! ! " ! #$ " % (* & & ' *+,' ! " ' ( ) # * & * '+,* * . /' 0 '+,* +1' , *0 ) +1' , '+,* ' ' +, **'+ ,0 ,' '+,* / * 0 '0 * *0 0 2, / * 0 )* * ) ' '+,* 0 ' 0 , *0 +, * ,' 0 ,' '+,* ' 0 , '+,* +,0 0 " ( ,' '+,* , 0 ' ) ' * +, ' - '+,* ) 0 / ' '* * '+,* ' , *0 0 +, * ( 0 ( 0 *0 *'+ ,0- ' ' ,' '+,* 0 3! 4 '+,* ∆ 0 )∆ 7 '+ , ' −− ≠ 6 -( 0 '+ , ' 56 *( 0 7 ' 5 −− 5 / *2, )6 5 6- ( % ∆ ) ' ' - −− 5 0 0, - ( ' - ,( '+ ∆ * −− - ( 5 - 6 5 6 )6 5 6 - ' ' 0, /' * 8 (*- −− -∆ )∆ ) * −− *( 2, ' ∆ −− 6 6 6 6 - 5 ) 5 −− −− 5 −− 5 9 ' 6 * ' 0 ' +, * *( 0 −− * + ' 6 ' ' : 6 ;,' ' +, * '+,* (* −− , ' +, * ' 0 ( , 0 - ) *, + −− ' ) - ' ' 3! !4- ∆ )∆ '+ , ' '+,* ∆ −− −− *( −− −− 2, ,( '+ 7 & . ' −− −− 5 56 6 56- −− /' !, + 5 −− )( −− - '+,* +, * ∆ 0 656 )∆ '+ , ' ( (* '0 * +,'0 - ' '0 ' , ' - 2, ∆ ', & 0 2, 5∆ ' ' ' * '+ , ' −− - 5 656 '+,* ∆ ) ,* ∆ 5 7 ' ,' ∆ 7 0 ' ∆ '+,* / ,' * * 0 ' '+,* / * 0 −− ( '0 ' #5∆ # 6 ) # 6 # 0 * '+,* #6 −− −− 5 #6 - ' * −− #5# - ' ' ' 0) * & 5 <) *, ) −− # *(,' ) 6 5 6 - 0 *( −− 0 - # +, * - ' '+,* & . ' −− 0 0 * -0 0 <0 * '+,* 6 * ' ' # −− 0 2, '+ , ' 0 # * 0 0, 2, - ( ( '0 ,* * '+,* −− *, ) # 3!!!4 '+,* ∆ 0 +, * )∆ 7 ( -( *(,' 6 56 - 0 '+ , ' - *2, *( / * ) ' '* 0 % '+,* * & '+,* % )0 7 '∆ −− ' * ∆ * ' ' ) −− 0 * ' ; ( - ' 0 6 ( '+,* > 8 (* * 6 '+,* > 0) ' ' ' 6 ? 6 5 @AB) 6 ? 6 ? 6 5 @AB0 0, 2, 6 5 6 ? 6 −− - ' ' 6≠ 6 ,( '+ ' −− 5 ( −− * ' * ) 5 −− '+,* −− 5 ' ∆ −− ' (*' −− 5 −− )∆ −− 0 0 ! + ' ) ' ' *, 0 * '+,* ' ' * * ' - 6 5 6 ) (* '0 *( ( ** ,& ' ' ,& <* +, ' 7 '+,*- , *2, 5 −− *(,' - )( −− (*' 2, 5∆ −− 5 0 ,' '+,* ,' '+,* 0) ' ,' 0 * '+,* > - 2, ' (, ( *&= ' & <& * 0 +, * ,' **' - 0 * 2, 7 ' - , 0 - 2, 6 5 6 0, /' * 0 ( ( '0 ,* 2, ( ' ( '+ , ' 3! !4 ,* ∆ 5∆ ' −− 5 ,' +, * * , '+,* 0 * 2, * , 0 * 0 '+,* 0 * '+,* ' ' ' 0 ' ) 0 0 &= '+,* 0 ,' ' 0 ' 0) 0* '+,* ' 7 0) ' ,' '+,*- , * 2, '+,* > ) 2, , *2, 0 * 0 '+,* ' ' ' ' '+,*- ) *0 7 '∆ ,( '+ 2, −− :6 ) 96 ∆ / *- 6 6 0 ( ( 0 7 ' - ,( '+ 6 96 0 ,* /'- 6 5 0 0 *0 0 ,' ∆ ' 0 −− 96 ? 