl)

HInCD(Hl,CDc(BCD)),c6:

FGCD
FGH1,H!C(MNI)

F-CDn(MNl)

=>K = A D n ( M N l )

•if"-' frnT .!

,
^ =>E = C D n ( M N ]
[EeNP,NPc(MNP)
^

V

b) . Giao diem ciia A D va mp (MNP).
Tim giao tuyen ciia m p ( A C D ) va mp(MNP).
Co

K

M€(MNP)
Me

AC,ACC(ACD)

Me(MNP)n(ACD)
Co

(l)

EeNP,NPc{MNP)
EeCD,CDc(ACD)

=>E6(MNP)n(ACD)

(2)

Tu (1) va (2): EM = ( M N P ) n (ACD)
^ F=ADn(MNP)(vi E M C ( M N P ) )

c) . Tim giao tuyen ciia mp (MPQ) va mp (BCD).
Gpi K = Q M n B C ( Q M , B C c m p ( A B C ) )

, (KeBC,BCc(BCD)
Co

KGQM,QMC(MPQ)

Tim giao diem cua AD vii mpiMNl)
Gpi K = F N n A D ( F N , A D c ( A C D ) ) , c6:

/in/- ,
<

EeCD

Gpi F - A D n E M ( A D , E M c ( A C D ) )

Tim giao diem ciia CD va mp(MNI)
Gpi F =

a) . Giao diem cua CD va mp (MNP).

(2),

Tim giao diem ctia BD va

1-6Hl,Hic(MNl)

viT:

He(MNl)n(ACD)

Tir (1) \i (2): H1 = (M NI) n (ACD)

FeBD

LOI GIAI

Co:

\

Tim giao tuyeit cua (BCD) lui (MNI)
HGMN,

'

H€MN,MNC(MNI)

=*H = BCn(MNI)

la c6: {
'
'
[HeBC, BCc(BCD)

d). Giao diem ciia A D va mp (MPQ).

Gpi E = C D n NP(CD,NP c mp(BCD)).

Tim giao diem ciia BC va mp(MNI)
Gpi H =

d) . Giao diem ciia CD va mp (MPQ).

Co

Pe (MNQ)
PeBD,BDc(BCD) ~

,
1

- iYa}?\'m^J^f^M^^^

^Ke(MPQ)n(BCD)

Pe(MPQ)n(BCD)

(3)

(4)

Tu (3) va (4): KP = ( M P Q ) n (BCD).
d). Tim giao diem c i i a CD v a m i i ^ T ^ V J H N T i M H O I N f i Tl lUAN


Gpi L = K P O C D ( K P , C D C ( B C D ) ) , C 6 :

LeCD

,

d). Tim giao diem P ciia SC va mp (ABM), suy ra giao tuyen ciia hai mp (SCD)
va (ABM).
Tim giao tuyen ciia hai mat phang (AMB) va (SAC). ,
Ta CO A va I la hai diem chung ciia hai mat phSng (ABM) va (SAC).

^

e). Tim giao diem ciia A D va mp ( M P Q ) .
Dau tien tim giao tuyen ciia m p ( A C D ) va m p ( M P Q ) .

ifc

*.
;

Vay: ( A B M ) n ( S A C ) = A I .

'A

a) . Tim giao diem N ciia duong th^ng CD va mp (SBM).

Gpi P la giao diem ciia A I va SC, thi P la giao diem ciia SC va mp(ABM).
Hai mp (SCD) va (ABM) c6 M va P la hai diem chung, vay giao tuyen ciia
chiing la PM.
Cho t i i giac ABCD va mot diem S khong thupc mp (ABCD ). Tren doan
AB lay mpt diem M ,tren doan SC lay mot diem N ( M , N khong trung
voi cac dau m i i t ) .
a) Tim giao diem ciia duong thSng A N voi mat phSng (SBD).
f

b) . Tim giao tuyen ciia 2 mat phSng (SBM) va (SAC).

b) Tim giao diem ciia duong thang M N voi mat phang (SBD).

Co M , L la hai diem chung ciia ( A C D ) va ( M N P ) .

11

,

Suy ra M L = ( M P Q ) n ( A C D )
Goi T = A D n M L ( A D , M L c ( A C D ) )

T = A D n ( M P Q ) (vi M L C ( M P Q ) )

Cho hinh chop S.ABCD c6 AB va CD khong song song. Gpi M la mot diem
thupc mien trong ciia tam giac SCD.

LOI GIAI

c) . Tim giao diem I ciia duong th^ng BM va mat phSng (SAC).
d) . Tim giao diem P ciia SC va mp(ABM), t u do suy ra giao tuyen ciia hai
mp (SCD) va (ABM).
f - v A V
a) Tim giao diem N ciia duong th^ng CD va mp (SBM).

^ ••V//f"

i€ANl€SPc:(SBD)

18

•

^

' ' A'^^

J-rYy^

A

M

,
^

'
^'

„'

\
\

^

\/''7

\

M-f---a'-'-P

\

b) Tim giao diem ciia duong thSng M N voi mat phaffg (SBD).

- - --^V-Vi,

\

•

Chpn mp phu (SMC) =) M N

•

Tim giao tuyen ciia ( SMC ) va (SBD)

Trong (ABCD), gpi Q = MC n BD, hai mat ph3ng (SMC) va (SBD) c6 S va Q
la hai diem chung nen ( SMC) n (SBD) = SQ.
•

Trong (SMC), gpi J = M N n SQ, c6:
JeMN

,

JGSQC(SBD)

c) Tim giao diem I ciia duong thing BM va mat ph5ng (SAC). M l s ' l !

. I = BMn(SAC).
l£SO,SOc(SAC) ^

,

J

//! \

\ " ~ J \7

, =>I = A N n ( S B D )

;

Gpi I = B M n S O ( B M , S O e (SBN)).

A

• Trong (SAC) gpi I = A N n SP, c6:

T u ( l ) v a (2) suy ra SO = ( S A C ) n ( S B N ) .

Co:

/
/

Suyra m p ( S A C ) n m p ( S B D ) = SP

0€AC,ACc=(SAC)

(2)

/

mat phang (SAC) va (SBD).

b) Tim giao tuyen ciia 2 mp (SBM) va (SAC).
Cac ban de y mat ph^ng ( S B N ) cung
chirrh la mat ph^ng (SBM).
Trong mp(ABCD) gpi O = AC n BN , c6:

Va Sfc£(SAC)n(SBN)

'fill;

J .

Vay S va P la hai diem chung cua hai

NeCD
"M
/cox.x ^ N = CDn(SBM).
N€SM,SMc(SBM)
^
'

(l)

Chpn mp phu (SAC) =3 A N .

Trong (ABCD) gpi P = AC n BD .

Gpi N = SM n CD(SM, CD c ( S C D ) ) , c6:

=^06(SAC)n(SBN)

•

Tim giao tuyen ciia ( SAC) va (SBD)

LOI GIAI

OeBN,BNc(SBN)

a) Tim giao diem ciia duong thang A N voi mat phang (SBD)

5.'11 ^ ' '

,
,
=>J = M N n ( S B D )
^

'

Cho hinh chop S.ABCD .Gpi O la giao diem ciia AC va BD . M , N , ? Ian lupt
la cac diem tren SA , SB , SD.
a) . Tim giao diem I ciia SO voi mat phang ( MNP )
b) . Tim giao diem Q ciia SC voi mat ph^ng ( MNP )

^

____
19


'
LOIGIAI
a) Tim giao diem I ciia SO voi mat pinang ( MNP )
Trong mp(SBD) goi : I = SOr>NP, co:
"leSO

,

,

l€NPc(MNP)

^

'

^ H,iti
'-.'S''"

b) Tim giao diem ciia A O va (BMN ):
•

Ta c6:B la diem chung cua (ABP ) va (BMN)
QGMN,MNC:(BMN)

Chon mp phu (SAC) 3 SC

Ta c6:

M € (MNP)

,

,

,

(3)
. ,

,

/
, ' ^ Q G ( A B P ) n ( B M N ) (4).
Q G A P , APc(ABP)
^
' ^
' ^ '

^

• Tim giao tuyen ciia ( SAC ) va (MNP)

'•[. \ ' ' " f

Tim giao tuyen cua (ABP ) va (BMMj

b). Tim giao diem Q ciia SC voi mp (MNP )
•

!

