  . ' ' '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