GEOMETRIA DESCRIPTIVA

1

KGHCGVTÃA FG^NTE\VE_A  \th`edefa ia tg|thfwnneþj vhvai h |atneai fg g~va hdta  |ht nwaiqwegt cgfeh) ~ej awvhte}aneþj g~nteva fg ih~ awvhtg~*  FGTGN@H^ TG^GT_AFH^ ¥ 4<1<) tg~|gnvh a ia  |tecgta gfeneþj gj g~|añhi |ht GFEVHTG^ GI CHTHNHN@AJH*

 Fe~gñh fg nwdegtva= Keicgt Aeig~ @*  Ec|tg~h gj \gtý „ \tejvgf ej \gtweaj*

 A ih~ qwg fg~ggj hdvgjgt wja nh|ea gignvtþjena gj ohtcavh [htf) \FO ý hvth)  |wgfgj nhcwjenat~g nhj gi awvht ãa g#caei h vgiåohjh* Vacdeåj ~e tgqwegtgj a~g~htãa gj Fedwmh Vånjenh) Fedwmh Cgnájenh) Kghcgvtãa Fg~nte|vea) gvn*

4

KGHCGVTÃA FG^NTE\VE_A  \th`edefa ia tg|thfwnneþj vhvai h |atneai fg g~va hdta  |ht nwaiqwegt cgfeh) ~ej awvhte}aneþj g~nteva fg ih~ awvhtg~*  FGTGN@H^ TG^GT_AFH^ ¥ 4<1<) tg~|gnvh a ia  |tecgta gfeneþj gj g~|añhi |ht GFEVHTG^ GI CHTHNHN@AJH*

 Fe~gñh fg nwdegtva= Keicgt Aeig~ @*  Ec|tg~h gj \gtý „ \tejvgf ej \gtweaj*

 A ih~ qwg fg~ggj hdvgjgt wja nh|ea gignvtþjena gj ohtcavh [htf) \FO ý hvth)  |wgfgj nhcwjenat~g nhj gi awvht ãa g#caei h vgiåohjh* Vacdeåj ~e tgqwegtgj a~g~htãa gj Fedwmh Vånjenh) Fedwmh Cgnájenh) Kghcgvtãa Fg~nte|vea) gvn*

4

 ÃJFENG   

\TGOANEH

NA\ÃVWIH E= GI \WJVH

1>

1*1*  1*1*  

Gi |wjvh

1*4*  1*4*  

\thugnneþj fg wj |wjvh

1*>*  1*>*  

1>

\iajh~ |tejne|aig~ fg |thugnneþj

1; 

1*6* Nhva) aigmacegjvh u a|atvacegjvh 1*7*  1*7*  

10

Fg|wtafh fg wj |wjvh

10

1*?* Ktaoenaneþj fg wj |wjvh |ht nhhtfgjafa~ 1*;*  1*;*  

15

\h~enehjg~ tgiavea~ fg fh~ |wjvh~

4<

1*0* _e~va~ Awpeieatg~ 1*5*  1*5*  

41 Tgkia~ fg e~edeiefaf

4>

1*1<*  1*1<*  

\thdigca~

NA\ÃVWIH EE= IA TGNVA

6>

4*1* Fgoejeneþj

6>

4*4* \thugnneþj fg wja tgnva

6>

4*>* \h~enehjg~ |atvenwiatg~ fg wja tgnva

66

 

Tgnva @hte}hjvai

 

Tgnva Othjvai

 

Tgnva fg \gtoei

67

 

Tgnva _gtvenai

 

Tgnva Jhtcai

 

Tgnva Htvh|gtoei

4*6* \wjvh qwg |gtvgjgng a wja tgnva

6; 

4*7* _gtfafgta Cakjevwf fg wja tgnva

6; 

4*?* _e~va fg |wjva fg wja tgnva

60

4*;* Htegjvaneþj h twcdh fg wja tgnva

65

4*0* Ejniejaneþj h |gjfegjvg fg wja tgnva

65

4*5* \h~enehjg~ tgiavea~ gjvtg fh~ tgnva~

7<

 

Tgnva~ qwg ~g ejvgt~gnvaj

7< >

 

Tgnva~ |ataigia~

 

Tgnva~ |gt|gjfenwiatg~

 

7< 71

Tgnva~ qwg ~g ntw}aj

74

4*1< \thdigca~

74

NA\ÃVWIH EEE= GI \IAJH

;7

>*1* Fgvgtcejaneþj fg wj |iajh

;7

>*4* \wjvh u tgnva |gtvgjgnegjvg~ a wj |iajh

;7

>*>* \h~enehjg~ |atvenwiatg~ fg wj |iajh

;? 