ChQn mp (ABP) 3 AO.

Tir (3) va (4) ( A B P ) n ( B M N ) = BQ

'

f ^v?) D

„\i::,

Goi I = BQ n A O ( vi BQ, AO c mp( ABP)), c6:

MeSA,SAc:(SAC)
MG(MNP)n(SAC)

(l)

l6SP,SPc(MNP)
Va
I e SO, SO c (SAC)

I G (MNP) n (SAC)

,

,

IeBQ,BQc(BMN)

^

'

r\(m\

Trong mp (a) cho hinh thang ABCD, day Ion A D . Goi I, J, K Ian lugt la cac
diem tren SA, AB, BC ( K khong la trung diem BC). Tim giao diem ciia:
a) IK va (SBD).

T u ( l ) v a (2) CO ( M N P ) n ( S A C ) = M I
•

leAO

Trong (SAC) goi Q = SC n M I , c6:

b) SD va (IJK ).

QeSC

c) SCva (IJK).

QeMI,MIc(MNP)

LOI GIAI
' ^ Q - SCn(MNP)

a). Tim giao diem cua IK va (SBD)

Cho t u dien ABCD. Goi M , N la hai diem tren AC va A D . O la diem ben
trong tam giac BCD. Tim giao diem cua :
a) . M N v a ( A B O )
b) .

Chon mp phu (SAK) chua IK .

Tim giao tuyen ciia (SAK ) va (SBD)
Taco: SG(SAK)r^,(SBD)

(l)

AO va (BMN )
LOI GIAI

a) Tim giao diem ciia M N va (ABO ):
•

•

Trong (ABCD) goi P = AK n BD.
Pe A K c ( S A K )
Co
P G BD c (SBD)

Chon mp phu (ACD) =) M N . Tim giao tuyen ciia (ACD ) va (ABO)

Ta c6: A la diem chung cua (ACD ) va (ABO)
Trong (BCD) goi P = BO n DC.

=>PG(SAK)n(SBD)

(2)

Tu (I) va (2) :=> (SAK) n (SBD) = SP

PeBO,BOc(ABO)
PeCD,CDc(ACD)

/

•

Trong (SAK), goi Q = IK n SP, c6:

Q = IKn(SBD)
QGSPC(SBD)

=^ P G (ABO) n (ACD)
Tir (1) va (2) ^

(2)

t

(ACD) n (ABO) = AP

- Trong (ACD), goi Q = AP n M N , c6:
QeMN
Q G A P , A P c ABO
20

MNn(ABO) = Q

b). Tim giao diem cua SD va (IJK ) :
•

Chon mat phang phu (SBD) chua SD. Tim giao tuyen ciia (SBD ) va (IJK).

Theo cau a) ta c6: Q e (SBD) r^ ( I J K )
Trong (ABCD), goi M = JK n BD
Tu(3)va(4)=v

(3)

M la diem chung cua ( I J K ) va (SBD) (4)

(IJK) n (SBD) = Q M . •

'

,


•

Trong (SBD) gQi N = Q M m SD.

T r o n g m a t phSng day ( A B C ) ke A N // E F ( N e B C ) .
BH

BF

,

i/VA

„^

T a c o : H F / / A N = : > : ^ = : £ ^ = 1 ^ BF = F N .
HA
FN
c). T i m giao d i e m ciia SC va ( I J K ) :

''

"'*

• C h p n m p p h u ( S A C ) Z3 S C . T i m giao tuyen cua ( S A C ) va ( I J K ) .
,

Col
C(?i

'

^

[l€SA,SAc:(SAC)

, ^ l 6

IJK

^

/

n SAC

E = A C n J K ( v i A C J K c ( A B C D ) ) . Vay

T u (5) va (6)

(5)
/ ^

>;

!,

T a c o : E F / / A N => — = — = 2
AE
NF
Ket luan:

, . =, . I

E e (IJK) n (SAC)

(IJK) n (SAC) = IE

'

1 uT
(6)

> ,

., r> <
' ;.. -

CN = 2NF.

FB

FB

FB

1

FC

FN + NC

3FB

3

i/iae)zvCc

b). T i m giao d i e m ciia K M va m p (ABC).

»• •< f » •

(• f

.

T a c o K M C ( I H K ) . Goi J = K M n E H ( E H , K M c ( I H K ) ) .

0/.="!'

• T r o n g (SAC), gpi F = I E n S C .
JeKM

Cho t u d i ^ n S.ABC. Goi I , H Ian l u p t la t r u n g d i e m ciia SA, A B . Tren canh
SC lay d i e m K sao cho CK = 3 S K .
a). T i m giao d i e m F cua BC v o i m p ( I H K ) . T i n h t i so

FB
FC

,

,

Thie't d i ^ n la phan c h u n g ciia mat phang (P) va h i n h ( H ) .
P A N G 3: T i m thiet dign ciia hinh (H) k h i cit bai mat phang (P). ^ '
'
Xac dinh thiei di^n la xac d i n h giao tuyen ciia m p (P) v o i cac mat ciia h i n h (H).
T h u o n g ta t i m giao t u y e n dau t i e n ciia mat phSng (P) v a i m p t m a t phang

b). G Q I M la t r u n g d i e m cua doan I H . T i m giao d i e m cua K M va m p ( A B C ) .

( a ) nao d o thupc h i n h ( H ) , giao tuyen nay de t i m duoc. Sau do keo dai

LOI GIAI

giao t u y e n nay c l t cac canh khac ciia h i n h ( H ) , t u do ta t i m d u p e cac giao

a). T i m giao d i e m ciia BC v o i m p ( I H K ) .

t u y e n tiep theo. Da giac gioi han boi cac doan giao t u y e n nay khep k i n

Ta t i m giao tu)'e'n ciia (ABC) va ( I H K ) trudc
Gpi E = A C n K l ( A C , K I c ( S A C ) ) , c 6 :

thanh m p t thiet dien can t i m .
T h o n g qua cvi the n h u n g bai tap sau thi cac b^n se hieu ro hom.
Cho h i n h chop S.ABCD. G p i M la m p t d i e m t r o n g t a m giac SCD.

[E€AC,ACC(ABC)

a) . T i m giao t u y e n ciia hai m p (SBM) va (SAC).
b) . T i m giao d i e m ciia d u o n g thSng B M va m p (SAC).

fH6(lHK)

c).

F = E H n B C ( E H , BC c ( A B C ) ) ,

LOI GIAI
c6:

F e BC

a). T i m giao t u y e n ciia hai m p (SBM) va ( S A C ) . ' 5 V ' ' ' *
Gpi

N = SM n C D ( vi S M , C D c

(SCD)).

V^y mat phSng (SBN) cung la mat phJing (SBM).

FeHH,EHe(lHK)^^ =

^C"M;

Gpi

0

Ggi D t r u n g d i e m ciia S C , ta c6 I K la d u o n g t r u n g b i n h ciia A S A D ,
T r o n g ACEK c6: —

AE

=

DK

= 2

= ^ C A = 2CK .

""^'^^^

Xac d j n h thie't dien ciia h i n h chop k h i cat b a i m p ( A B M ) .

T u (1) va (2) => E H = ( A B C ) n ( I H K )
Gpi

* '-^'f'

Ta c6:

= ACnBN(vi AC,BNC(ABCD)).

OeBN,BNc(SBN)
OG

AC,ACC(SAC)

^

rr'i'


=^Oe(SAC)r^{SBN)

(l)

VaSe(SAC)n(SBN)

Trong mp (BCD), goi I = BD n H O .

(2)

JleBD

^^^^^

T u (1) va (2): ( S A C ) n ( S B N ) = S O .

^°'|leHO,HOc(MNO)

'''

b) . T i m giao diem cua duong thSng B M va mp (SAC).