9 ) −− 6 '+,*- ) *(,' 6 5 ( −− 0 −− *2, 6 0 ' 5 −− 0 ∆ '+,* > ) ' ' ' - 6 96 ! , −− (' −− -0 ., −− 6 :6 ' ' '+,* )& 0 2, (* '0 * −− 5 2, 65 6 6 96 - −− : ∆ ' ' ∆ ' −− - −− −− −− : −− 9 −− - (* '0 * ( −− *0 ? −− / ,* ., * (, 0 ) 2, * ( −− 5 9 ,' (,' * ) 6 −− - 0 0 * 56 - −− *2, 6 ? −− 9 9 6 - ' ' −− ' ( * * ' * 2, 0 ' , ,' .+, + = , '0 * ' * +* ' + 0, 0 ) * ( ! * , /' 0 *( * ' ' ' ' * ' , /' 0 * .+, ' >(* * 0 0 *< * ) ' . , * 0 /' ( '0 ' *( * ' 0 ,* % C *0 * = 0 0 ' , /' 0 * .+, ) % 0 0 2, *< '0 * ( '0 0 ' . & ' * .+, ' * ( ( 0 0 ( 00 ! +* ' , ' 0 ' , ' + = ( < ,' ,* 2, ,' ,* 2, 2, ( ( ,' (,' ) ,' 2, ( ( 0 (,' 0 0 ' * +* ' (, 0 *< ' '+,' ( /'7 ' (, 0 ' ' , + ' - ' (, 0 ' , *< ,* ' ' 0 0 * +*' * ( (0 0 0 0 ,' ' * ,'. ' 0 0 00 ' ( ,* - * ( ( ' , * + ' +, * ,' + ' 0 0 - ' ,' 00 ) ( 0 ,' (,' 0 0 !""#$ %&! &'(& ' ,' ' ,' > , 0* +, * −− + ' −− " ' !& &)! ),' ' ) ,) *'+ ,0 ( (' % ' ) ' ' −− * + ' ' %)% + ' ' ,' (,' −− *0 0 ,8 * , 0 ' * ( 0 * ) 2, 0 ,8 ,' + ' (' % ' ) % '* ,'. ' * 0 - # ' ' '0 * ' ) 0 D !""#$ %&! * '!+ " ' !& &)! %)% 0 ,' '+,* 6 - ,' ),' (*' α 0 ' 0 ( * - 2, ' , ,' '+,* 6 ' &= - ,' *0 * - ' 00 +, * ) ' '0 'α ' ' ' 0 ,8 ,' ,'. ' - 2, * *0 0 6 ' (,' ) # ' ' '0 * , 0* ( 0 ,8 ,' 0 ,8 ,' ,'. ' ' ' ' ) ' * 0 2, * ' ,'. ' ' ,' (,' E (' % * ( ' ) % ' 00 - 0 ,8 ,' ,'. ' ' ' 'E ,'. ' * ,'. ' ' ' 0 (,' ' 0 (,' 6 ' * (*' α )* * * E ' ' E E * '+,* , 0 ' . - −− −− −− E6 E E 5 6 ' ' 5 !""#$ %&+,! 0 ,' 0 −− E ' . ' ' + ' −− 5 E ) 5 6 −− −− 5 E E - 0 0 '0 - ( * 2, - )*, + * ' ,'. ' * 0 <0 -*0 ' 0 '* 0 −− , ' *(,' ' 0 # )* ' ) 0 < −− + ' −− /' 0 E ' +, * * 0 ' 0 * 0 - & 2, E * -∆ 5 ∆E 0 ,8 * ,'. ' ' ' ' ' 0 (,' *(,' 0 # - (, * 0 ,'. ' '* 0 < ' ) )E ! ' ' * 0 E - E) - 2, ' , ,' (' % ' * ( ' ) (' % ' * ( ' ) ,'. ' ' ' 0 (,' 8 % '+,* ∆ E % * * *2, - ' ' , , ,' 0 * * −− −− 5 E) −− 5 −− ,( ' 2, < ,' ,'. ' ' ' < ,' ,'. ' ' ' ' ' ' ∆ * '+,* , 0 & , /+&() "+* # " ! , -* E ) #* '!+ %&+)% %)% 0& # & % & "!& )&+ & !+)% &+) # 1 )"#"! 