libro-triangulos

WJTEBY GRJNJKE M+ 'MR_MJKF%

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SYFILBNJ? K³: K³:: :

2 +

   , mjlmrlb w+ ]b`ÿk bl `yjdemf 

SYFILBNJ? K³13

Bk lj de`ryj, mjlmrlb w J% 43³ I% 13³ M% =3³ A% 93³ B% :3³

J% 53³ A% :33³

9 +

I% ::3³ B% 43³

M % :93³

4 +

SYFILBNJ? K³2 K³23 3

Bk rk qyejk`rlf e}ý}mblb} JIM 'JI>IM% , bk JM u IM }b riemjk lf} srkqf N u K yb}sbmqetjnbkqb , qjl zrb IN>IK u     ³+ Mjlmrlb lj    +


Abl `yjdemf, mjlmrlb w+

J% 93³ A% 92³ 0 +

I% :3³ B% :2³

M% =3³


]b`ÿk bl `yjdemf, lj nbaeaj abl êk`rlf zrb  u  b} 43³+ Mjlmrlb w+ dfynjk  J% 92³ A% 23³

=+

I % =3³ B% 93³

M% =2³

SYFILBNJ? K³= K³== =

1+

Bk bl `yjdemf, mjlmrlb w+ J% 93³ I% :2³ M% =3³ A% 99³=3´ B% 90³

J% 93³ A% =2³

I% 92³ B% 13³

M% =3³


Bk rk qyeêk`rlf JIM }b riemjk lf} srkqf} S , Z u Y , bk

JI ,

IM u MJ

yb}sbmqetjnbkqb+ ]e

SZ>ZI, YZ > ZM , JS > SY u n SYZ > 73³+ Mjlmrlb n IZS+ J% 93³ A% :3³

I % :2³ B% :7³

±Qfaj ne dî}emj kf sj}j ab rkj `bfnbqyîj²¨+²Ybkå Ab}mjyqb}²

M% 13³

Lj SYB Qr jmjabnej¨

@BFNBQYEJ SLJKJ

7 +

   A% 


J%

Bk rk qyeêk`rlf JIM 'JI > IM% bk JM IM }b riemjk lf} srkqf} A B yb}sbmqetjnbkqb, qjl zrb? IA > AB n AIM.n IJM > =3¿+ Mjlmrljy n BAM+ £

I% 



M%  

 B% 

ª

ª

J% 12¿ A% 93¿ 5+

I% =2¿

K³:93 3 :9 + SYFILBNJ? K³:9

Bk rk qyejk`rlf JIM , }b qyjjk lj} mbtejkj} ekqbyefyb} JS u IZ , ab nfaf zrb JZ>IZ , JS>SM,     ³ u        + Mjlmrlb lj    +

M% 92¿ B% =3¿


J% 93³ A% 13³

Bk bl `yjdemf nf}qyjaf mjlmrljy bl tjlfy ab w+

I% 92³ B% 12³

M% =3³

K³:=3 3 :=+ SYFILBNJ? K³:=

      Bk lf} ljaf}   ab rk qyejk`rlf JIM }b riemjk lf} srkqf} S , Z u Y yb}sbmqetjnbkqb qjl zrb8 JY>YS u MY>YZ u     ³, mjlmrlb lj nbaeaj abl êk`rlf abqbynekjaf sfy lj} ie}bmqyemb} ie}bmqyemb} ab lf} êk `rlf} JSZ u SZM+ J% 23³ A% 43³ J% 12³ A% =2³

I % 22³ B% :2³

M % 93³

Bk lj de`ryj Mjlmrljy w

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

I% 1 3³

M% 12³ B% 22³

K³::3 3 ::+ SYFILBNJ? K³::

Bk bl `yjdemf



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&&

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B

8

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M,

¢

M >



B  R * :=3y ¢

J% 23³ A% 4 3³

M% 42³

K³:13 3 :1+ SYFILBNJ? K³:1

K³:33 3 :3+ SYFILBNJ? K³:3

Mjlmrlb w, }e

I% 03³ B% 12³

       + Mjlmrlb w+

% 93³ % =3³ M% :2³ % 13³ B% =2³

K³:23 3 :2 + SYFILBNJ? K³:2

Bk bl `yjdemf, JI>JM>SM u IS>SZ+ Gjlljy w+ J% =3³ I% 23³ M% 43³ A% 03³ B% 73³

±Sybme}f b} bkmfkqyjy lf ekdekeqjnbkqb `yjkab bk lf ekdekeqjnbkqb sbzrbúf, sjyj }bkqey lj syb}bkmej ab AEF]²¨+mrmjkf AEF]²¨+mrmjkf