'

Trong mp(BCD) goi ] = C D n H O , c6:

'' '

I

•jeCD

Gpi H = B M n S O ( v i B M , S O c ( S B N ) ) , c 6 :

[J € H O , H O c ( M N O )

H€BM
•H . S0.SO c ( S A C ) " "

I = BDr^(MNO

H

r:^J = C D n ( M N O ) .
= ""^ "

'•'

'"^

c). Tim thiet dien cua mp(OMN) voi hinh ch

c) . Xac djnh thiet dien cua hinh chop khi cat boi mp (ABM).

Theo each dung diem o cau a) va b) thl :

Goi I = A H n S C ( v i A H , S C c: ( S A C ) ) .

(ABC)n(MNO) = MN, (ABD)n(MNO)

Tim giao tuyen cm hai mat phang (ABM) va (SCD).

( A C D ) r ^ ( M N O ) = N J , ( B C D ) n ( M N O ) = IJ.

A H , A H c (ABM)
/
, ''^16
l€SCSCc(SCD)

,

' ,
,
SCD n ABM)
\ \
\

16

, ,

. f r.

(s)

Vay thiet dien can tim la tu giac MNJI.
Cho

Va M e ( S C D ) n ( A B M ) (4)

'UUbvX

;|

T u (3) va (4): ( S C D ) n ( A B M ) = IM.

tu dien S . A B C . Goi M € A S , N e ( S B C ) , P e ( A B C ) , khong c6 duong
thang nao song song.

a) Tim giao diem cua M N voi (ABC), suy ra giao tuyen cua ( M N P ) va (ABC).
b) T i m giao diem cua A B vol ( M N I ' ) .

Goi J = I M n S D ( v i I M , S D c ( S C D ) ) .

,
t«V

Vay ( S A D ) n ( A B M ) = AJ , va ( S B C ) n ( A B M ) = BI
Ket luan : thiet dien can tim la tu giac ABIJ .

c) T i m giao diem cua NI^ voi (SAB).
d) T i m thiet dien ciia hinh chop cat boi mat phang ( M N P ) .
LOI GIAI

a). T i m giao diem cua M N voi (ABC)

5

Cho tu di^n A B C D . Tren AB, A C lay 2 diem M, N sao cho M N khong

Chpn mat phang phu (SAH) chua M N . /

song song B C . Goi O la 1 diem trong tarn giac B C D .

Tim

mp(SAH)

a) . Tim giao tuyen (OMN) va (BCD).

a) . Tim giao tuyen ( O M N ) va (BCD).

Co:
[HeMN,MNc(MNO)
^
'

mnl^A.^^a
(2)

/

. ,j
• —

HeBC,BCe(ABC)

T u (1) va (2)

(2)

(SAH) n (ABC) = A H .

Gpi l = M N r ^ A H ( v i I V I N , A H c ( S A H ) ) .

^

T u (1) va (2) suy ra ( B C D ) n ( M N O ) = H O
b) Tim giao diem cua D C , BD voi (OMN).

(l)

HeSN,SNc(SAH)

^He(SAH)n(ABC)

|HeBC,BCe(BCD)

Vaco 0 € ( B C D ) n ( M N O )

mp(ABC).

H = SN n BC .

LOI G I A I

(l)

va

cua

Trong mp(SBC), goi:

c) . Tim thiet dien ciia mp(OMN) voi hinh chop.

^^W.(BCD)n(MNO)

tuyen

Ae(ABC)n(SAH)

b) . Tim giao diem cua D C , B D voi (OMN).

Trong mp ( A B C ) , gpi H = M N n B C , c6:

giao

*T


Co:

leMN

le AH,AHc(ABC)

I = MNn(ABC)

Co:

Tim giao tuyen cua (MNP) va (ABC).
: , Taco: P e ( M N P ) n ( A B C )
Va I e (MNP) n (ABC)

Ke(lJK)n(ABC)

(4)

H€(lJK)n(ABC)

Tim giao tuyen ciia mp(IJK) va mp(SBC).
Co J € (IJK) n (SBC)

K = ABn(MNP).

(1)

^ JDeHK,HKc(lJK)

c) . Tim giao diem ciia NP vai (SAB).

^ |DeBC,BCc(SBC)

G

Co

V:.//-

E = HK n A C ( H K , A C c ( A B C ) ) .

GQi K = A B n P l ( v i A B , P I c ( A B C ) ) .

K€PI,PIc(MNP)

•.'0.::/ .{\\

HK = (lJK)n(ABCJ

GQi D = H K n B C ( H K , B C c ( A B C ) ) ;

b) . Tim giao diem ciia AB vai (MNP).

Co

H€MN,MNCZ(ABC)

• H = IJo(ABC)

b. Tim giao tuyen ciia mp(IJK) va mp(ABC).

(3)

Tu (3) va (4) =^ (MNP) n (ABC) = P I .

KeAB

HelJ

L;™K,MKC(SAB)-^ =

=>D€(lJK)n(SBC)

(2)

T u ( l ) v a ( 2 ) = > DJ = (lJK)n(SBC)

™"P'^'')-

„^

Tim giao tuyen ciia mp(IJK) va mp(SAB).

d) . Tim thiet di^n ciia hinh chop cat boi m|t phang (MNP).

Co 1 6 (IJK) n (SAB)

Goi J = B C n P l ( v i B C , P I c = ( A B C ) ) .
,/fS

GQi P = S C n J N ( v i S C J N c : ( S B C ) ) .

Ket luan theo each dung diem ciia nhung cau a tren ta c6 thiet dien can tim
la tii giac MPJK.
Cho t i i dien S.ABC. Goi \ ] , K Ian luot la 3 diem nam trong ba mat phSng

fFeDJ,DJc(lJK)
,
, ,
]
\€ IJK n SAB
FeSB,SBc(SAB)
v / v

I
-

a) Tim giao diem ciia IJ voi mp (ABC).

|LGFI,FIe(lJK)
'

•
^

EGHK,HKC(IJK)

''

EeAC,ACc:(SAC)

Gpi M = S I n A B ( S I , A B c ( S A B ) ) , N = SJnBC(SJ,BCc(SBC))
T u d6 suy ra M N la giao tuyen ciia mp (SIJ) va mp (ABC).

|LeSA,SAc(SAC)

.LG(lJK)n(SAC)

(5)

E = H K n A C ( H K , A C c ( A B C ) ) , c6:

LOI GIAI
a). Tim giao diem ciia IJ voi mp (ABC).

(4)

Tim giao tuyen ciia mp(IJK) va mp(SAC).
GQi L = F I n S A ( F I , S A c ( S A B ) ) , c 6 :

b) Tim giao tuyen ciia mp (IJK) voi cac mat ciia hinh chop. T u do suy

i

T u (3) va (4): FI = (IJK) n (SAB)

(SAB), (SBC), (ABC).

GQi H = I J n M N ( l J , M N c ( S I j ) ) ,

J-..'

GQi F = DJnSB(DJ,SBc(SBC)) , c6:

Vay giao tuyen ciia (MNP) va (SBC) la JN.

ra thiet dien ciia mp(IJK) cat boi hinh chop.

•l! (XI i

(3)

' "^"^

Ee(lJK)n(SAC)

(6)

T u (5) va (6): LE = ( I J K ) n (SAC)
Ket luan: T u each tim giao diem cua mat phSng (IJK) voi cac canh ciia hinh
chop S.ABC, suy ra thiet di?n ciia mp(IJK) ck boi hinh chop la tii giac DFLE.


Cho hinh chop S.ABCD c6 day ABCD la hinh binh hanh tarn O. Goi M , N , I
Ian lugt nam tren ba canh AD, CD, SO. Tim thiet di§n ciia hinh chop voi mat
phang (MNI).
LOI GIAI
Trong (ABCD), goi: ] = BD n M N ,

P H l / O N G PHAP CHLfNG MINH BA D l / O N G T H A N G D O N G QUY:
Ta tim giao diem cua hai duong thang trong ba duong thang da cho, roi
chung minh giao diem do nam tren duong thang thii ba. Cu the nhu sau:
Cach chung m i n h 3 duang t h i n g a, b, c dong quy tai mot diem, f
Chon mot mat p h i n g (P) chua duong thang (a)va ( b ) . G o i I = ( a ) n ( b )

K = MNnAB, H = MNnBC '

Tim mot mat p h i n g (Q) chua duong thang (a), tim mot mat phang (R)

Trong (SBD), goi Q = IJ n SB

chua duong t h i n g (b),saocho (c) = ( Q ) n ( R ) :

Trong (SAB), goi K = KQ n SA

Vay: 3 duong t h i n g (a),(b),(c) dong quy tai diem I .