2& & "#) 3 −− −− −− −− E? 9 ' !, + < ,'. ' −− )* ' 0 −− !""#$ %&+ )/#&" # .%&! * '!+ 0 ,' '+,* 6 - < ,' ,'. ' 0 ' - 2, * *0 0 * '+,* ' (,' ) E( & ' ' * , 0* ( 0 ,8 ,' ,'. ' ' ' ' ) ' ' ' E 0 ,'. ' ' ' ,' (,' ! * <0 * '+,* 08 8 0 2, 0 '+ , ' (&%# -%&+)(&%#) #.%&! &'(& −−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α * ' +* ) ( *( *' D> M >M 3α4 0, * ! , 0 , 0 * ,* ( '0 * + + 2, 0 ,8'0 - /* , ,' , 0 0 0 +, * * 0 ,' ,* 0 0 ( * 0 & . ' *, ' ( ' - & * ) 00 ' ,' + ' ( 0 , &0 ' ' *& *( * 3&0 ' ' ' ' +, *4 ! ( *0 0 0 * , 0 , 0 * ! '0 '' .' * 0 * +* N N- *% 0 0 * ' '0 ' <0 ' '+,' , /' * + ' . ' ' ' !""#$ %&, +4 ' ' +* ) ( -' (,0 ( ,(/ , % + ' ' ' ' 0 J, 2, 0 * ,0 , 0* ,* ,* ., (*' ' ( 0 ( 0 *'1 π; - π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≡ T T≡ T T- ≡ T ) 9 E, ( 2, - ' '0 ' - * '+,* ) T T T ' '+ , ' ' ', /' 0 ,' 2, 0 0 /' ) 1 % 0 (* * L * '+,* ) T ' '+ , ' - ' ' * '+,* ) T T T ' '+ , ' ( 2, 7 LF 0 2, * '+,* ) T '' '+ , ' ,( '+ 2, ) T' ' '+ , ' - (,0 '0 0 ,' (,' 0 * + ' *2, ≡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e lo anterior se deduce que las tres medianas de un triángulo se cortan en un único punto, que recibe el nombre de baricentro. < ,' '+,* 0 ,' 0 * &= 0 * '+,* < ,' ( ** **0 . '0 ,' ', & '+,* T T T 4 , * 2, ' ' ,' *, 0 - ' ' 0 <0 T T T 4 2, * 0 0 ,' '+,* ' ' ,' (,' '0 ) 4- 0 0, 2, * ' ' ' * *, 0 ,' '+,* ' ' ,' 1' (,' *(,' 0 0 * 2, ' ' ' * *, ** ortocentro ' ( *+ ' '& > 0 2, ' ' ,' ,'. ' 0 % ,'. ' ' 0 2, * ,'. ' ,' 4 , 2, ,' '+,* , *2, ( (, 0 ' ,' 4 4 ' ,) ,' , 0 * '& > 2, ' (, 0 ' ' ' '+,' 6 4 4 4 , 2, ' , * 2, 0 * ( 00 * '+,* ' * ( '0 ' ' '+,* ) T T T 0 ) T T T ' ≡ T T ) 6 ≡ 6T ≡ T T) ≡ T T 6 ≡ T T ) ≡ 6T ' '+,* '+ , ' ( 2, ,'. ' * 2, '0 ' * .