B+


@BFNBQYEJ SLJKJ

J% 53³ A% :93³

SYFILBNJ? K³:4 K³:43 3

Bk rk qyejk`rlf JIM }b qyjj lj} nbaejkj} IN u JS qjl zrb    ³ u    ³, mjlmrlb   + J% =0³ A% 02³

I% 43³ B% 23³

Bk rk qyejk`rlf JIM , }b riemj bl srkqf K bk lj  , qjl zrb MK>JI u syflfk`jmeýk abl ljaf   u       ³, }e lj} nbaejqyemb} ab  }b ekqby}bmjk bk S , mjlmrlb lj nbaeaj abl êk`rlf MKS+ I% =7³ B% 12³

M% ::3³



M% 2=³


J% 22³ A% 23³

I% :33³ B% :=3³

J% :33³ I% 53³ M% 52³ A% 73³ B% ::3³ SYFILBNJ? K³9: K³9:3 3

Bk bl `yjdemf, SM>MZ+ Mjlmrlb w(u+

M% 1:³


  ]b`ÿk bl `yjdemf,  u  }fk ie}bmqyemb}  ekqbyefyb} u  b} jlqryj abl qyejk`rlf JIM+ ]e       + Mjlmrlb YQ+ J% :31³ A% :=7³

I% :37³ B% :29³

M% :17³


Bk bl `yjdemf, JI>IA u IM>IB , mjlmrlb+

J% A% 

I%    B% 

J% :3³ I% 93³ M% 13³

M%

A% 12³ B% 23³


]b`ÿk bl    

`yjdemf

    ³+

Mjlmrlb? SYFILBNJ? K9= K9=3 3

Bk rk qyejk`rlf JIM , JI>1, IM>7 u JM>:3 sfy  u   sbysbkaemrljyb} j bl tåyqemb M }b qyjjk  lj} ie}bmqyemb} ekqbyefy ab J u bwqbyefy ab I , yb}sbmqetjnbkqb+ Gjllb BD+ J% 3 ,2

I% 3 ,43

M% 3,02



@BFNBQYEJ SLJKJ

A% 3 ,73

B%:

A% 2=³ 9 §

B% 02³ 9 §



Mjlmrljy w8 § JIM? bzrelêqbyf8 AB > = , MA > 9 = u JB > 1

Bk rk qyejk`rlf JIM , JI>1, IM>7 u JM>:3 sfy  u   sbysbkaemrljyb} j bl tåyqemb M }b qyjjk  lj} ie}bmqyemb} ekqbyefy ab J u bwqbyefy ab I , yb}sbmqetjnbkqb+ Gjllb BD+

J% :3³ I% :2³ M% :4³ A% :9³ B% :1³

J% 3 ,2 A% 3 ,73

I% 3,43 B%:

M% 3,02

K³953 3 :4 + SYFILBNJ? K³95

Bk lj de`ryj nf}qyjaj , mjlmrljy IZ+ ]e BS>1+ SYFILBNJ? K³92 K³923 3

Bk bl `yjdemf, JI>1'SZ%+ Mjlmrlb w+

J%: A% 92 J% =3³ A% 2=³

I% =0³ B% 97³

M% :1³

I%9 B%=

M%: ,2

SYFILBNJ? K³=3 K³=33 3

Bk lj de`ryj8 JI>IM , mjlmrlb bl tjlfy ab w+ SYFILBNJ? K³94 K³943 3

Bk bl `yjdemf nf}qyjaf mjlmrlb IB }e lj ae}qjkmej  b} :3 u     ³+ ab B j 

J% 94,2³ A% 0 ,2³

I% :7,2³ B% :7

M% 12³

SYFILBNJ? K³=: K³=:3 3

J%4 A% 

I%2 B%7 

   u JI>MA+ Mjlmrlb bl ]b`ÿk bl `yjdemf,  nbkfy tjlfy bkqbyf ab ½ ±+

M%7


J% 92³ Bk rk qyejk`rlf ybmqêk`rlf JIM, ybmqf bk I, bk lj ybèýk bwqbyefy u ybljqetf j  }b riemj S , qjl zrb    ³, SJ>JM u 2'JI%>='SI%+ Mjlmrlb lj   + J% =0³