Trong (SBC), goi P = Q H n SC
Vay : thiet dien la ngu giac MNPQR

(a),(b)c:mp(P)
(a)n(b) = I
m p ( P ) n m p ( Q ) = (a) ^ ( a ) n ( b ) n ( c ) = I
mp(P)nmp(R)-(b)
mp(Q)nmp(R)=:(c)

Cho hinh chiSp S.ABCi"). Goi M, N , I ' Ian liiot la tnmg diem lay tren AB
A D V a SC .
Tim thiet dien ciia hinh chop voi mat phang (MNP)
LOI GIAI

Cho t u dien S.ABC. Tren SA, SB, va SC Ian luot lay cac diem D, E, F sao cho
DE cit AB tai I , EF cit BC tai j , FD cit CA tai K. Chiing minh ba diem I , J, K
thang hang^
LOT^GIAI

Trong (ABCD), goi:

I = A B n D E ( AB,DE c ( S A B ) )

E = M N n DC,
F = MN

Co:

BC

Trong (SCD), goi Q = EI' n SD

l6DE,DEc(DEF)

Trong (SBC), goi R = FP,",SB

Ie(ABC)n(DEF)

Vay : thiet diC-n la ngu giac MNPQR

D A N G 4: Chung minh ba diem t h i n g hang, ba duang t h i n g dong qui,

Muon cliiing minh ba diem A, B, C thang

thi suy ra ba diem A , B, C nam tren giao
tuye'n ciia (a) va ((i), nen chung thang hang.

Ke A C , A C C ( A B C )

K6(ABC)n(DEF)

* ^'

thupc hai mat phang phan biet (a) va (p),

(AC,DFC(SAC))

KeDF,DFc:(DEF)

chung minh mpt diem thupc mgt duang t h i n g co d i n h ^

hang, ta chirng minh ba diem do Ian krot

(l)

K-ACnDF
Co

Phuang phap:

AB,ABe(ABC)

(2)

J = BCnEF(BC,EFc(SBC))
Co

]€BC,BCc(ABC)

=^Je(ABC)n(DEF)

(3)

JeEF,EFc(DEF)
T u (1) (2) va (3) suy ra 3 diem I , J, K t h i n g hang.
29


Cho t i i di#n ABCD c6 G la trpng tam tam giac BCD. Gpi M , N, P km iugt
la trung diem ciia AB, BC, CD.
a) Tim giao tuyen ciia (AND) va (ABP).
nvtib-.i-

b) Xac djnh giao diem J = M N o (SBD)
• Chpn mp phu (SMC) chua M N .
Tim giao tuyen ciia (SMC ) va (SBD):

b) Goi I = AG n MP, ] = CM n A N . Chung minh D, I , J t h i n g hang.
.

^

, /

,

Trong (ABCD) goi K = MC n BD.
Hai mat phiing (SMC) va (SBD) c6 hai

LOIGIAI

diem chung la S va K.

a) . Tim giao tuyen cua (AND) va ( A B P ) .

Vay: (SMC) n (SBD) = SK

A£(ABP)O(ADN)(I)

• Trong (SMC), goi J = M N n SK .

Taco G = B P n D N , c 6 :

Vay J = M N n ( SBD)

JGeBP,BPc(ABP)
I

c) Chung minh I , J , B t h i n g hang

[GeDN,DNc(ADN)

g

Ta CO : B la diem chung cua (ANB) va ( SBD)

; =>Ge(ABP)n(ADN)(2)

l€SO,SOc(SBD)
V

B

b) . Chung minh D, I , J thang hang
I = AGoMP, AGc(ADG),
=^l€(ADG)n(DMN)

,=5l6(SBD)n(ABN)

le AN,ANc(ABN)

T u (1) va (2) A G = ( A B P ) n ( A D N )

j6SE,SEc(SBD)

MPC(DMN)

' JeMN,MNc(ABN)'

(3)

^

'

^ .^^ ' ^

Je(SBD)n(ABN)

(3)

T u (]) (2) (3) suy ra ba diem B , I , J thang hang.

i ^ J = C M n A N , A N c ( A D G ) , CM c ( D M N ) => J e ( A D G ) n ( D M N )
De(ADG)n(DMN)

. ^ l;:J*ai/

(4)

(s)

rtMii^^

Tu (3), (4), (5) SLiy ra ba diem D, I J thuoc giao tuyen cua hai mat phing
(ADG) va (DMN).
Ket luan vay ba diem D , I , J thang hang.

'

Cho hinh binh hanh ABCD. S la diem khong thuQc (ABCD), M va N Ian
lugt la trung diem cua doan AB va SC.
a) Xac djnh giao diem I = A N n (SBD)

it]

b) Xac djnh giao diem J = M N n (SBD)

. v

a)

Tim giao diem K = IJ n (SAC)

b.) Xac djnh giao diem L = DJ n (SAC)
c)

if')

Chung minh A , K , L , M thang hang
LOI GIAI

a) Tim giao diem K = IJ n (SAC)
• Chon mp phu (SIB) chua IJ
Se(SBl)n(SAC)

(l)

Trong (ABCD) gpi E = AC n B I , c6:

LOI GIAl
• Chon mp phu (SAC) chua A N .

cat BC tai O va OJ c i t SC tai M .

• Tim giao tuyen cua (SIB ) va (SAC)

c) Chung minh I, ], B thcing hang.

a) Xac dinh giao diem I = A N n (SBD )

Cho tu giac ABCD va S g (ABCD). Gpi I, J la hai diem tren A D va SB, A D

.

EeACc(SAC)

''

E € BI c (SBI)

' , y\

Tim giao tuyen cua (SAC ) va (SBD):
Trong mat phang (ABCD) gpi O la giao diem ciia AC va BD. Hai mat p h i n g
(SAC) va (SBD) c6 hai diem chung la S va O.

-AA } 'o iXtpii

Vay: ( SAC) n (SBD) = SO
• Trong (SAC) goi I = A N n SO. Vay: I = A N n ( SBD)

\

i
'[

Ee(SAC)n(SBl)

T u (1) \'a (2) (SBI) n (SAC) = SE .

Ql^Z)

Trong (SIB) gpi K = IJ n SE .
Ta

KelJ
CO •

K€SE,SEc(SAC)

K = IJn(SAC)
31


b) X a c d i n h giao d i e m L = DJ o
•

(SAC)

C h p n m p p h u (SBD) chua DJ . T i m giao tuye'n ciia (SBD ) va (SAC)

S e (SBD) n ( S A C )

(3)

)

T r o n g ( A B C D ) goi F = A C n BD .

fiv('

Jl/rl)

leBN

• '

•

T i r ( 3 ) v a ( 4 ) : (SBD) n ( SAC) = SF

giao tuye'n cua ( S M D ) va

Trong (ABCD), goi

I'dnlm

'iniui'")

c) C h u n g m i n h A ,K ,L , M thang hang

Co

LeDJc(A]0)
LeSFc(SAC)
M6jOc(AJO)
M G SC c ( S A C )

V^y giao tuye'n cua c h i i n g la SK .
•

T r o n g (SMD), g p i J = M N n SK.

(3) , c6:
Ta c6: •

:>KG(SAC)n(AJO)

KGSE,SEC(SAC)

Co

- • ' ' ' ' • ' /

va (SMD) c6

hai d i e m c h u n g la S va K .

Lcsr,snc(SAC)=''-:°"^<=^=)

JK€lj,IJc„(AJO)

(SAC)

K = AC n DM.

H a i m a t p h 3 n g (SAC)

Ta CO A G ( S A C ) n ( A J O )

V

C h p n m p p h u ( S M D ) chua M N .