 

\iajh @hte}hjvai

 

\iajh Othjvai

 

;?  ;; 

\iajh fg \gtoei

;; 

 

\iajh _gtvenai

;0

 

\iajh Jhtcai

;0

 

\iajh Htvh|gtoei

>*6* Tgnva~ jhvadig~ gj gi |iajh

;5 ;5

 

Tgnva @hte}hjvai fgi |iajh

;5

 

Tgnva Othjvai fgi \iajh

0<

 

Tgnva fg \gtoei fgi \iajh

0<

>*7* Htegjvaneþj fg wj |iajh

0<

>*?* \iajh fg najvh

01

>*;* _gtfafgta cakjevwf fg wj |iajh

04

>*0* Tgnva fg cápeca |gjfegjvg

04

>*5* Ejniejaneþj u |gjfegjvg fgi |iajh

0>

>*1<* \thdigca~

06

NA\ÃVWIH E_= \ATAIGIE^CH U \GT\GJFENWIATEFAF

111

6*1* Nhjfenehjg~ fg |ataigie~ch gjvtg tgnva~ u |iajh~

111

   

Tgnva |ataigia a wj |iajh

111

\iajh~ \ataigih~

6*4* Nhjfenehjg~ fg |gt|gjfenwiatefaf  

114 11>

Tgnva |gt|gjfenwiat a wj |iajh

6*>* \ht wj |wjvh vta}at wj |iajh |gt|gjfenwiat a wja tgnva fafa 6

11> 116

6*6* \iajh~ |gt|gjfenwiatg~

116

6*7* \iajh cgfeavte}

11? 

6*?* \ht wj |wjvh vta}at wj |iajh |gt|gjfenwiat a wj |iajh fafh u a wja tgnva fafa

11? 

6*;* Fe~vajnea cá~ nhtva fg~fg wj |wjvh a wja tgnva

11; 

6*0* \thdigca~

11; 

NA\ÃVWIH _= EJVGT^GNNEHJG^

115

7*1* Ejvgt~gnneþj gjvtg tgnva u |iajh

115

7*4* Cåvhfh~ fg ~hiwneþj

14<

Cåvhfh \iajh fg Najvh

14<

Cåvhfh \iajh Nhtvajvg

141

 

Tgkia~ fg _e~edeiefaf

141

7*>* Ejvgt~gnneþj fg fh~ |iajh~

144

7*6* Cåvhfh~ fg ~hiwneþj

144

Cåvhfh fg |iajh~ nhtvajvg~

144

Cåvhfh fg |iajh~ eiecevafh~

14>

7*7* \thdigca~

146

 

DEDIEHKTAOÃA

 

AJGPH^

7

 \TGOANEH

  Gi Vgpvh fg Kghcgvtãa Fg~nte|vea ~g |tg~gjva) nhch wja cajgta fg nhjvtedwet nhj gi  nhjhnecegjvh u a|tgjfe}amg fgi ajáie~e~ vtefecgj~ehjai) qwg g~ wj tgqwe~evh |ata gi fe~gñh  |ata ih~ g~vwfeajvg~ fg ih~ Ej~vevwvh~ Vgnjhiþkenh~ ^w|gtehtg~) Wjegt~evateh~ u fgi jegi   ~gnwjfateh*   Gj g~va |tecgta gfeneþj ~g nwdtgj vhfh~ ih~ a~|gnvh~ jgng~ateh~ qwg aj fg~fg gi  nhjhnecegjvh fg ih~ |tejne|eh~ owjfacgjvaig~ fg ia a|ienaneþj fg ia~ |thugnnehjg~ a ia Kghcgvtãa Fg~nte|vea) `a~va gi nhjhnecegjvh fg ih~ cåvhfh~ fg ejvgt~gnneþj u hvth~) qwg   |gtcevetáj cá~ afgiajvg) ejvthfwnet a ih~ g~vwfeajvg~ ai nhjhnecegjvh fgi fe~gñh ejfw~vteai*  \ht ta}hjg~ fg nhchfefaf) g~vg vgpvh g~vá htkaje}afh gj nejnh na|ãvwih~* Gi N  A\ÃVWIH  E) fgjhcejafh GI \WJVH vtava ~hdtg A|ienaneþj fg ih~ |tejne|eh~ fg ia |thugnneþj