+, '+,* ' '+,* & . ,' , * 2, - 0 ,' 0 , &= (,' 0 ,' *( *+ ' ' ,' ,'. ' 3 ,'. ' 4 , 2, ' ,' ( *+ ' +,* * 0 0 0 , *0 ' ' ,' 1' (,' - 2, 2,0 0 0 * &= 0 *( *+ ' +,* - ** ' 0 *( *+ ' 4 0, 2, ( ( * ,' ,' ,'. ' ,' ( *+ ' +,* * (, +, ' (,' % - * '+,* ' 6 ) 6T ( & ' 0 * +, ' ( 0 '0 , ' 7 2, * 7 ' 0 ,' '+,* '& > ' &= , 2, 7 4 ,' (,' 0 * <0 0 % '+,*- ' ' * 0 ' 0 * 2, ' ' ' * *0 0 * '+,* ' +, * ( ' - E ,' (,' 2, 2,0 0 * *0 0 * '+,* - ' ' E '* <0 0 % '+,* Esto se expresa diciendo que la bisectriz de un ángulo convexo es el lugar geométrico de los puntos que equidistan de los lados del ángulo. 4 , 2, 0 0 ,' '+,* ' ' ,' (,' 4 , 2, * 0 ,' '+,* ' ' , ' El punto de corte de las tres bisectrices recibe el nombre de incentro. ,' ( ' + ' 8 9 '0 * ( 2, % )2, +,* +, ( , ' 2, * '+,* / , +* ) ( ,' % > + ' ' * +,* @ 0 ,' : ,'. ' - ' , ' +* ) ( ,' '+,* 2,* ,' * ,'. ' ; ,' (,' > ,' 4 , 2, 0 0 ' > ' *, 0 * *'+ ,0 4 λ*0 ' 0 3 0 - 03 - 45 λ4 P 2,= , * 2, *, <0 0 0 2, λR * , 0 *'+ ,00- P > 0 0 * , ' *'+ ,00R 4 E ∈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∈ - T∈ 2, 0 ' 0 * H 2, *% > + ' ,* ' ' '0 ' , ' * .+, - 0 , 8 * ' /' 0 * ( ( '0 ,* <0 ) * ' /' 0 *', & (* +, ) ' '0 # 0 ' (*+ 0 0 , 0 - 0 ' 2, - E ) ( & ' EET T T - . & ' - +,* ' 2, * '+,* ) /* 0 T∈ # - 0 * ,'. ' 9 0 ,' ( *+ ' '& > 0 2, ,' ,' ,'. ' 3 ,'. ' ' 4 0 *0 '+ ' * ,'. ' 4 , 2, *' ' 0 ,' '+,* ' 0 ,' ,'. ' ' ' * 4 ' ,) ,' , 0 * '& > ' *2, ' (, 0 ' ' '+,' ,'. ' 4 , 0 * 0 .' /' 0 ' 0 ,' ( *+ ' +,* % ( 0 2, 0 % (,' ' 0 ,' ,'. ' 2, ,' * ( *+ ' , 2, 0 % ' =' ' 0 ,' ,'. ' ' ' *( *+ ' ( '0 < ' 0 ' & 00 .* 0 ,' 0 * . ' +, ' 4 ' 0 '+,* '+,* * %( ', ) 2, , *2, 0 * 4 (, 0 ' , ,' '+,* 0 *0 0 H- A) 4 (, 0 ' , ,' '+,* ,) '+,* '* 00 * 0 04 (, 0 ' , ,' '+,* ,) *0 '* 00 * 0 .4 ' ,' (, 0 ' ,' ,'. ' +4 ' '+,* ' 2,* ' (, 0 ,' ,' ,'. ' %4 ' ,' '+,* 2,* * (,' ' ' - ' ' ) ,' ' 4 ,' '+,* ' * , ** 0 ' # =' *, - ' ' 84 ,' '+,* ' * , ** *, # =' <- ' ' : ; , ,'. ' , (, < *0 ! P, ' &= 2, ' 0 '+,* ' ) ,' 2, - (, 0 * ,' ,' , 0 * '+,* * '+,* +, * * 0 * 0 ,' '& > & *0 '& > ' ,'. ' - ' ' 0 * * , ,'. ' -) , 2, 7 6 (,' ' '+ , ' 0 A@A - P , ' * '+,* ' *. ' ,' '+,* 0 '+,* ) '+,* 0 * 02, * %( ', ' ) 0 (,' 0 ,' 0* (*' 3 4 6 ) 6 4 E∈ - ' ' E ' 0' / * / * 0 0 0 * '+,* ' 0 ,' ( *+ ' ' * 