on tap toan 11 tron bo ne



. ' ' 'ABC A B C

= = =
uuur r uuur r uuur r
' , ,AA a AB b AC c
 !!
"#$%&
uur
AI
'$%&
r r r
, ,a b c

("#$%&
uuur
AO
'$%&
r r r
, ,a b c
$)*"( !!
"#$%&
uuur
AG
'$%&
r r r
, ,a b c
$)"
∆
' ' 'A B C


+,-.
( ) ( )
= + = +
uuuur uuuur uuuuur uuur uuuuur
1 1
' ' ' ' ' '
2 2
MN AC A B AB A C
$)/01233!
!!
',-.
( )
= + + +
uuur uuur uuur uuuur uuur
1
' '
4
AO AB AB AC AC

( ) ( )
   
 
   
AI AB AC a b a c a b c
= + = + + + = + +
uur uuuur uuuur r r r r r r r
( ) ( )
( ) ( )
( )

 


 
 


 
AO AC AB a c b
AO a c b
AO AC AB a c b

= + = + +


⇒ = + +


= + = + +


uuur uuuur uuur r r r
uuur r r r
uuur uuur uuuur r r r
( ) ( )
 
  
 
  
  

AG AA AB AC a a b a c
a b c
= + + = + + + +
= + +
uuur uuur uuuur uuuur r r r r r
r r r

( ) ( )
= + = +
uuuur uuuur uuuuur uuur uuuuur
1 1
' ' ' ' ' '
2 2
MN AC A B AB A C

 !"#$$%&%%'

( ) ( )
 
           
 
         
AC A B AB A C AC A B AC A C
AC AB A C A B B C B C
+ = + ⇔ + = +
⇔ − = − ⇔ =
uuuur uuuuur uuuur uuuuur uuuur uuuuur uuuur uuuuur
uuuur uuuur uuuuur uuuuur uuuuur uuuuur

4563 43 7

a
6376 767636 689$84
:"
∆
ABC

,-.
⊥ ⊥
,SA BC SB AC
(,-.
( )
⊥
SH ABC

;#4<63$5=>3 ?
#() !"#$&&*#!+
,&-,./#,
⊥
&$0
⊥
&./#&
⊥
,$
*12$
⊥
,&
*#!+
SN BC
BC SH
AH BC

⊥

⇒ ⊥

⊥

#34/#55'!5' 0678*9:0;<<&= *#
c
r
a
r
b
r
*12$&
⊥
,0
3*>!?#,/#
( )
⊥
SH ABC
!'*@+!A#,$B:CD$&E'
*#!+
( )
HS ABC
⊥
./#$0F!G!"#$,HD$&E
,/#+!A#,$B:CD$&E+!A#$0,$
( )




!5.



b
AH b
SAH
a
SA a
= = =

I9/+!A#,$B:CD$&E3 
5+α +!.#5!5

!5.

b
a
α
=
@A4563 B4C"*D
( )
⊥
SA ABCD
637
·
= °
120BAD


;#EF4 B$6
(6,-.
( )
⊥
OH ABCD

;#EF46 $B
#IF$&6F5./#
AC BD
⊥
( )
SA ABCD
⊥ ⇒
$F!G!"#,HD$&6E
,/#+!A#!J3KL
L
3*#!+M0N3F!"##O!,$./#0M,$

( ) ( )
SA ABCD OH ABCD
⊥ ⇒ ⊥
!6$&./#+!A#,&6+!A#,&$&
3P
L
F#O!,$&#O!Q!?R$
                           
·
= °
30BAC


= = = =
SA SB SC SD a

!"#$%
( )
⊥
SO ABCD

&'(")*+
,-"./012#"34!"#$%
( )
⊥
MN SBD

5'(67")
#IFM5 !"#$&6S,$-,&-,-,6H
( )
SO AC
SO ABCD
SO BD
⊥

⇒ ⊥

⊥

3*#!+
( )
SO ABCD
⊥

./#MF!G!"#,HD$&6E
#34/#55'!5' 0678*9:0;<<&= *#
F
·
L
LBCA =
./##O!$6#O!T./#


a
CO =
·
( )
·
L

!5. L

OC
SCO SCO
SC
= = ⇒ =
'I9/+!A#,D$&6E3L
L

!*#!+
( )
( )
( )
( )

SO ABCD SO BD
BD SO
BD SAB
DB AC
BD SAB
MN SAB
MN AC
⊥ ⇒ ⊥
⊥

⇒ ⊥

⊥

⊥


⇒ ⊥



P
()0F!G!"#MH,&
*#!+
( )
AC SBD AC HO
⊥ ⇒ ⊥
'U5RCM05R
Q+!!!"#$,&
*#!+#O!,M&#O!Q!?RM./#M0-


a
8
∆
ABC
"19:194
":;
·
= °
120BAC
<"=>
=
3SA a
,"
?"@4A:
!"#$%
( )
⊥
AK SBC

&'(")"=>%*+*+
'(67")
#*#!+
( )
SA ABC SA BC
⊥ ⇒ ⊥
0$N!#5!"#$&./#
AH BC
⊥
( )