I% 45³ 9 §

M% =3³

I% 97³ M% =2³ A% =0³ B% =5³



@BFNBQYEJ SLJKJ



Bk rk qyejk`rlf JIM , }b riemj S bk , Y bk  u Z bk  + ]e JI>IS>SZ>ZY>YM u   b} bl njufy tjlfy bkqbyf sjy , mjlmrlb lj 

Bk rk qyejk`rlf JIM }b qyjj lj nbaejkj JN u       +]e JS>SN u lj jlqryj  ,      ³, mjlmrlb lj    

 J% =0³ 9 A% :7³ 

J%1³ A%=³

I%9³ B%2³

M%4³

I% 2=³ 9 B% 12³ 9

M% :4³






]e JI>7 u DZ>28 mjlmrlb IM

SYFILBNJ? K³== K³==3 3

Bk bl `yjdemf MA>:7 , mjlmrlb lj }rnj ab lf} tjlfyb} bkqbyf} zrb srbab qfnjy JI+

J%7 A% :2

J% :5 A% 99

I% 93 B% 9=

M% 9:

I% :3 B% :9

M% :4


Bk lj de`ryj, mjlmrlb w SYFILBNJ? K³=1 K³=13 3

Bk bl `yjdemf, JI>IM, mjlmrlb w+

J% 03³ I% 23³ J% 12³ A% 02³

M% 73³ A% :33³

I% =3³ B% :2³

M% 43³


B% 42³

JI>:3 u IS>98 mjlmrlb+ SYFILBNJ? K³=2 K³=23 3

 Bk rk qyejk`rlf JIM , lj nbaejqye ab   ekqby}bmj j  , bk Z, qjl zrb JI>9'ZM% u     ³ + Mjlmrlb lj     J% =0³ A% =0³ 9 ¥

I% 2=³ 9 B% =3³ ¥

M% :4³

J% =3³

I% :2³

M% :7³=3´



@BFNBQYEJ SLJKJ

A% 94³=3´

B% 99³=3´ J% 43³ A% 27³


 Bk lj de`ryj 

     u    gjlljy w+

I% 49³ B% 42³

M% 41³



J% 03³ A% :33³

I% 73³ B% ::3³

M% 53³

J%I%M% A%B%

SYFILBNJ? K³1: K³1:3 3

Bk rk qyejk`rlf ybmqêk`rlf JIM ybmqf bk I }b qyjj lj mbtejkj ekqbyefy , qjl zrb JK>='KI% u        mjlmrlb   + J% =3³ A% 12³

I% 94 ,2³ B% =0³

J%= I% 9 ,2

Bk rk qyejk`rlf ybmqêk`rlf JIM , ybmqf bk I , }b  sbysbkaemrljy qyjj lj ie}bmqye ekqbyefy IA u   j lj syflfk`jmeýk ab  'G bk aemgj syflfk`jmeýk%8 }e IA>GA, mjlmrlb lj    +



I% =0³ B% =0³ 9

M% = ,2 A%1 B% 1 ,2

M% 2=³




SYFILBNJ? K³1= K³1=03 03

]b qebkb b} qyejk`rlf JIM , JI>:3 u IM>93, }b qyjj lj ie}bmqye ekqbyefy IA , }e    ³, mjlmrlb lj ae}qjkmej abl srkqf nbaef ab  j + J%= A%1

]e JM>2 u BD>=+ Mjlmrlb JA

M% 2=³


J% 99³=3 A% 2=³ 9


I%9 B%:

Gjlljy bl tjlfy ab w , }e JB>IM u IB>BM+

M%2


Bk rk qyejk`rlf JIM , }b riemj bl srkqf A  , qjl zrb     bwqbyefy ybljqetf jl ljaf  ³ u     ³, mjlmrlb lj    }e jabnê} }b }jib zrb JI>IM>MA+

J% :7³ A% :9³

I% 93³ B% :3³

M% :2³


@B

BQ

Lj SYB Qr

SL

SYFILB SYFILB

SYFILB SYFILB

K³

Bk rk qy ejk`rl jk`rlf e}f}mbl }f}mblb} JIM ab ij}b JM, JM, }b qyjj lj mbte mbtejkj ekqby efy JA qjl zrb IA>JM A>JM u lj mrlb lj       ³+ Mjlmrl