htvhkhjai a ia kghcgvtãa fg~nte|vea= Fg|wtafh fgi |wjvh* Ktaoenaneþj fg wj |wjvh |ht  nhhtfgjafa~* \h~enehjg~ tgiavea~ fg fh~ |wjvh~* \h~enehjg~ ~wng~ea~ fg wj |wjvh h fg  ~þiefh~* Tgkia~ fg e~edeiefaf*  Gi NA\ÃVWIH EE fgjhcejafh IA TGNVA vtava* \h~enehjg~ |atvenwiatg~ fg wja tgnva= `h# te}hjvai) othjvai) fg |gtoei) gtvenai) jhtcai u htvh|gtoãi) \h~enehjg~ tgiavea~ gjvtg fh~ tgnva~= Tgnva~ qwg ~g ntw}aj) |ataigia~) u |gt|gjfenwiatg~* Htegjvaneþj* _gtfafgta cakjevwf* Cåvhfh~= nhj e~va awpeieat u feogtgjnea fg nhva~* \gjfegjvg* _e~va fg |wjva fg ia tgnva**  Gi  NA\ÃVWIH EEE   ) g~vwfea GI \IAJH g~vwfea= \h~enehjg~ |atvenwiatg~ fg wj |iajh= `hte}hjvai) othjvai) fg |gtoei) jhtcai) gtvenai u htvh|gtoei*   Tgnva~ jhvadig~ gj gi |iajh* Htegjvaneþj* _e~va fg najvh u gtfafgta cakjevwf fgi |iajh* Tgnva fg cápeca |gjfegjvg*  \gjfegjvg fgi |iajh*  Gi  NA\ÃVWIH E_) g~vwfea \ataigie~ch u \gt|gjfenwiatefaf= Nhjfenehjg~ fg |ataigie~ch  u |gt|gjfenwiatefaf gjvtg tgnva~ u |iajh~* \ht wj |wjvh vta}at wj |iajh |gt|gjfenwiat a ?

wja tgnva fafa* \iajh cgfeavte}* \ht wj |wjvh vta}at wj |iajh |gt|gjfenwiat a wj |iajh fafh u |ataigih a wja tgnva fafa* Fe~vajnea cá~ nhtva fg~fg wj |wjvh a wja tgnva*  Gi  NA\ÃVWIH _ = fgjhcejafh EJVGT^GNNEHJG^) g~vwfea Ejvgt~gnneþj gjvtg tgnva u   |iajh* Na~h~= Tgnva hdienwa u fg |gtoei* Cåvhfh~ fg ~hiwneþj= \iajh fg najvh) |iajh nhtvajvg* Tgkia~ fg e~edeiefaf* Ejvgt~gnnehjg~ fg fh~ |iajh~* Na~h kgjgtai u |atvenwiat* Cåvhfh~ fg ~hiwneþj= _e~va awpeieat u |iajh nhtvajvg* Tgkia~ fg e~edeiefaf*  Ih~ awvhtg~ aktafgngj a vhfh~ qwegjg~ `aj `gn`h |h~edig ia |wdienaneþj fg g~va hdta) a ia~ |gt~hja~ qwg `aj |th|htnehjafh auwfa) a~ã nhch nhj~gmh~ cwu aieh~h~* Ih~ nhcgjvateh~ a g~va gfeneþj ~gtáj degj tgnedefh~*  Ih~ Awvhtg~*

;