( )
( )
AH BC
BC SAH
SA BC
BC SAH
BC AK
AK SAH
⊥

⇒ ⊥

⊥

⊥ 

⇒ ⊥

⊂


;F!G!"#$H,0./#
AK SH
⊥
( )
AK SH
BC AK AK SBC
BC SH H
⊥



⊥ ⇒ ⊥


∩ =

3
( )
( )
( ) ( )
( ) ( )
·
(
)
·
( )
·
 

AH ACB
SH SBC
ABC SBC SH AH AHS
SBC ABC BC
SH AH BC
⊂


⊂

⇒ = =


∩ =


⊥

L
#  VL
SA
H H
AH
= = ⇒ =
#34/#55'!5' 0678*9:0;<<&= *#
*#!+$05RQ+!!!"#,$&9/W!A#,$&3#
GA4563 B4C3 BD4
·
= °
60BAD

=
3
2
a
SA
H
6I5=>3 B?J$)"
∆
ABD

,-.

( )
⊥
BD SAC
;#66
(
α
4>6 B?$>3 B?;#
α
tan
;#KL<B$63
#IF0F!G!"#,HD&6E./#,0
⊥
&6
$&6F5./#$
⊥
&6
( )
SH BD
AC BD BD SAC
SH AC H
⊥


⊥ ⇒ ⊥


∩ =

$&6F5!R#+!
·

L
VLBAD
=
H#O!$&6
#O!T!R#'
 
S
V 
a a
OH OA OC
= = =
  

  
  

  
  
      P
'
       
P

P  P
   


a a a a a a
SH SA AH AO
SH a

a a a
SC SH HC AO
a
SC
     
 
= − = − = − = − =
 ÷  ÷  ÷
 ÷
 ÷  ÷  ÷
 
     
⇒ =
 
= + = + = +
 ÷
 
⇒ =
3*#!+
( )
( ) ( )
( ) ( )
·
( )

P V
# ' P


SAC BD

SAC ABCD AC OH SO
SAC SBD SO
SH
a
HO
a
α
α
⊥ 

∩ = ⇒ =


∩ =

= = =
MA4563 4C
∆
ABC
NDO
( )
⊥
SA ABC
637

,-.
( )
⊥
BC SAI
(;#KLP3H5=>6 ?

;#4<5=>6 ?$>3 ?
#*#!+
( )
SA ABC SA BC⊥ ⇒ ⊥
DE
$&#O!T< !"#&H$<
⊥
&DE
*>DEDE./#&
⊥
D,$<E
3()0F!G!"#$HD,&E
*#!+
#34/#55'!5' 0678*9:0;<<&= *#
( ) ( )
( ) ( )
SBC SAI
H SI
SBC SAI SI
⊥ 

⇒ ∈

∩


XY#O!Q,$<!+
    
     
 

a
AH
AH AI SA AH a
= + ⇒ = ⇒ =
!*#!+
( )
( ) ( )
( ) ( )
( ) ( )
( ) ( )
·
(
)
·
( )
¶
¶
( )
¶
L
 

# L




BC SAI
ABC ABC BC
SBC ABC SI AI SIA

SBC SAI SI
ABC SAI AI
SA a
SIA SIA
AI
a
⊥ 

∩ =

⇒ = =

∩ =


∩ =

= = = ⇒ =
QRA4563 
( )
⊥
SA ABC

∆
ABC
NH6I H
3I6$
= =2 3, 2SA a AB a

,-.

( )
⊥
AH SBC

(;#4<5=.>6 ?$>3 ?
;#KL<63$ 
#*#!+
( )
SA ABC SA BC⊥ ⇒ ⊥
DE
$&#O!T< !"#&H$<
⊥
&DE
*>DEDE./#&
⊥
D,$<E
( )
( )
BC SAI
SA AH
AH SAI
⊥ 

⇒ ⊥

⊂


 a
0F!G!"#$H,<H

AH SI⊥
( )
SA AH
SI AH AH SBC
SI BC I
⊥


⊥ ⇒ ⊥


∩ =

3
( )
( ) ( )
( ) ( )
( ) ( )
( ) ( )
·
(
)
·
( )
¶ ¶
( )
¶
 
  S # 




BC SAI
ABC ABC BC
SA a
SBC ABC SI AI SIA SIA SIA
AI
SBC SAI SI
a
ABC SAI AI
α
⊥


∩ =

⇒ = = = = = ⇒ =

∩ =


∩ =

*5+α+!.#5!5# α-
!W5Z!O!A#,$&[5R$<-
 a
#34/#55'!5' 0678*9:0;<<&= *#P
QQ.4563 4C
∆
ABC

$8"$)3 7 7
( )
⊥
SA ABC
637
3
,-.
( )
⊥
BI SAC
(;#EF4<O5=>63?$>6 ?
;#KL<6 $3
#34/#55'!5' 0678*9:0;<<&= *#V