0 2, ., +,* - P , ' ' * ' R 2, ' ,' * ,' , 0 * +, * * , 0 * , 0 0 0 +' * R ' ,& , ' ,(* '+,*- * , 7 ,'. ' 0 A ,' (,' 0 , 0 ' &= ' R0 *,' 0 - ' ' * 2, ' 0 )0 '+,* '0 ) ' E ,' DA 6 ) 6 ' E 4 E∉ - ' ' * ' 6 '- 0 0, 2, 7 V* 1' * & ) - * 0* 0 (,' ' + ' ) R ,) ,' '+ , ' (,' 0 E 0* (*' 3 4 0 > 0 arco capaz 0 '+,* 6 *' 7 0 ,' α )0 > 8 ' ,'. ' '+,* ' ) (* ,0α- P( 0 ),' '+,* 0 ' 0 ,' *0 - * '+,* (, 6 E * 2, * '+,* *2, ( ' , ,' +, * * V % ( <0 '+,* 0 *0 - )* *, * & 9 ' * .+, 40 8 ' 2, 7 ) ' ( ( '0 ,* ) '+ , ' ( ( '0 ,* ) '+ , ' - # ) ' ( ( '0 ,* E )# (,' , 2, 7 4 # 5 3 ? 4O 4 (,' 0 0 E 4 # 5 EO '8,' (* ,0 0 % *0 - E) E 0 0 ' : ! +, ' , ' ' . 0 * .+, 4- ' * 2, % ' C *0 * +,' ( 0 * ' +, * 4 * ( 0 '+,* 2, ( ' '+ , ' 4P ,* 0 ( 0 '+,* (, 0 ( 2, ' '+ , ' R 4 0, 2, * '+,* ) ' '+ , ' P C0 '0 /' * 0 0 ( * ≡ SR ' ' + & - C * *+1' ( 0 * ' 2, - 0 .+, ( ( 0 . 2, ' ., ' +, * - ' ( 0 0, 0 % '+ , ' ; ' * .+, 4 , 2, #- - 6< ' ,) ,' 6 ' ,) ,' 2, 0 A° 6 ' 6 ,) ' ' 2, 7 # (,' ' *' 0 PE,=' ( / * '+,* +* ) '+,* ' 0 0 0 * ' '& > ,'. ' ≡ * 2, ( ' -) ' * (,' <0 R )* 0 + ' * 0 * *'+ ,0 0 * ( ,' ( **+ ,' , 0 * (, 0 ' ' ,' ≡ * - ' 2, (' ,' '+,* 0 * '+,* 0 ,' '+,* )* *0 2, * '+,* , 2, - 2, ) ,' '+,* ' ( ** - )2, D 66 ' ,) ,' '+,* 0* + ' - 0 '0 67 4 2, ' ( '0 ' 3 **2, . '4 4 )10 ' ** 68 0* ( ' * , 2, * ' ( ' 0 * .+, - # '+,* ' ),0 0 * ' 69 ' 0 * *'+ ,0 0 **0 *' ' 0 0 '+,*- ,' 0 ' , *2, (, 0 ( *'+ * 0 ' ' 0 '+,* ' ' ( ,) ,' ( **+ , ,' ' ( *0 0 / ' ' + ' , ,' - '+,* 0 ' - -* ' 2, * * 2, 0 ) 0 0 *0 ) * ' )# (,' ' O0 * ' 0 ' ( '00 0 0 0 ' , 0 0 ' 0 * *'+ ,0 0 ,' 0 , *0 )0 , 0 + ' * ' * +, ' .+, - 0 ' * '. /' 0 0 0 ' '+ , ' * '0 ' C *0 (, 0 (,' 0 ' /' 0 - ( ' '+,' * & '+,* 6: 0 )* *'+ ,0 , 0 * *0 2, * 8 '* '+,* ' 2, '+,* (* ,00 * '+,* ,. ' ( C 0 * 2, '+ , ' 0 ( 2, * '+,* & ' 0 *' /' 0 + ' '+ , ' 0 D * .+, ' * 2, * 0 0, * '+ , ' 0 * '0 ' '+,* % ) ' 0 ' ,. ' C *0 - C 0 *+,' 2, - 8,' ' * 00 - ( 6; 0 ,' 0 * *0 0 ,' , 0 0 ' ,) ' '+,* '+,* +, * - 0 %( ', +, * **0 0 * , 0 0 - 2, * ' * ' & ' V% 0 ' V) V% ., V3 '0 * .