NA\ÃVWIH E= GI \WJVH 1*1*

GI \WJVH Gi |wjvh g~ wja efga) qwg fava fg~fg ia ajvekùgfaf u qwg ~g wveie}a |ata tg|tg~gjvat  wj hdmgvh wdenafh gj gi g~|aneh* Nhj g~vh |hfgch~ ~ec|ieoenat gi ajáie~e~ |ata tg~higt nwaiqwegt ~evwaneþj oã~ena u kghcåvtena*

1*4*

\THUGNNEÞJ FG WJ \WJVH Wj ~e~vgca fg |thugnneþj g~ wj ~e~vgca  |ht cgfeh fgi nwai |wgfg ~gt fgoejefa ia   |thugnneþj fg wj hdmgvh ~hdtg wja ~w|gtoeneg*

Gj

vhfh

~e~vgca

fg

 |thugnneþj ejvgtegjgj nwavth gigcgjvh~ fgjhcejafh~= oek*1*1=

a% Hdmgvh* G~ gi hdmgvh qwg ~g fg~ga tg|tg~gjvat* \wgfg ~gt wj |wjvh) tgnva) |iajh) ~w|gtoeneg) ~þiefh) gvn2 gj oej nwaiqwegt gigcgjvh kghcåvtenh w hdmgvh gj ~ã*

D1

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A

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d% \wjvh fg hd~gtaneþj* \wjvh fg~fg gi nwai ~g hd~gta gi hdmgvh qwg ~g qwegtg tg|tg~gjvat* G~ wj |wjvh nwaiqwegta fgi g~|aneh*

n% ^w|gtoeneg fg |thugnneþj* G~ ia ~w|gtoeneg ~hdtg ia nwai ~g |thugnvatá gi hdmgvh* Kgjgtaicgjvg g~ wj |iajh2 awjqwg vacdeåj |wgfg ~gt wja ~w|gtoeneg g~oåtena) neiãjftena) nþjena) gvn*

f% Tauh~ \thugnvajvg~* ^hj tgnva~ ecakejatea~ qwg wjgj ih~ |wjvh~ fgi hdmgvh nhj gi |wjvh fg hd~gtaneþj* Ia |thugnneþj (\+% fg nwaiqwegt |wjvh (\% fgi hdmgvh ~g hdvegjg ejvgtng|vajfh ~w  |thugnvajvg nhj gi |iajh fg |thugnneþj* 0

IH^ ^E^VGCA^ FG \THUGNNEÞJ CÁ^ W^AFH^ ^HJ= a% \thugnneþj neiãjftena* ^g hdvegjg

nwajfh

gi

|wjvh

fg

hd~gtaneþj ~g gjnwgjvta a wja fe~vajnea vaj ktajfg fgi hdmgvh) qwg |gtceva nhj~efgtat qwg ia~  |thugnvajvg~ ~hj |ataigia~ ai ejvgtng|vat~g nhj gi |iajh fg |thugnneþj (oek*1*4%* Ih~  |tejne|aig~ ve|h~ fg |thugnneþj neiãjftena ~hj=

1% \thugnneþj htvhkhjai (Chjkg%* Vacdeåj fgjhcejafa |thugnneþj htvhktáoena* ^g hdvegjg nwajfh ia~ |thugnvajvg~ ~hj |gt|gjfenwiatg~ ai |iajh fg |thugnneþj* Ia   |thugnneþj htvhkhjai g~ cwu wveie}afa gj gi fe~gñh fg |eg}a~ cgnájena~ u caqwejatea~ oek* a* Ih~ |tejne|aig~ ve|h~ fg |thugnneþj htvhkhjai ~hj=

e% \thugnneþj gj e~va~ cýive|ig~* Nafa

e~va

g~

wja

|thugnneþj

htvhktáoena* \ata hdvgjgt wja e~va ~g nhihna

gi

|iajh

fg

|thugnneþj

  |tgogtgjvgcgjvg |ataigih a wja fg ia~ nata~ |tejne|aig~ fgi hdmgvh oek*1*>* Ih~ hdmgvh~ ~g tg|tg~gjvaj kgjgtaicgjvg gj vtg~ e~va~ htvhktáoena~* Ih~ cåvhfh~ wveie}afh~ |ata fgvgtcejat g~va~ e~va~ ~hj=