K³

]b rie rie rk srkq srkq S bk bl bl ekqby e y ab rk qy e k`rl k`rl JIM q l zrb JS>JI JS>JI>IM, n JMS>=3¿ u n M JS>:3¿+ M l rl y l y l n IJS

J%09³ J%09 ³ A%44³ A%44³

J% 13¿ A% 93¿

SYFILB SYFILB

%

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jmjabnej¨

%

%

3



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:

9

:

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I% =3¿ B% =4¿

SYFILB SYFILB

ekqby e yb} IN u MK , ab nfaf zrb JK>KM, JK>KM, IN> IN>IM u I K>JN, K>JN, mjl mjlmrl mrlb lj nIJM+ 1

J% 2=³ 2=³ A% =3³ =3³

I% =0³ =0³ B% 12³ 12³

SYFILB SYFILB

M% 43³ 43³

K³

2

I% :9y B% =3y

J% :2³ :2³ A% 12³ 12³

I% 92³ 92³ B% 02³ 02³

SYFILB SYFILB

K³

J% 94³ 94³=3 A% =:³ =:³12³ 12³

M% :2y

I% :7³ :7³=3´ B% =4³ =4³12´

SYFILB SYFILB SYFILB SYFILB

K³

4

7

nJIM>43 IM>43y, Mjlmrl mrlb nIJM+ J% =3y A% 43y

I% 12y B% 53y

K³

J% :93³ :93³ A% :13³ :13³

M% 02y

I% :33³ :33³ B% :=2³ :=2³

SYFILB SYFILB SYFILB SYFILB

M% 95³ 95³=3´

Bk rk qy ejk`rl jk`rlf JIM, }b rie riemj rk srkq srkqf ekqby efy S qjl zrb lj         b} ie}bmq          }bmqy e ab   + Mjlmrl mrlb lj    +

Bk rk qy e k`rl k`rlf JIM JIM }b rie riemj bl bl srkq srkqf S JI u }b qyjj SZ I I 'Z bk I %+ ]e JS>SZ>ZM JS>SZ>ZM u 0

M% =3³ =3³

Bk rk qy ejk`rl jk`rlf JIM ybmq ybmqf bk I, }b qyjj lj mbte mbtejkj ekqby efy JB, JB, qjl zrb lj       mrlb w+    + Mjlmrl

Bk bl ekqby efy ab rk qy e k`rl k`rlf JIM }b rie riemj bl srkq srkqf S, qjl zrb nIJS>03y, nMJS>:3y, nJIS>=3y u nSIM>93 IM>93y+ Mjlmrl mrljy nSMJ+ J% :3y A% 93y

K³

Bk rk qy ejk`rl jk`rlf JIM }b rie riemj rk srkq srkqf S bwq bwqby efy ybl ybljqetf qetf jl ljaf JM JM+ ]e?     mjlmrl mrlb lj nbae nbaeaj abl abl ³     ³ mjl êk`rl êk`rlf JI JIS }e }ebkaf jabnê}? jabnê}? IM> IM>MS>SJ+

JIM }b qyjjk lj} mbte mbtejkj}

1

=

M%47³ %47³

M% 23¿

K³

Bk rk qy e k`rl k`rl

I%03³ %03³ B%49³ B%49³

M% :23³ :23³

K³

K³

Bk rk qy e k`rl k`rlf ybmq ybmq k`rl k`rlf JIM ybmq ybmqf bk I, }b qebkb qebkb zrb nIJM>:2y+ Bk JI }b rie riemj bl srkq srkqf S zrb ae ae}qj ab JM bl afil afilb ab lf zrb ae ae}qj I ab S M + Lj nbae nba e aj abl ab l êk`rl êk`r l f SMI S MI b}? b} ? M

Bk rk qy ejk`rl jk`rlf ybmq ybmqêk`rl êk`rlf JIM, ybmq ybmqf bk I8 bk  }b rie riemj lf} srkq srkqf} N u K qjl zrb             , lrb`f }b qyjj  sbysbkae  'G bk   % ab nfaf zrb sbysbkaemrl mrljy j  NM>9'G NM>9'GM M%+ Mjlmrl mrlb lj   +

J% :2y A% =3y

J% =4³ =4³ A% 09³ 09³

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I% 93y B% 13

M% 92y

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I% :7³ :7³ B% =:³ =:³

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M% 21³ 21³

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SYFILB SYFILBNJ? NJ? K³


Bk rk qy eêk`rl êk`rlf ybmq ybmqêk`rl êk`rlf JIM ybmq ybmqf bk I }b qyjj lj ie}bmq }bmqy e ekqby efy JS u bk JM }b rie riemj bl bl srkq srkqf Z qjl zrb SM SM > MZ+ Mjlmrl mrljy n JSZ+