Tim

T r o n g (SBD) goi L = DJ n SF .
Ta CO :

• I-BNo(SAC)

l€SO,SOc(SAC)

b) T i m giao d i e m J = M N r\ SAC)

• "

Vay F la d i e m c h u n g t h u hai cua hai mat phMng (4)
'

Ta c6: •

J G M N

,

JeSK,SKc(SAC)

^

>J = M N n ( S A C )
V,"

c) C h i i n g m i n h C, I , J thang hang

f-l

Theo each t i m d i e m 6 n h u n g cau tren ta c6 ba d i e m C, I , J la d i e m c h u n g ciia
LG(SAC)n(AJO)

(5)

hai mat p h a n g ( B C N ) va (SAC) => Ba d i e m C, I , J c i i n g thupc giao tuyen ciia
hai m a t phcing (BCN) va (SAC). Ket luan C, I , J thiing hang.

^MG(SAC)n(AJO)

Cho h i n h chop S.ABCD. Goi M , N , P Ian l u o t la t r u n g d i e m ciia SA, SB,

(6)

SC. Goi !-: = A B n C D , K = A D n B C .

T u ( 3 ) , ( 4 ) , ( 5 ) , ( 6 ) siiy ra bcVn diem A, K, L, M c i m g thuoc giao tuye'n ciia

a) T i m giao tuye'n cua ( S A C ) n ( S B D ) , ( M N P ) n ( S B D ) .

hai mat p h a n g (SAC) va (AJO).

b) T i m giao d i e m Q cua d u o n g thang SD v o i m a t phang ( M N P ) .

Vay: A, K,

c) G o i H = N M n P Q . C h u n g m i n h 3 diem S, H , E t h ^ n g hang.

M thang hang.

Cho tiV giac A B C D va S i (ABCD). Go! M , N la hai d i e m tren BC va SD.

d) C h u n g m i n h 3 d u o n g thang SK, Q M , N P d o n g q u i .
LOI GIAI

a) T i m giao d i e m I = B N n ( SAC)
b) T i m giao d i e m J = M N n ( SAC)

a). T i m giao tuye'n cua ( S A C ) n ( S B D ) .

rix:) /t j / • •»

c) C h i r n g m i n h C , I , J thang hang

/

o

T r o n g m p ( A B C D ) gc?i O = A C n B D , c6:
OG

LOI GIAI
a) T i m giao d i e m I = B N n ( S A C )
•

,„ K ;

; . • •; i

OGBD,BDC(SDB)
•I
=>OG(SAC)n(SBD)

C h o n m p p h u (SBD) chua B N

• T i m giao tuyen cua (SBD ) va (SAC)

SG(SAC)n(SBD)

T r o n g ( A B C D ) goi O = A C n BD.

•

T r o n g (SBD), gpi I = B N n SO.

(l)
(2)

T i l ( 1 ) va (2) ( S A C ) n ( S B D ) = S O .

/ H a i m a t phang (SAC) va (SBD) c6 hai d i e m c h u n g la S va O. V ^ y giao tuye'n

cua c h i i n g la SO.

AC,ACe(SAC)

U=

, t,l

T i m giao t u y e n ciia ( M N P ) n ( S B D )
T r o n g m p ( S A C ) goi

F = M P n S O , co:

, i riiffrt

•

- ^


Trong

rFeMP,MPc:(MNP)
,
, ,
/
, ^=>Fe(MNP)n(SBD)
[FeSO,SOc(SBD)
^
/ V

;

Co:

(3)
V;

N€(MNP)

,

,

,

^

NeSB,SBc(SBD)

^

'

^

'

/,

, ,

^

^"

la d u a n g

••"->•'

' A ' j f b -jifi it

I t

t r u n g b i n h c u a tarn giac.
Tu

d o s u y ra F D la d u o n g

t r u n g b i n h cua t a m giac A I K .
=> D t r u n g d i e m ciia A K .

A ) ;:5noTT

Ket l u ^ n

b) T i m giao d i e m Q cua d u o n g thSng SD v o l m $ t phSng ( M N P ) . - 5 ' ' ' '^'''^
G
I M

=> S I = I = F A .

i . twvn,pf„;T

'

Tu(3)va(4) (MNP)n(SBD) = NF.

ASDFCO

N F , S D c: ( S B D ) ) .

KD
KA

2

Trong m p (ABCD),
gpi E = A N n C D .

QeNF,NFc(MNP)^^

bDn^MNlj

Trong m p (SCD) gpi: P = E M n SC.

^

Tir d o s u y r a thiet d i ^ n ciia
c) G p i H = N M n P Q . C h i m g m i n h 3 d i e m S, H , E t h ^ n g h^ng.

h i n h chop S.ABCD b j c i t b o i

H =MNnPQ,MNc(SAB),PQc(SCD)

=> H e ( S A B ) n ( S C D )

ir:>

E =ABnCD,ABc(SAB),CDc(SCD)

=> E e ( S A B ) n ( S C D ) ( * * )

in

Se(SAB)n(SCD)
Tu

(*)

mSt ( A M N ) la t i i giac A M P N .

(***)

( * ) , ( * * ) , ( * * * ) s u y ra ba d i e m

S , H , E thupc giao t u y e n ciia h a i m | t

p h i i n g (SAB) va (SCD) nen ba d i e m S, H , E t h i n g hang.

,

/

d) C h u n g m i n h 3 d u o n g thang SK, Q M , N P d o n g q u i .
Gpi G = M Q n N P ( v i M Q , N P c ( M N P ) )

N g o a i ra ( S A D ) n (SBC) = SK
'

^ G e SK

thang BC va B D , M la d i e m tren doan A C . G i a s i i k h o n g t o n tai song
(6)
^ ^

a) T i m giao d i e m ciia d u o n g thang A B va m a t phang ( M P Q ) . Suy ra giao
d i e m N ciia d u o n g thang A D va mat phang (MPQ) .

{?)

b) PQ cat C D tai I . T i m giao tuyen ciia hai m a t phang ( M P Q ) v o i m a t phang
( A C D ) . N h a n xet g i ve vj t r i ciia M , N , I?

Cho h i n h chop S.ABCD c6 day la h i n h b i n h hanh. G p i M la t r u n g d i e m

c) DP va CQ c i t nhau tai E, M Q va N P cJt nhau tai F . Chiinp. to A, E, F thang hang.
LOI GIAI

cua canh SD, I la d i e m tren canh SA sao cho A I = 215 .

Gpi giao d i e m K cua I M v o i mat p h i n g (ABCD). T i n h t i so KD/KA. Gpi N
la t r u n g diem BC. T i m thiet di?n cua h i n h chop S.ABCD c i t boi mat ( A M N ) .
1

LOIGIAI

Trong m p ( S A D ) gpi K = I M n A D .

'' ' i

K G I M

,

. :'A8|:(£)

Dvmg

a) T r o n g m p ( A B C ) g p i H = A B n M P , c6:
[HeAB

^
,
,
, =^H = A B n ( M P Q ) .
HePMc(MPQ)

Ta c6: H va Q la h a i d i e m c h u n g ciia
.;V(J)MT

Vi
ASAD

hai m a t p h a n g ( M P Q ) va (ABD), nen
giao tuyen ciia c h i i n g la d u o n g thang

d u p e ve lai 6 h i n h 2.
DF//KI(FGSA).

j

i

'

'

-.|,/. ,, i\,,,,

song t r o n g h i n h ve ciia bai toan.

T i f ( 5 ) , ( 6 ) , (7) suy ra 3 d u o n g t h i n g SK, Q M , N P d o n g q u i .

j

'•1 o:,

Cho t i i d i ^ n A B C D . G p i P va Q Ian l u p t la nhCrng d i e m tren hai doan

(5)

G€MQ,MQc(SAD)
/
, ^=^Ge SAD n(SBC
GeNP,NPc(SBC)
^
^ ^
'
.

Hinh 2

-• -i iv,^ QAH)qm

gmiV

H Q . H Q c i t A D tai d i e m N , t h i N la
giao d i e m ciia A D va (MPQ).

«


A e (SAC) n ( A M N )

b) M va I la hai diem chung cua hai mat phSng (MPQ) va (ACD).
Vay giao tuye'n cua (ACD) va (MPQ) la duong t h i n g M I .
Vi N 6 (MQP) o (ACD) =^ N € M I . Vay ba diem M , N , I th^ng hang .
c) Vi ba diem A , E , F la ba diem chung cua hai mat ph^ng (ADP) va (ACQ).
• " Nen chiing thupc giao tuye'n cua (ADP) va (ACQ).
Ket lu^n ba diem A , E, F th3ng hang .