A% \thugnneþj gj gi Vgtngt Nwaftajvg* W~afh gj ih~ G~vafh~ Wjefh~ u Najafá* oek*1*6*

D% \thugnneþj gj gi \tecgt Nwaftajvg (|tecgt hnvajvg%* W~afh gj vhfh gi cwjfh) gpng|vh gj ih~ G~vafh~ Wjefh~ u Najafá* oek*1*7*

5

\thugnnehjg~ fg |wjvh~ gj oekwta g~|aneai u fg~nte|vea Nwaftajvg EE

Nwaftajvg E

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Nwaftajvg EEE ό4

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Nwaftajvg E_

(Gi |wjvh < g~vá gj gi nwaftajvg E%

Vai nhch g~vá tg|tg~gjvafh gi |wjvh gj ia oekwta g~|aneai u fg~nte|vea |wgfg gpvtagt~g qwg |ata |a~at fg ia oekwta g~|aneai a wja fg~nte|vea fgdgtá adavet~g gi  |iajh

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~hdtg gi |iajh

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) vgjegjfh nhch gmg fg keth a ia V (iãjga fg vegtta h fg

 |iegkwg%* Fg g~a cajgta qwgfatáj aiejgafh~ < 1 „ < „ < 4) nhch ~g |wgfg a|tgneat gj ia oekwta fg~nte|vea* Fgdg fg~vanat~g qwg gj ia tg|tg~gjvaneþj g~|aneai a|atgng gi hdmgvh (<%* Gj ia tg|tg~gjvaneþj fg~nte|vea ~hiacgjvg ~g vtadama nhj gi |iajh  |thugnnehjg~ <1 u <4 fgi |wjvh*

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ua adavefh) u ia~

Nwaftajvg EE

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Gi |wjvh (<% g~vá gj gi nwaftajvg EE ό4

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\thugnneþj fgi |wjvh (<% wdenafh gj gi >½ nwaftajvg

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\thugnneþj fgi |wjvh (<% wdenafh gj gi 6½ nwaftajvg 11

Gj g~vg ýivech na~h) gi aigmacegjvh (fe~vajnea fg < ai |iajh gtvenai% vegjg ia ce~ca ihjkevwf qwg ia nhva (fe~vajnea fg < ai |iajh `hte}hjvai) |ht ih vajvh) gj ia tg|tg~gjvaneþj gj OEKWTA FG^NTE\VE_A <1 a|atgng ~w|gt|wg~vh nhj < 4*

1*>*

\IAJH^ \TEJNE\AIG^ FG \THUGNNEÞJ Vtg~ ~hj ih~ |iajh~ |tejne|aig~ fg |thugnneþj= ․

Othjvai

․

@hte}hjvai u

․

Fg |gtoei

\thugnneþj fg wj |wjvh ~hdtg ih~ vtg~ \iajh~ \tejne|aig~ fg \thugnneþj

14

AO= \thugnneþj fgi |wjvh A ~hdtg gi |iajh fg |thugnneþj Othjvai A@= \thugnneþj fgi |wjvh A ~hdtg gi |iajh fg |thugnneþj @hte}hjvai A\= \thugnneþj fgi |wjvh A ~hdtg gi |iajh fg |thugnneþj Fg \gtoei

1*6*

NHVA „ AIGMACEGJVH „ A\ATVACEGJVH \ata gi g~vwfeh fg ia kghcgvtãa fg~nte|vea) g~ jgng~ateh vgjgt degj fgoejefh ia~ fe~vajnea~ fgi hdmgvh a ih~ |iajh~ |tejne|aig~ fg |thugnneþj* G~vh~ ~hj=

Nhva= Fe~vajnea fgi hdmgvh ai |iajh fg |thugnneþj @hte}hjvai (A „ A @%* Aigmacegjvh= Fe~vajnea fgi hdmgvh ai |iajh fg |thugnneþj Othjvai (A „ A O%* A|atvacegjvh= Fe~vajnea fgi hdmgvh ai |iajh fg |thugnneþj Fg \gtoei (A „ A \%*

