angle trisection - final english

13
7/29/2019 Angle Trisection - FINAL English http://slidepdf.com/reader/full/angle-trisection-final-english 1/13 Koleh Trmshjtmbo Kt tfh pklh :4< (`hebw sjfhgk) , \kotzhe ksshrts tfkt trmshjtmbo ms ko uosbevhd prb`ehg `hjkush ms `kshd tb tfh hquktmbo (x:!:"4x#$"4k%&) wfhrh ju`mj hquktmbos krh obt trmshjthd `' ruehr kod dmvmdhr klh :4< * jko sfbw 'bu tfkt tfh k`bvh ju`mj hquktmbo ms dhrmvhd +rbg k rst lrkdh hquktmbo k  x  x  jbs 0 + 7  wmtf twb (-) uo.obwo +kjtbrs (x,k) kod wh jko sbevh tfh prb`ehg m+ wh wmee kdd boh gbrh hquktmbo (rst lrkdh wmtf - uo.obwo +kjtbrs) wfmjf jko fkvh ghkomol tb bur lhbghtrmj sjfhgh  /fho wh ohhd k shjbod lrkdh hquktmbo (`hebw sjfhgh 1 klh :4;) klh :4; wfhrh ms prbvhd tfkt tfms hquktmbo ms sbevk`eh `' ruehr kod jbgpkss kod wfmjf * od ohkr tb tfh hod b+ tfh krtmjeh Tfh mdhk

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Page 1: Angle Trisection - FINAL English

7/29/2019 Angle Trisection - FINAL English

http://slidepdf.com/reader/full/angle-trisection-final-english 1/13

Koleh Trmshjtmbo

Kt tfh pklh :4< (`hebw sjfhgk) , \kotzhe ksshrts tfkt trmshjtmbo ms ko

uosbevhd prb`ehg `hjkush ms `kshd tb tfh hquktmbo (x:!:"4x#$"4k%&)

wfhrh ju`mj hquktmbos krh obt trmshjthd `' ruehr kod dmvmdhr

klh :4<

* jko sfbw 'bu tfkt tfh k`bvh ju`mj hquktmbo ms dhrmvhd +rbg k rst lrkdh

hquktmbo k x xd    jbs0+7  wmtf twb (-) uo.obwo +kjtbrs (x,k) kod wh jko sbevh

tfh prb`ehg m+ wh wmee kdd boh gbrh hquktmbo (rst lrkdh wmtf - uo.obwo

+kjtbrs) wfmjf jko fkvh ghkomol tb bur lhbghtrmj sjfhgh

 /fho wh ohhd k shjbod lrkdh hquktmbo (`hebw sjfhgh 1 klh :4;)

klh :4;

wfhrh ms prbvhd tfkt tfms hquktmbo ms sbevk`eh `' ruehr kod jbgpkss kod

wfmjf * od ohkr tb tfh hod b+ tfh krtmjeh

Tfh mdhk

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 /fh +kjt tfkt wh jko tk.h bur prbtrkjtbr kod tb dmvmdh ko' koleh tb tfrhh hqukes `ut

kelh`rkmjkee' tb fkvh ko uosbevhd hquktmbo jbghs klkmost wmtf tfh jbggbo shosh

 /fhbrhtmjkee' m+ ko koleh jko obt dmvmdhd wmtf tfh tfrhh tfho tfh trmshjthd koleh jko

obt hxmst 0br hxkgpeh m+ ko koleh b+ dhlrhhs jko obt dmvmdhd wmtf tfh tfrhh tfho

tfh koleh b+ tfrhh dhlrhhs jko obt hxmst b gko' kolehs jko obt hxmst tfhrh+brh tfhjbrrhspbodmol oug`hrs jko obt hxmst

Zhkrjfmol ibr tfh sbeutmbo

* tbb. k trmshjthd koleh kod +rbg tfh sjfhgh * trmhd tb od sbgh kelh`rkmjkee'

hquktmbos wfhrh tfh' wmee dhrmvhd gh tb tfh sbeutmbo b+ tfh prb`ehg

1:4

4:

0.40

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0

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 ∐+7 

  

   ∐+7+7   d  p x p x

 p

 x x

 p

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k x xk x xd 

8bth9 /fh k`bvh hquktmbo gk' `h dhrmvhd tfrbulf tfh whee!.obwo trmlbobghtrmj

rhektmbosfmp

⇐∐7⇐∐7   

  

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d kkk

1::   00::00

7+∐⇐∐7⇐  d  p x p x x xpdp

0modmol tfh rbbts9

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)(1 0;440;   x x x x x x   +∐7⇐7++

00

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⇐   ⇐7∐++⇐∐7∐∐∐   1::   00

00;

0

;

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$)   ⇐0

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+∐∐∐7   ()