"-^

Cho hinh chop S.ABCD vdi ABCD la hinh binh hanh. Gpi M la diem bat
ky thupc SB, N thupc mien trong tam giac SCD.
^
,_

oft

(s^e)n(AMN)

(2)
'

T i r d ) va ( 2 ) : ( S A C ) n ( A M N ) = A E . ^•>«
GpiK = S C n A E ( v i A E , S C c ( S A C ) ) , c6:

'4,

•

1«

Tim giao diem cua SD va mat phang (AMN):

^ '»'*

Ta CO K va N la hai diem chung cvia hai mat phSng (AMN) va (SCD).
V^y ( A M N ) n ( S C D ) = K N

Gpi P la giao diem ciia K N va SD .
Suy ra P cung la giao diem ciia SD va mp(AMN).

LOIGIAI
a) Tim giao diem ciia M N va mp (ABCD).

\

c) Tim giao diem cua SA va mat phang (CMN).
Chpn mat ph^ng (SAC) chua SA. Tim ( S A C ) n ( C M N )

Gpi I = SN n CD(vi SN,CD c (SCD)). Chpn mat phSng (SBI) chua M N .
s

Taco C e ( S A C ) n ( C M N )

(3)

Theo cau b) E = SO n M N (SO, M N c (SBl)), c6:

Ta CO B va I la hai diem chung cua hai
mat ph3ng (SBI) va (ABCD) .

E€SOc(SAC)

Vay: (SBl)n(ABCD) = BI
Goi H = M N n Bl(vi M N , BI c (SBl))'*^A:
Ta

EeSO,SO<=(SAC)

EeMN,MNc(AMN)

^^^^'^
,
,=^K=SCn(AMN).
KeAE,AEc(AMN)
^
^

BAI TAP TONG HOP CHl/ONG I

a) Tim giao diem cua M N va mp (ABCD).
,
,
,
,
b) T i m S C n ( A M N ) , S D n ( A M N ) .
c) r i m S A n ( C M N ) .

va

(l)

,

^ ,^

^

(.\

Tu (3) va (4) (SAC) n ( C M N ) - CE

[HeMN

Gpi Q = SA n CE(vr SA, CE c (SAC)).

CO

[H6BI,BIc(ABCD)

, ^ Q = SAn(CMN). .
Ta c6: ^ ^ ^ ^
Q6CE,CEc(CMN)
^
^
^

=>H = M N o ( A B C D ) .
b) Tim S C n ( M A N ) .

!t d m I

g

^

Cho hinh chop S . A B C D c6 day A B C D la hinh thang voi A B song song C D . O
la giao diem ciia 2 duong cheo, M thupc S B . ' M ' i

Dau tien ta tim giao tuye'n cua mat phc^ng (SAC) va (SBI)

a) Xac dinh giao tuyen ciia ( S A C ) n ( S B D ) ; ( S A D ) n ( S B C ) .

Gpi O = AC n BI (vi AC, BI c (ABCD)).

b) Tim giao diem ciia S A n ( M D C ) ; S O n ( M D C ) .

Ta c6: S va O la hai diem chung ciia hai mat phling (SAC) va (SBI).
Vay: (SBl)n(SAC) = SO.
„
Gpi E = SO n M N (vi SO, M N c (SBl)).
Chpn mat p h ^ n g (SAC) chua SC.
Tim giao tuyen cua (SAC) va (AMN).

,. .

^ . , ,

,

' '

'

•

LOIGIAI

a) Xac djnh giao tuyen ciia ( S A C ) n ( S B D ) .

'

''

Ta CO S la diem chung thu nha't va O la diem chung t h u hai ciia 2 mat phang
( S A C ) va ( S B D ) .

(0

ci

37


t)) Tim £,iao diem K cua N M va mp(SBD).

V l y (SAC)o(SBD) = SO

Chpn mp (ABN) chua M N .

Xac djnh giao tuyen cua {SAD)r.(SBC) .
Se(SAD)n(SBC)

Tim giao tuyen cua mp (ABN) va mp (SBD)

(l).

IeSO,SOc(SBD)

Trong mp(ABCD) g p i H = A D n BC, c6:
He A D , A D e ( S A D )

A

;

=>lG(ABN)n(SBD)

H€BC,BCc(SBC)
=>H€(SAD)n(SBC)

•tmi «?'«•> a

' l€AN,ANc(ABN)
(3)

B e ( A B N ) n ( S B D ) (4)

(2)

Tir (3) va (4) BI = (ABN) n (SBD);

T u (1) va (2): (SAD) n (SBC) = SH .
b) Tim giao diem cua S O n ( M D C ) ;

K =BInMN.

SA n ( M D C ) .
rr^,

G(?i I = S O n D M ( v i S O , D M c (SBD)) , c6:

U

:, fuwi^ A\

• -KM

c) T m h t i s o

—

GQI Q trung diem cua A I .
GQiJ=-SAnCl(viSA,CIc(SAC)),

Ta CO A Q = QI = I N (Vi I la trong tarn cua tam giac SAC).

c6:

Ta c6: M Q la duong trung binh cua tam giac AABI

J ; a C I e ( M C D ) - ^ = ^^^(^^^)Cho hin!. chop S.ABCD c6 ABCD la hinh binh hanh tarn O. Gpi M , N Ian
lug-t la trung diem cua AB, SC.
a) T i m l = A N n ( S B D ) .

d) Theo each tim giao tuyen ciia cau b) thi 3 diem B, K, I thiing hang.

b) T i m K = M N n ( S B D ) .
c) Tmh ti so

d) Chung minh B, I , K thang hang. Tinh ti so

Cho hinh chop SABC. Gpi K, N la trung diem ciia SA, BC. Diem M thupc

IK

SC, SM = | M C .

LOIGIAI
a) Tim giao tuyen ciia mp(SAC) va mp(SBD).
Se(SAC)n(SBD)

OfcBD,BDe(SBD)

(l)

i

-.B:*!-'•

a) Tim thiet di^n ciia hinh chop v6i mp (KMN).
, LA
b) M p (KMN) cat AB tai L. Tinh ti so
LB

•'he}

=>Oe(SAC)n(SBD)

Tir (1) va (2) SO = (SAC) n (SBD)

38

ID

Ta c6: trong A N M Q : IK = ^ Q M
Va trong A A B I : Q M = - B I
IB = 4IK « — - 4 .
^
2
IK

KN

0€AC,ACc(SAC)

=^ M Q // B l .
Ta co: IK la duong trung binh ciia tam giac A N M Q .
V|y K la trung diem ciia M N .
KM ,
Suy ra
= 1.
'
KN

(2)

LOI GIAI
a) Tim thiet di?n ciia hinh chop voi mp (KMN).

Trong mat phSng (SAC), gpi I la giao diem ciia K M va AC .

Goi I - SO n A N (so, A N c (SAC)).

Trong mat phSng (ABC), gpi L la giao diem cua I N va AB. Ket luan thiet

Suyra N = A N n ( S B D )

dien can tim la M N L K .

'

39


BJ_BK
= 2 =>BJ = 2JP =j>CI = 2JP.
JP
KD
Tir do suy ra DP la duong trung binh cua tam giac CEJ.
Suy ra D trung diem ciia CE .
M*-''

b) M p ( K M N ) c i t A B tai L . T i n h ri so

Trong mat phiing ( S A C ) , ke A E // K M vai E thupc S C .
Ta CO K M la duang trung binh ciia tarn
giac S A E nen M trung diem cua SE.

b) Tim giao diem F = A D n (IJK). Chung minh FA = 2FD.

Vi IE, A D c ( A C D ) . G 9 i F = I E n A D .

