Bài tập quan hệ song song trong không gian - Võ Công Trường

QUAN HỆ SONG SONG TRONG KHÔNG GIAN Võ Công Trường

2023-2024 3 0983.900.570

( ) P ( ) mp P ( ) P

, A B ( ) P

A ( ) P A ( ) P ( ) P

A ( ) P A ( ) A P 

B ( ) P B ( ) P

( ) P ( ) B P 

, A B AB

AB , A B

, B, C A ( ) ABC

d ( ) P

( ) d P  ( ) P d 

QUAN HỆ SONG SONG TRONG KHÔNG GIAN Võ Công Trường

2023-2024 4 0983.900.570

d ( ) P ( ) Q

( ) P ( ) Q ( ) ( ) d P Q = 

, B, C A ( ) mp ABC

( ) ABC

A a A ( ) ( ) , hay , mp A a A a

, a b ( ) , mp a b

1 2... n A A A ( ) 

S ( )  S 1 2... n A A A

n 1 2 2 3 1 , ,..., . n SA A SA A SA A n

1 2... n A A A

1 2 . ... n S A A A

1 2 . ... n S A A A

S

1 2 2 3 1 , ,..., n SA A SA A SA A

1 2... n A A A

1 2 , ,..., n SA SA SA

1 2... n A A A

A, B, C, D , D, DB, BCD ABC AC A D ABC

D ABC A, B, C, D

, , D, , D, D AB AC A BC C B

, D, DB, BCD ABC AC A

QUAN HỆ SONG SONG TRONG KHÔNG GIAN Võ Công Trường

2023-2024 5 0983.900.570

( )  ( ) 

( )  ( ) 

=  M a b ( ) ( ) M    

.S ABCD ABCD

M SA ( ) SAC ( ) SBD ( ) SAB ( ) SCD ( ) SBC ( ) SAD ( ) BCM ( ) SAD

( ) CDM ( ) SAB ( ) BDM ( ) SAC

( ) mp ABCD O AC BD = 

( ) ( ) O SAC SBD  

( ) ( ) S SAC SBD   ( ) ( ) SO SAC SBD = 

( ) mp ABCD F AB CD = 

( ) ( ) F SAB SCD  

( ) ( ) S SAB SCD   ( ) ( ) SF SAB SCD = 

( ) mp ABCD E BC AD = 

( ) ( ) E SBC SAD  

( ) ( ) S SAB SCD   ( ) ( ) SE SBC SAD = 

( ) ( ) M MBC SAD   ( ) ( ) E BC AD E MBC SAD     

( ) ( ) ME MBC SAD = 

( ) ( ) M MCD SAB   ( ) ( ) F AB CD F MCD SAB =    

( ) ( ) MF MCD SAB = 

( ) ( ) M BDM SAC   ( ) ( ) O BDM SAC  

( ) ( ) MO BDM SAC = 

( ) IBC ( ). JAD

( ) IBC

( ). DMN

( ) ( ) S SAC SBD  

E

F

O

A D

C

B

S

M

QUAN HỆ SONG SONG TRONG KHÔNG GIAN Võ Công Trường

2023-2024 6 0983.900.570

( ) ( ).    I AD I JAD IBC

( ) ( ).     J BC J JAD IBC

( ) ( ). =  IJ IBC JAD

( ) ABC =  E DM IB

( ) ( ).   E DMN IBC

( ) ACD =  F DN IC

( ) ( ).   F DMN IBC

( ) ( ). =  EF DMN IBC

( ) 

( ) 

( ) ( ) a b I a b I

      =  = 

( ) 

( ) 

( )  ( ) 

d ( )  ( ) 

( ) 

( ) 

( ) ( ) ( ) ( )

b

d b I

d b I



  

    =   =   = 

SABC SA M SC N MN AC O ABC ( ) OMN AC BC

AB

( ) mp SAC K MN AC =  ( ) MN OMN 

( ) K AC OMN = 

( ) mp ABC H OK BC =  ( ) OK OMN 

( ) H BC OMN = 

( ) mp ABC G OK AB =  ( ) OK OMN  ( ) G AB OMN = 

b

a α I

d

b β

α

I

H G

K A

C

B

S

O

M

N

QUAN HỆ SONG SONG TRONG KHÔNG GIAN Võ Công Trường

2023-2024 7 0983.900.570

.S ABCD M SCD

SM ( ) ABCD

AM ( ) SBD

( ) SCD N SM CD = 

( ) CD ABCD  ( ) N SM ABCD = 

( ) SAN AM

( ) ABCD E AN BD = 

( ) ( ) E SAM SBD   ( ) ( ) S SAM SBD  

( ) ( ) SE SAM SBD = 

( ) SAN F SE AM = 

( ) F AM SBD = 

( ) ( ) I DE AB I DEF ABC =   

( ) ( ) J EF BC J DEF ABC =    

( ) ( ) K FD AC K DEF ABC =    

b

a

c

K

β α

A

B

C

F

E

A D

C

B

S

N

M

QUAN HỆ SONG SONG TRONG KHÔNG GIAN Võ Công Trường

2023-2024 8 0983.900.570

( ) 

( ) 

( ) MNP

( ) ( ) MNP ABCD MN  =

( ) ABCD Q NP CD = 

K NP BC = 

( ) mp SBC E SB KM = 

( ) mp SAD . F SD QM = 

( ) ( ) ( ) ( ) , MNP SCD MQ MNP SBC MK  =  =

( ) ( ) ( ) ( ) , MNP SAB NE MNP SAD PF  =  =

( ) MNP

a CE a = . DF a =

( ) MEF

( ) mp ABC

( ) MNE AC I  =

( ) mp ABD

( ) MNE AD J  =

( ) MNE AB M  =

( ) mp MEF MIJ 

.S ABCD O AC BD , M N , SA SC

MN ( ) SAC

O ( ) SAC ( ) SBD

M SA  ( ) SA SAC  ( ) M SAC 

N SC  ( ) SC SAC  ( ) N SAC 

( ) MN SAC 

( ) , O AC AC SAC   ( ) O SAC 

( ) , O BD BD SBD   ( ) O SBD 