044;0;;   1   x x x x x   7⇐7++

\h jko obt jbgh tb .obwo sbeutmbos `hjkush tfh' jbotkmo tfh uo.obwo eholtf 0 x

 /fh ohxt lbke ms tb trkosekth tfh sbeutmbos mo wfmjf kolehs jbrrhspbod sb tfkt wh jko

drkw k jbojeusmbo tfkt wmee fhep us mo sbevmol tfh prb`ehg

 (4) ⇐   0

0

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Kod `k`k`k   smojbsjbs.smo)smo(   ∐7∐  rhsuets9

   

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k p x  (>)

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() ⇐0

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(8bth9 \h jbrrhekth `bws ! sbeutmbos `kshd bo tfh trmkoleh∬

 HBS  wfhrh p

k

00smo

  ΘΙ 7 )

\h b`shrvh tfkt tfh krj∭

 x (`kshd bo bur sjfhgk) jbrrhspbods tb koleh k wmtf

momtmke koleh :k

 /fh krj;;

∭ x  jbrrhspbods tb tfh koleh b+ $-&!k sb tfh momtmke koleh ms : ($-&!k) %

:;&!:k kod

tfh krj;0

∭ x  jbrrhspbods tb tfh koleh ! ($-& # k) sb tfh momtmke koleh ms : (!) ($-& #

k) % ! (:;& : k)

 /fh sbeutmbos b+ bur ju`mj hquktmbo krh tfh $": b+ tfh strmols b+ tfh jbrrhspbodmol

kolehs :=,:;&!:=,!:;&!:=

5ksmol b+ bur sbeutmbo wh fkvh9

^ Wk xk x xd    jbs.0;.jbs0 0   +7+7  ($&)

)W;01jbs(0;.^);01jbs(0 ;;;;;;   k xk x xd    ∐+7∐+7  ($$)

)W);01(jbs^0;.()W;01(jbs^0 ;0;0;0   k xk x xd    +∐+7+∐+7  ($-)

Jbojeusmbos

*+ wh wmee ebb. jkrh+uee' wh fkvh ko hquktmbo rst lrkdh (   k x xd    jbs0+7 ) wmtf twb (-)

uo.obwo +kjtbrs Kjjbrdmol tb bur gktfs .obwehdlh, wh ohhd ko kobtfhr boh

smgmekr hquktmbo (tb `h mojeudhd tb tfh sjfhgh) sb tb jko sbevh tfh prb`ehg

* suppbsh tfkt tfh sthps b+ rmsmol tfh hquktmbo tb k ju`mj hquktmbo whrh obt rmlft, +br

hxkgpeh9

( ) :: `z kx y`z kx y   +7⇐+7  wfhrh x,' uo.obwo +kjtbrs

 /rmshjtmbo b+ ko koleh ! beutmbo

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\h jbosmdhr tfkt wh fkvh k trmshjtmbohd koleh ks tfh +beebwmol lurh (0mlurh $)

wfhrh jmrjehs   ),(),(   '  H LLL   7 kod  p DH  DL   77

0mlurh $

 /fho wh `rmol k emoh  DH  K@ //  (0mlurh -) wfhrh tfh' 'mhed tfh +beebwmol dktk

;d B@ 7 ,   0d BL 7 ,   K@L@ ⊨  kod wh jko hksme' sfbw tfkt0

; kL   7∬

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0mlurh -

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7smol tfhsh dktk wh wmee tr' tb od ush+ue hquktmbos wfmjf wmee fhep us tb od tfh

sbeutmbo b+ tfms prb`ehg

   

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  7⇐7 

  

  

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smo   sb wh fkvh 9

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0rbg (4),()

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⇐7+∐∐+∐⇐   10

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 x  kod;0

 x  /fh rbbts dbo2t lmvh tfh

sbeutmbo b+0

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\h tk.h tfh hquktmbo (4) kod wh krh tr'mol tb od tfhmr rbbts9

⇐7+∐+∐++∐   10

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Amhtk hquktmbos (\h wmee ush tfh ohjhsskr' hquktmbos boe') 9

;r  #   0

r  #   :r  #   ;4   kr    7∐7

λ 

ΰ ;4   kr   7⇐   (<)

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04:04:;40;:0;  kr r r r r r r r r r r r    77+++

λ 

κ    (>)

 /fh shjbod sbeutmbo ms b`vmbus tfkt ms0

smo;

kr   7

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8/@9 * fkvh jfhj.hd kod kee tfh rbbts sktms+' kod tfh 4 hquktmbos b+ Amhtk

()   ⇐7∐∐∐⇐7∐∐⇐   1..0

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trmlbobghtr'

 @ x x   ∐7 :4   :

\h jko wrmth tfms trkos+brgktmbo ks   @ xr    ∐7 :4  :  kod  xr    7  sb tb fkvh tfh oke

hquktmbo (;) 5ut wfkt rbbts r wh gust ush tb fkvh tfh rbbts ;; x  kod

;0 x   J

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;d B@ 7

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0br r%x (ks wh fkvh ghotmbohd `h+brh)

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10

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