Doan S C dugc chia lam 5 phan, S M
chiem 2 phan, M C chiem 3 phan suy
ra C E chiem 1 phan.
Trong tarn giac C I M c6
^

*

EM

=

AI

=2

Ma I E c ( l J K ) ^ F = A D n ( l J K ) .
Xet trong tam giac ACE c6 F la giao
/
diem ciia 2 duong trung tuyen A D va
P
EI. Suy ra F la trpng tam ciia AACE.
Vay FA - 2FD
c) Tim thiet di^n ciia tir di#n ABCD voi mp (IJK). Xac djnh hinh tinh ciia thiet
dien.

,
'

=>CA = - A I .

2

Trong A A B C ke D N // A B ( D € A C ) .
Vay: D N la duong trung binh cua A A B C nen D N = ^ A B
Trong tarn giac I D N c6

_ AL _ 4
ID D N 5

DN = - A L
4

Taco:

(1)

IAL c> 2AB = SAL

2(LA + LB) = SLA « 2LB = SLA

LA

2

a) Tim E = CD n ( I J K ) . Chung minh DE = DC .
(IJK). Chung minh FA = 2FD.

c) Tim thiet dien cua t u di?n ABCD voi mp (IJK). Xac djnh hinh tinh ciia
thiet di?n.
LOIGIAI
a) Tim E = C D n ( l J K ) .
E = CD n JK(CD, JK e (BCD)), c6:
E€CD
•EeJK,JKe(lJK)="^ = ^ ^ " O J ^ )
Chung minh DE = D C .
Trong ABCE ke DP // EJ.
Trong ABDP c6 JK // PD nen:

A 3 •.) i

( I J K ) n ( A B D ) = KF; ( I J K ) n ( A C D ) = FI

Trong ADAB:

LB ~ 3
Cho t u di?n ABCD. Goi I , J la trung diem cua AC, BC. Lay K thupc canh
BD sao cho BK = 2 K D .

b) Tim giao diem F = A D

( I J K ) n ( A B C ) = IJ; ( I J K ) n ( B C D ) = JK

Thiet di^n can tim la tu giac IJKF.

(2)

T u (1) va (2) c6:

iAB =

B

DK
DB

-5F.i^KF//AB
DA 3

Trong ACAB, IJ la duong trung binh

0)
IJ // AB

(2)-

Tir (1) va (2) suy ra tir giac IJKF la hinh thang.
Cho tu di?n S.ABC. Tren SB, SC Ian lugt lay 2 diem l, J sao cho IJ khong
song song voi BC. Trong tam giac ABC lay mpt diem K.
a) . Xac djnh giao tuyen ciia 2 mp (ABC) va (IJK).
b) . Xac djnh giao diem ciia AB, AC va (IJK).
c) . Tim giao tuyen ciia (SAB) va (IJK).

'

d) . Tim giao diem cua BC, IJ voi mp (SAK).

rinifl oilD

' " nnxb J y c l Urui n'ki 88

e) . Xac djnh thiet dien ciia mp (IJK) voi tit dien S.ABC.
LOI GIAI
a). Xac djnh giao tuyen ciia 2 mp (ABC) va (IJK).
Gpi D = I J n B C ( v i IJ, BC c (SBC)), c6:
r^De(lJK)n(ABC)
K€(lJK)n(ABC)

(l)
(2) .

Tir(l)va(2) (lJK)n(ABC) = D K .

D6lJ,IJc(lJK)
DeBC,BCc(ABC)

il A

H,


b) . Xac djnh giao diem cua A B , A C va (IJK)
Goi E va F Ian lupt la giao diem ciia
AB, AC voi DK ( vi AB , A C , DK ciing
t h u Q C mat ph^ng (ABC )). Ngoai ra DK
lai t h u Q C mat phJng (IJK):
Vay: AB n mp(lJK) = E; A C n mp(lJK) = F
c) . Tim giao tuyen ciia (SAB) va (IJK).

Vliy:

phiing

(SAB)

va

(IJK)

b) . Tim giao tuyen ciia mp (MNO) vol cac
m i t phSng (SBC) va (SAD).
G(?i F = B C n H O ( B C , H O e ( A B C D ) ) , c 6 :

A 4,

fFeBC,BCc(SBC)
,
^ ,
, , ,
^
' , = ^ F 6 M N O n(SBC ( l
F6HO,HOc(MNO)
^
' ^
' ^ '

nen:

(SAB)n(lJK) = IE.

N€(MNO)n(SBC)

d) . Tim giao diem ciia BC, IJ voi mp (SAK).

,

Trong mp(ABCD), gpi G = A D n HO ,c6:

,

,
, =>G = B C n ( S A K ) .
GeAK, AKc(SAK)
^
'
'in I

Goi L = S G n I j ( v i SCIJ c(SBC)), c6:
L

6 IJ

LeSG, SGc(SAK)

(2) .

Tir ( 1 ) va (2) ( M N O ) n (SBC) = FN .

Ggi G = A K n B C ( v i A K , B C C ( A B C ) ) , C 6

[G € B C

H e M N , M N c (MNO)

=>H = A B n ( M N O ) .

Ta CO I va E la hai diem chung ciia hai
mat

He Ap

fG€AD,ADc(SAD)
\)nSAD
GeHO,HOc(MNO)

3)
^

Co: M e ( M N O ) n (SAD)

•L = I J n ( S A K ) .

'

'

^ '

(4).

Tu (3) va (4) ( M N O ) n (SAD) = MG .
c) . Xac djnh thie't dien ciia (MNO) voi hinh chop S.ABCD.

e) . Xac dinh thie't dien ciia mp (IJK) voi tu dien S.ABC.

Theo each dyng diem 6 nhirng cau tren, ta c6:

Theo each dung diem 6 nhii-ng cau tren ta c6:

'nit AO

( M N O ) n ( A B C D ) = GF; ( M N O ) n ( S B C ) = FN

(IJK) n (ABC) = EF; (IJK) n (SAC) = FJ

X -A iV

( M N O ) n ( S A B ) = N M ; ( M N O ) n ( S A D ) = MG

(IJK)n(SAB) = IE; (IJK)n(SBC) = JI

Vay: thie't dien can tim la t i i giac MNFG.

Vay: thie't di^n can tim la t u giac IJFE.
Cho hinh chop S.ABCD c6 ABCD la hinh thang, day Ion la AB. Tren SA,
SB Ian lugt lay 2 diem M , N sao cho M N khong song song voi AB. Gpi
O = AC n DB

d) . Chung minh 3 diem S, K, E thang hang.

E = A D n B C , A D C ( S A D ) , BCc(SBC)
=>E6(SAD)n(SBC)

K = G M n FN, G M c (SAD), FN c (SBC)

b) . Tim giao tuyen ciia mp (MNO) voi cac mat (SBC) va (SAD).

=> K 6 (SAD)n(SBC)

c) . Xac djnh thie't di^n cua (MNO) voi hinh chop S.ABCD.

S€(SAD)n(SBC)

d) . Goi K la giao diem ciia hai giao tuyen 6 cau b, E = A D n BC . Chung minh
3 diem S, K, E thgng hang.

;

(»)

a) . Tim giao diem ciia duong thang AB voi mp (MNO).

'^^^A ^ J O S A :m

(* *)

-f
•

:

...

Tu (*)(* *)(* * *) suy ra ba diem E, K, S thupc giao tuyen ciia hai mat phang
(SAD) va (SBC) nen ba diem E, K, S thang hang.

LOIGIAI

Cho t i i di?n ABCD. Gpi M la trung diem AB, K la trpng tam ciia tam giac

a). Tim giao diem ciia duong thSng AB voi mp (MNO).
Gpi H - A B n M N ( v i AB, M N c (SAB)) .

^

(s|i) (£)'>ivr^r)w

ACD.
£). Xac djnh giao tuyen ciia (AKM) va (BCD).
43


b) . T i m giao diem H cua M K va mp (BCD). C h u n g minh K la trpng tarn cua
tam giac A B H .

| F € N P C ( B C D )

c) . Tren B C lay diem N , tim giao diem P, Q ciia C D va A D vai mp ( M N K ) .

t/

^

Ke't luan ba duong thang M Q , N P , B D dong qui tai diem F.

LOIGIAI

C h o hinh chop S . A B C D c6 day A B C D la hinh binh hanh. Gpi G la trpng

a). Xac dinh giao tuyen ciia ( A K M ) va ( B C D ) .

tam ciia tam giac S A D , M trung diem ciia SB

Mat phang ( A B G ) cung chinh la mat phang ( A K M ) .