+, 0 * ( +' +, ' 4 4 , 2, * '+,* J '+,* / * 4 U *(,' 0 0 * ) -) W *(,' 0 0 * ) , 2, * ( *+ ' W U ,' , 0 0 4 ' ),0 0 * ( 0 ' - 0 0, 2, * (,' J- ) ' *' 0 04P ' *' 0 * (,' J- - )SR 7< 0 ,' 0 * * ' & ' V% 0' ' *' 0 - ' ' * *0 0 ,' , 0 0 ' ,) ' '+,* +, * - 2, * ' V) V% ., V3 '0 * .+, 4 , 2, * (,' J- - ) S '+,* ' '+,* PE,= * /' % ) ' * ,* 0 0 * 8 DQ )= R D 7 Mosaicos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β , 2, '+,* ' -) 4 ' * ,) '+,* ' 4 *'1 0 ** 2, ' ' 0 * ( * * ' * +,' 0 ( *+ ' +,* α>β⇔ ' 9 ( ' '+,* ' 0 * 0' ( & 0 ( *+ ' +,* '+,* , 2, 0 ') - ,' ' *, ' *0 ) ( & *'1 +,* 0 ) +,* 2, + ' *2, ' - ) ' '+,* ' 0 *0 0 *( *+ ' ( *+ ' +,* 0 - D- - G *0 P > ' QQAOI- H - AAO I- HG + 0 R * 0 0 0 ,' &= 0 ,' 0 * 0 : X H- 5 ⇔ 5H ) 3LD4X (, 0 ' ' , ' 0 , *0 0 ( ( +,* ' ( *+ ' ' * & α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perpendiculares que se cruzan7 ' ( ** ** 1* ,* 0 ' ( 0 .' ( ( '0 ,* ' ( ( '0 ,* * 4 ' ( ( '0 ,* * - ' ' ( ( '0 , * ** =' 2, , *2, ,< ' ' 0 ' ( ** 04 ' recta se dice perpendicular a un plano * 0 ,' 0 * ' ' 0 ' 0 % (*' ' '0 ' , ' , >( ' ' * ( 2, 0 - ',' 0 ' ,* 0 2, '8 , )2, * ' ' ( ( '0 ,* 0 0 ' )(*' - ' - ( ** ' - ( ** ' (*' ' )(*' # *(,' 76 .+, *(,' & ' ( 0 0* + ' 0 0 * 4 , 2, 4 , 2, * 4 D4 , 2, * ( + ' ' # ' ) * '+,* '+,* -) - # ) # - ) - E- * '+,* ' * 2,* '+,* ' ,0 - # - # ) ' 2,* )2, '+,* , ' λ * *'+ ,00 ,' 0 ' 0 * '+,* )µ * *'+ ,00 ,' 0 ' 0 * '+,* # 3' 0 ,' 0 '+,* 0 * 0 ' -( 2,* - ' ' +, **'+ ,04 , 2, µ =λ/2 +1' * # DG G4 * ( ' ,'. ' ' 0 ' # ) 0 * *'+ ,00 * + ' # , 2, * (,' -E) DH =!&()%& & +!"#$ %&+&>& "#"# %&%&& (# )"#$ %& * 0 8 ,' ' ,' 0 ' ,' &= ' , ' 0 ( *+ ' ' ' 7 α>β⇔ ' 9 - "!) # &# ,' ( *+ ' *, ' ' α HA D QA +,* G A@ ' *, ' A I QAAO I +' *2, & 0 α @ Q G DA A DD +1' * D #%&+ H AO GA G G GH ' 7 &'!+) & 2, % ) * 0 0 0 ,' &= #* '!+ ' 7 < D D D8 E ⇔ E8 + ... + α = HA ' QAX α : @A( 5 ? - - - 0 0 '0 0 0, 2, QA3L 4? HA X HA- )( * ) ' 0 *'1 ' ' 2, 0 * * ' E 8@ E 7@ E 6@ E @ % D A@AOI Q@AO ( , +4' ' '0 ' , ' ' , ' ' ,' &= * +, ' 7 H (, 0 ' ' 0 ' α1, α2, ..., α L1, α - &'!