Ekq Ekqby efynbkq fynbkqb j rk qy ejk`rl jk`rlf e}ý}mbl }ý}mblb} JIM afkab lj      ³ }b rie riemj bl b l srkq srkqf N8 qjl zrb     ³ u      ³, mjl mjlmrl mrlb lj    J% 99³ 99³ A% 93³ 93³

jmjabnej¨

I% =3³ =3³ B% :7³ :7³

J% 12³ 12³ A% 13³ 13³

M% :3³ :3³

I% 43³ 43³ B% 2=³ 2=³

M% =³ =³


Bk rk qy ejk`rl jk`rlf JIM,     ³ u     ³ , lrb`f }b rie riemj bl bl srkq srkqf B ab lj  nbae nbaejqy e ab  qjl zrb     ³ mjl mjlmrl mrlb lj   


Bk rk qy ejk`rl jk`rlf ybmq ybmqêk`rl êk`rlf JIM, ybmq ybmqf bk I, }b qyjj lj mbte mbtejkj ekqby efy JN ab nfaf zrb          u lj ae}qjkme jkmej ab ab I j   b} lj mrjy qj sjy qb ab lj lfkèq fkèqra ra ab NM8 mjl mjlmrl mrlb lj     J% 93³ 93³ A% :2³ :2³

I% =3³ =3³ B% :7³ :7³=3

J% 73³ 73³ A% :93³ :93³

I % 53³ 53³ B% ::2³ ::2³

M% ::3³ ::3³

M% 92³ 92³


Bk rk qy ejk`rl jk`rlf e}ý}mbl }ý}mblb} JIM JIM      ³, ekqby efynbkq fynbkqb }b rie riemj bl srkq srkqf S qjl zrb JI>JS>IM, >JS>IM,      ³ u     mjlmrl mrlb lj    + ³, mjl J% 93³ 93³ A% =0³ =0³

I% 13³ 13³ B% 19³ 19³

M% 12³ 12³


Bk rk qy ejk`rl jk`rlf ybmq ybmqêk`rl êk`rlf JIM, ybmq ybmqf bk I,  bwq bwqby efynbkq fynbkqb u ybl ybljqetf qetf j  }b rie riemj bl bl srkq srkqf S qjl zrb lj      ³,       JI b} e`rjl `rjl jl afil afilb ab lj ae ae}qjkme jkmej ab S   u JI  j  +mjl +mjlmrl mrlb    + J% 93³ 93³ A% :4³ :4³

I% 92³ 92³ B% :7³ :7³

M% :2³ :2³

½]e rk aîj qb aj lj lfmryj ab llfyjy, llênjnb kf qb syfnbqf gjmbyqb }fkybîy8 sbyf srbaf e` f+ llfyjy mfkq e` f+ ]e jl `ÿk aîj yb}rbltb} dr ` jyqb, jyqb, kf arab} bk lljnjynb kf qb syfnbqf sba eyqb zrb qb e` f+ zrbab}8 sbyf srbaf dr ` jynb jynb mfkq e` f+ ]e rk aîj qb aj lj lfmryj ab kf gjiljy mfk kja eb, llênjnb bk b}b nfnbkqf, syfnbqf zrbajynb iebk zrebq eqf8 Sbyf }e rk aîj nb lljnj} u kf yb}sfkaf, tbk mfyy ebkaf j ne bkmrbkqyf, qjl tb uf qb b}qb kbmb}eqjkaf±¨+wjte wnmw wnmw


Bk lj ybè ybèýk ekqby efy ab rk qy ejk`rl jk`rlf bzrel bzrelêêqbyf JIM }b rie riemj bl srkq srkqf S, qjl zrb     ³8 lrb`f }b qyjj bwq bwqby efynbkq fynbkqb jl qy ejk`rl jk`rlf JSM JSM lf} qy eêk`rl êk`rlf} bzrel bzrelêêqbyf} JSB u SM SM + Mjlmrl mrlb lj   + S

J% 53³ 53³ A% :23³ :23³

y ±S y C

I% :33³ :33³ B% :73³ :73³

M% :93³ :93³

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libro-triangulos

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