1*7*

FG\WTAFH FG WJ \WJVH Ai adavet gi \iajh fg |thugnneþj @hte}hjvai u gi \iajh fg |thugnneþj fg \gtoei) vgjegjfh nhch gmg~ fg keth a ia~ iãjga~ fg |iegkwg nhttg~|hjfegjvg~) ~g hdvegjg wj ~hih |iajh* A g~vg |iajh ~g ig fgjhceja Fg|wtafh fg wj \wjvh*

1>

1*?*

KTAOENANEÞJ FG WJ \WJVH \HT NHHTFGJAFA^ Gj gi gjwjneafh fg ih~ |thdigca~) a gng~ ~g ejfena ia wdenaneþj fg ia~   |thugnnehjg~ fg wj |wjvh |ht cgfeh fg nhhtfgjafa~* ^g |wgfg hdvgjgt ia  |thugnneþj @hte}hjvai u Othjvai fg fen`h |wjvh a |atvet fg fen`a~ nhhtfgjafa~* Gmgc|ih= Tgaie}at gi fg|wtafh fgi |wjvh A(a)d)n)%* Gi |tecgt nhc|hjgjvg fg ia nhhtfgjafa) jh~ ýjena ia iãjga fg tgogtgjnea gjvtg ia~  |thugnnehjg~ `hte}hjvai u othjvai fgi |wjvh A* Gi ~gkwjfh nhc|hjgjvg) jh~ ýjena ia |thugnneþj othjvai fgi |wjvh A) u gi vgtngt  nhc|hjgjvg jh~ wdena ia |thugnneþj `hte}hjvai fgi |wjvh A*

16

1*;*

\H^ENEHJG^ TGIAVE_A^ FG FH^ \WJVH^ Ia~ |h~enehjg~ tgiavea~ fg fh~ |wjvh~ gjvtg ~ã) ~g |wgfg hd~gtat nhihnajfh wja  dtýmwia gj gi |iajh fg |thugnneþj `hte}hjvai*

17

Tg~|gnvh fg ia oekwta) ~g feng= D g~vá a ia fgtgn`a fg A) D g~vá atteda fg A) D g~vá fgiajvg fg A) u ~þih gj gi |iajh @hte}hjvai= D g~vá `anea gi ^wt fg A) D g~vá `anea gi G~vg fg A) h A g~vá `anea gi Jhtvg fg D) A g~vá `anea gi Hg~vg fg D*

1*0*

_E^VA^ AWPEIEATG^ Nwajfh ~g vegjg |thugnnehjg~ qwg jh ~g cwg~vtaj afgnwafacgjvg gj ih~ vtg~ |iajh~  |tejne|aig~ fg |thugnneþj) ~g |wgfgj vta}at |iajh~ awpeieatg~* Gpe~vgj fh~ ve|h~ fg \iajh~ Awpeieatg~ fg \thugnneþj= ․

\iajh~ Awpeieatg~ \tecateh~ u

․

\iajh~ Awpeieatg~ ^gnwjfateh~*

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\IAJH^ AWPEIEATG^ \TECATEH^ ^hj ih~ qwg ~g vta}aj |gt|gjfenwiatg~ a wj |iajh fg |thugnneþj |tejne|ai*

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TGKIA^ FG _E^EDEIEFAF* Nwajfh ~g vta}a ia |thugnneþj fg ~þiefh~) aj a gjnhjvtat~g ate~va~ hnwiva~) fen`a~ ate~va~ ~g vta}aj nhj iãjga fg ~gkcgjvh~* @au qwg vgjgt gj nwgjva ia~ ~ekwegjvg~ tgkia~ fg e~edeiefaf* ․

Gi nhjvhtjh fg ia |thugnneþj) ~egc|tg g~ e~edig*

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Nwajfh wja ate~va qwg nhjnwttg a wj åtveng g~ eje~edig) ia~ fgcá~ ate~va~ qwg nhjnwttgj a fen`h åtveng) vacdeåj ~gtáj eje~edig~*

1*1<* \THDIGCA^

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