+) & ,) '+,* ' α1 + α2 + ... + α L1 + α = HA HAX α : @A( 0 HA X HA) HA: @A - 0 0 '0 C D8 HA ? α ? 0 63D 0 ' ' α )β ( & ' &'!+) @()- & ! * '!+ * +, ' - ' * 2, #)+&%&% %&! 1A #"&B)- , +4' ' 0 ', +,* 0 ' ) *0 - ) , '+,* ' ()- & &+ ?(& %&+ )% %&+, + 4' ' ' *0 - * '+,* ' H 0 /* % )2, & 2,= , ' +' & 2, *'1 0 ( *+ ' +,* 2, '+,* - * #!)"# & , #/+& =!& &,!&%& ,+) &) ' E8 E @6 E <@ @ @ E <@ @ ' ' 0 ,' 0 * ' F ,* ' DI E 8@ E 8 E 7@ E @6 E 7@ E ' ( HA? HA? HA? αD ? αG 5 HA ' QAX αD - αG : @A 0 '0 ' ' , /'- 8,' * '+,* ' , ' ' *&= ' * +, ' '.+, ' 7 0 0, 2, αD 5 αG 5 QA 0 , 0 0 - (,0='0 E 7@ E 6 ' , D HA? αG 5 HA '+,* ,' % > + ' 0 '0 ' 2, * .+, 2, ' , ' *&= 8,' '* E 6@ E <@ @ @ E 6@ E < ' '0 ' α ? α ?α ? αD 5 HA ' QAX α : @A( α 5 QA( 0 ! '.+, /' (, E 6@ E 5 - - -D 0 '0 ' *,) 2, D@ '0 ' HA ? α ?α ? αD 5 HA ' QA X α : @A ( 5 - -D , 2, α ?α ? αD 5 %) +) 2#'! ) & ) & "!)% )% • α 5 A@ * '+,* ' ' 0 ,' ( ' + ' +,* - α ? αD 5 Q , 2, QAX α - ' ' QA X α - αD X A - ) ' ' , ' α 5 αD 5 QA ( & * 0 * '+,* - α ? αD Y Q ** 0 ( '0 2, & +)" 2#'! )"#$ ,!&%&& ) ! ,& *' &'!+ ) • α 5 A * '+,* ' ' 0 ,' % > + ' +,* - α ? αD 5 @A ' QAX α : @A( 5 -D 0 '0 ' 2, α 5 αD 5 QA- )* +, ' 0 '.+, ' ' AA- • α 9 A * '+,* ' ' 0 ,' ( *+ ' ' ' α ? αD : @A ' QA X α : @A ( 5 -D7 ( *0 0 +,* , /' ' ,' '1 ( * 0 *0 ) 2, H5 D- 5 - ' ( ' E 6@ E ' '0 ' HA?α ? αD 5 HA ' QAX α : @A( 5 -D • α 5 QA * '+,* ' ' 0 ,' , 0 0 - αD 5 GA U,' * 0 '+,* ' , ' ,' , 0 0 ),' 0 0 + ' • α 5 A@ * '+,* ' ' 0 ,' ( ' + ' +,* - αD 5 - & * 2, ' ( '0 * 00 0 * '+,* ' ' 0 ,' ( *+ ' +,* • α 5 A * '+,* ' ' 0 ,' % > + ' +,* - αD 5 A U,' * 0 '+,* ' , ' 0 %>+ ' +,* • * ( ( * 0 α ) αD ' ' *- '* ' 2, 0 (* * ,0 ( 5 D- 5 D E 6@ E ' '0 ' (, 0 0 D E @ E <@ @ D E @ E< ' HA? αD 5 HA ' QAX αD : @A , /'- ' 0 ( - DQ ' α ? α ? α 5 HA ' QAX α : @A ' 5 - 0 '0 0 0, 2, " ( (*F# ( &+ ?(& %&"!)% )% =!&" "! & & ! 