,^

^ .4 -

-.

Tim giao tuyen cua (ABG) va ( B C D )

a) T i m giao diem N ciia M G va mat phSng ( A B C D ) .

GQiG = A K n C D

b) C h u n g minh ba diem C , D, N thang hang, va D trung diem ciia C N .

(vi

f

LOI GIAI

A K , C D C { A C D ) )

fG6AK,AKc(AKM)
,
.
,
.
CO:
;
/=^G6(AKM)n(BCD
GeCDXDc(BCD)
^
' ^
'
Va B e ( A B G ) n ( B C D )

Trong mat phang (SBE) chua M G , gpi N la giao diem ciia M G va B E .

, .
(l)
^ '

Vi B E thupc mat phSng ( A B C D ) , nen N thupc mat phSng ( A B C D ) .
Trong mat phSng (SBN) ke E F // M N ( F thupc S B ) .
Trong tam giac S E F c6 M G // E F nen c6:

(2)

T u (1) va (2) ( A B G ) n ( B C D ) = B G .
b) . Tim giao diem H ciia M K va mp ( B C D ) .

SM

SG

MF

GE

= 2 =>SM = 2 M F « B M = 2MF

Vgy F trung diem ciia B M .

Trong m p ( A B G ) gpi H = M K n B G , c6:

Trong A B M N c6 E F // M N nen c6:

HeMK,
^
^
/ ^ , ^ H = MKn(BCD)
H € BG, BG c (BCD)
^
'

BF

BE
=

,
„^
= 1 => B E = E N ,

C h u n g m i n h K la trpng tarn ciia tam giac A B H .

FM
EN
Vay E la trung diem ciia BN .

Vi K la trpng tam cua tam giac A C D nen

De dang chiing minh A A E B = ADEN(c.g.c)

K chia doan A G thanh ba phan bang nhau.

A B E = E N D . Hai goc nay bang nhau theo truong hop so le trong nen

Goi L la diem do'i xung ciia K qua G .

AB // D N , ma A B // C D . Vay ba diem C , D , N th^ng hang .

Vay K la trung diem cua A L .
'

^

T u do suy ra F thupc giao tuyen B D ciia hai mat phSng ( A B D ) va ( B C D ) .

d) . C h u n g minh 3 duang thang M Q , N P , B D dong qui.
^fe*--

| F e M Q e ( A B D ) ^ ^

E D la duong trung binh ciia tam giac N B C suy ra D la trung diem ciia C N .

Trong A A B L , M K la duong trung binh

Cho hinh chop S . A B C D , day la hinh binh hanh A B C D c6 tam la O. Goi M

cua tam g i a c .

la trung diem cua S C .

Ta c6: A B G L = A H G K (g.c.g) => B G = H G

a) . Xac djnh giao tuyen ciia (ABM) va (SCD).

Vay K la trong tam cua tam giac A B H .

b) . Gpi N la trung diem ciia BO. Hay xac djnh giao diem I ciia ( A M N ) voi
SI
SD. C h i i n g minh rang | ^

c) Tim giao diem P, Q ciia C D va A D voi mp ( M N K ) .

Trong m p ( A B C ) goi E = M N n A C .
Trong m p ( A C D ) duong thSng E K cSt C D va A D Ian lupt tai P va Q , thi P va
Q chinh la giao diem ciia C D va A D voi m p ( M N K ) .
d) . C h u n g minh 3 duong thSng M Q , NP , BD dong qui.
Trong m p ( M N K ) gpi F = M Q n N P , vi:

2
^ '

,

boi mat phang ( A M N ) .

a)

a i,,^'u
j j i u

u<)...}
A

I

J

^'^'^^ "^'^^ "^^''^ '^'"'^ chop S . A B C D cat

LOI GIAI
Xac djnh giao tuyen ciia (ABM) va (SCD).

Me(ABM)o(SCD)

*

ABC(ABM),CDC(SCD)

/ ;

=>(ABM)n(SCD) = M H ( M H / / A B / / C D )

b). Xac d j n h giao d i e m I cua ( A M N ) v o i SD.

/

' \

/

/-4

Taco (SAC)n(SBD) = SO.
I

., I C

a) T i m giao d i e m I ciia M N v o i ( B C D ) . T i n h t i so'
CQI

l-^t/

/

\

'leMN

i'.f ^j^vi^i'

^

/ } ^ f < ^ ' ^ \ ' ' ' ' ' ' /

Vi

I e CD , CD c

(BCD)

=>I = M N n ( B C D )

Gpi K = A M n S O ( A M , S O c ( S A C ) ) .

Tam giac A C I dup-c ve lai n h u h i n h 2

T i m giao t u y e n ciia ( A M N ) va (SBD) *
1&c6:\

I = M N O C D ( M N , C D C ( A C D ) ) .

Ne(AMN)
,
.
' ^
, =>N€(AMN)n(SBD)
N€BD,BDc(SBD)
^
'

,
^

TuDkeDG//IM(Ge

c

/(l)

AC).

. AM
TrongAAGDco:—=

.
'

»

AN .
— . 2

f-4i;*:'^*-'-^

uHl 3 8 i V

^'

•

A M = 2MG .
Tac6\

[KeAM, A M C ( A M N )

,

^
=>KG(AMN)n(SBD)
K€BD, BDc(SBD)
^

.

(2)
/

,

.

autT
V
/

, .

V;

^

-

la d u o n g t r u n g b i n h ciia tam giac C M I ,
suy ra D la t r u n g d i e m ciia C I

T u (1) va (2) suy ra ( A M N ) n ( S B D ) = N K .
N K c i t SD tai d i e m I , thi 1 chinh la giao d i e m cua mat p h J n g ( A M N ) v o i SD.
T r o n g mat p i l i n g (SBD), t u O d u n g OP // N I (P € S D ) .

no D P J
T r o n g A D N I c6 OP // D l nen co — = — = - = 2 => D P = 2PI
ON
PI
1
T r o n g ASOP c6 K I // OP, nen c6: | ^ -

Vay: G t r u n g d i e m ciia C M suy ra D G

(1)
^

= ^ = 2 ^ SI = 2PI

IC
Ke't luan: — = 2 .
ID
b) T i m giao d i e m J ciia B D v o i ( M N Q ) .
T,'nhtis6'i^,lQ.
JD
JI

(2) 1

(K la t r o n g tam A S A C ) .

Goi J = Q I n B D ( Q I , B D c

Vi

T u (1) va (2) suy ra — = ^ '
^
ID 3

^

Thiet dien cua h i n h chop S.ABCD cat boi mat phJing ( A M N ) .

u . i u i„i

Goi L la giao d i e m cua A N va BC. Ke't luan thiet dien can t i m la t u giac A L M I .

Cho t u dien A B C D . Tren A D lay N sao cho A N = 2 N D , M la t r u n g d i e m ciia
A C , tren BC lay Q sao cho BQ = 1 BC .
.;.! JinurlD . 0 3 •
IC
a) T i m giao d i e m I cua M N v o i (BCD). T i n h t i so ^ •
b) T i m giao d i e m J cua BD v o i ( M N Q ) . T i n h ti so — , — .
'
^'
]D JI
LOI GIAI

-—--•'i-

(BCD)).

JeBD

J-BDn(MNQ) .

JeQI , Q I C ( M N Q )

Tam giac BCI d u g c ve lai n h u h i n h 3.

^

Goi E t r u n g d i e m ciia BC, t u E ke d u o n g thSng song song v o i Q I cat BD, IC
Ian l u o t tai F va H .
Ta ccS QJ la d u o n g t r u n g b i n h cun tam giac BEF => BJ = JF
T r o n g A C Q I c 6 EQ
: ^ : .H^I^ 2

=^CH - 2 H I

o

C D + D M = 2H1

^

DH + HI + D H = 2 H l o H l =2DH ,

T r o n g .\DIJ c6 :

DI + DM = 2HI

DF _ D l i

1

V] ~ H I ' 2

T u (1) va (2) suy ra:

ID

3

DF = - F J

(2).

'? Or-