1A #"&& ! • α 5 QA- α ? α 5 IA ' A@X α : @A ' 5 L ' * '.+, /' ( ,' ( ' + ' +,* - * ( *+ ' +,* 0 ' ,' '+,* ' 0 H + 0 - 2, *0 ,' ( *+ ' +,* 0 A*0 ' '0 ' , ' ' ' 2, ' 0 &= 0* ' , ' ,' ( ' + ' - ,' , 0 0 ),' /+ ' ' * 0' * - * 0 0 0 ,' ( ' + ' * , /' 2, +' * 2, , * .+, ,* (, ( * L ' * '.+, /' ( ,' % ( + ' +,* - * ( *+ ' +,* 0 ' ,' '+,* ' 0 QQAOI + 0 - * 2, ( * (, 0 ( ' * * 0 *( ' ( L ' *&= ' , ,' % > + ' +,* - * ( *+ ' ,' 0 0 + ' L ' *&= ' , ,' /+ ' +,* - * ( *+ ' /+ ' L ' * '.+, /' ( ,' ( *+ ' 0 ) '1 0 *0 2, @- *2, 0 ' ' '1 0 *0 2, @ ) ' 0 0 - * ,0 (* , '0 ' * '.+, /' ( ,' , 0 0 • α 5 A@ 3'+,* ' 0 (' + ' +,* 4- α ? α 5 G ' A@ X α : @A ' 5 L ' *&= ' , (' + ' +,* - * ( *+ ' 0 ,' 0 + ' +,* L ' * '.+, /' ( ,' % > + ' - * ( *+ ' '0 2, ' ,' '+,* ' 0 + 0 - ( * L ' * '.+, /' ( ,' ( *+ ' 0 ) '1 0 *0 2, H- *2, 0 ' ,' '+,* ' 0 + 0 0 A@ A- ( ' * 2, ) ' ' (*0 ' ' ,' * ,0 (* , '0 ' *&= ' ' , ' ,0 0 ) *+1' ( ' + ' • ' ' , ' , 0 0 ' ( ' + ' - α ? α ? α 5 HA ' AX α : @A ' 5 - *+ 2, α 5 A( 5 - ! 0 ' '.+, ' 2, ( '( 5 - 5A ' * +, ' GA D E @ E *'1 0 '+,* 2, ' , ' ' ,' &= ,' - α ? α 5 AA 0 α : @A- ")%) α# %&/& & ()- =!& < • ,' 0 * α QAAO I- * 0 AAOI- 2, ( '0 * '+,* ' 0 ,' ( *+ ' +,* 0 D *0 - )* , /' 2, ( 0, * 2, , * .+, +, ' - )2, '0, ,' ' +,* • ,' 0 * α G 3 00 0 '+,* ' '+,* ' 0 ,' ( *+ ' +,* 0 D *0 4 0 ,' ! ' * .+, , /' 2, ( 0, - (, 0 ( /+ ' 4- * ' - ' *+ * • ,' 0 * α DD 3 00 0 '+,* ' 0 ,' 0 + ' 4- * '+,* ' 0 ,' ( *+ ' +,* 0 G *0 4- ( * , /' 2, ' ' ( 0 , /' . * • ' 0 α (, * 1' ( ) +, *2, GA *0 02, ( ' ( * , 0 5 - 5 70 00 0 0 ( 0, HG 3 00 0 ' *< 0 GH 3 00 0 ' *+ * AA- α 5 GA ( 5 + ' ),' '+,* G D E @ E ' (*' ' '0 ' , ' • • • , /'- 0 *0 HA? α 5 HA ' QAX α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• 0 0, 2, ' * ,0 ( 0, ' 2, * ( * 1' '.+, /' ' * * *(*' ' * 2, *< 0 %' '0 0 ( * '.+, ' * 0 0 0 ,' &= 0 * ,* +,* 7 3 L L L L L 4- 3DLDLDLD4- 3HLHLH4- ),' ' & *0 ( * 7 3 ALGLG4 ( +,* 0 , * ' * 0 '.+, ' ' G ,0 '00 & 2, ' ( - ' 2, .' * ' *. /* . * ' +,* % 0 * * '* 0 '.,+, ' 2, ' * ' '