angle trisection - final english
TRANSCRIPT
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
7/29/2019 Angle Trisection - FINAL English
http://slidepdf.com/reader/full/angle-trisection-final-english 2/13
/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
0smo.0;.0jbs0 00:
0
:0
0 7+∐⇐∐7
∐+7
∐+7+7 d p x p x
p
x x
p
x x x x
k x xk x xd
8bth9 /fh k`bvh hquktmbo gk' `h dhrmvhd tfrbulf tfh whee!.obwo trmlbobghtrmj
rhektmbosfmp
⇐∐7⇐∐7
∐7⇐∐7
0
:
:
::
: :6
40
:
04
0.:
00smo4
0smo:
0
:smo
p
x xd
p
x
p
x
p
x
p
x
p
d kkk
1:: 00::00
7+∐⇐∐7⇐ d p x p x x xpdp
0modmol tfh rbbts9
7/29/2019 Angle Trisection - FINAL English
http://slidepdf.com/reader/full/angle-trisection-final-english 3/13
)(1 0;440; x x x x x x +∐7⇐7++
00
0;0;
0
0;40;
0
404;0; :)(:)(: p x x x x p x x x x x p x x x x x x ∐7+∐⇐∐7++⇐∐7++ ⇐
⇐ ⇐7∐++⇐∐7∐∐∐ 1:: 00
00;
0
;
00
00;
0
; p x x x x p x x x x
⇐ ⇐+∐»∐
70
;04 00
0
0
00
;
p x x x x
$) ⇐0
;04 00
0
0
00
;;
p x x x x
+∐+∐7 (4)
-) ⇐0
;04 00
0
0
00
;0
p x x x x
+∐∐∐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
0
0;; :;00 x p x x ∐+∐7 6=3 p
x
p
xk
000smo 077
0
)0(.:
0
0smo0 0
0
0
;;
x pk
p
x
∐+
∐
7 (;)
p
x p
p
x p
k
0
)0(0
0jbs
00
0
0
∐7
∐
7 (<)
0rbg (;),(<) ⇐
∐7+
∐7
0smo
0
;
0jbs
0
:0
0
0jbs0.:
0
0smo0
;;
kk p
k p
k p
x
Kod `k`k`k smojbsjbs.smo)smo( ∐7∐ rhsuets9
∐7
091smo0;;
k p x (>)
7/29/2019 Angle Trisection - FINAL English
http://slidepdf.com/reader/full/angle-trisection-final-english 4/13
() ⇐0
)0(.:
0
0smo0 0
0
0
;0
x pk
p
x∐
∐∐
7 ,(<) ⇐
0
0jbs0.:
0
0smo0
;0
k p
k p
x ∐∐
7 ⇐
⇐
+∐7
+∐
0
91smo0
0
smo
0
;
0
jbs
0
:0
k p
kk p ()
(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
7/29/2019 Angle Trisection - FINAL English
http://slidepdf.com/reader/full/angle-trisection-final-english 5/13
\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∬
7/29/2019 Angle Trisection - FINAL English
http://slidepdf.com/reader/full/angle-trisection-final-english 6/13
0mlurh -
\h sht 0 xBD 7 , : xBK 7 , ; x KD 7 , BS v 7 kod PLS L@LH 777 0
7smol tfhsh dktk wh wmee tr' tb od ush+ue hquktmbos wfmjf wmee fhep us tb od tfh
sbeutmbo b+ tfms prb`ehg
+7+70
jbs00
k p x SD xv ($)
0
:
0
:jbs
x
xk7
,
p
Pk
00
smo 7
p NH xd 7+∐ :; (-)
7/29/2019 Angle Trisection - FINAL English
http://slidepdf.com/reader/full/angle-trisection-final-english 7/13
7⇐7
0smo.
0smo
k P NH
P
NH k
0
:
0
:jbs
x
xk7
7⇐
0
:jbs0:
k x x
b0
smo.0
:jbs0;
k P
k x pd ∐+7 (:)
\h sht Kk7
0
:jbs
0
jbs00
k p x SD xv +7+7 ($)
?bg`momol tfh k`bvh wh fkvh9
⇐∐∐+7 p
P P K
k pv pd
0.).
0jbs.(;
0jbs)
0smo(.;
k pK
k P p Kvd ∐7∐∐∐
\h sht vKd s ∐7 ;
)0
smo;(0
smo0
smo0)0
smo(00
jbs)0
smo( 0000000 k
K pk
Pk
pP pk
P p s sk
pKk
P p s ∐7+∐+∐∐⇐∐7∐∐⇐
⇐∐7+∐++∐⇐0
00000
0
0000
440.0
000
p
P K p K p
p
P P
p
P pP p
p
P sP sp s wh guetmpe' wmtf 0
4 p
⇐7+∐+∐++∐⇐ 1444464 0000440040:00 p P K K p P P p p p sP sp p s
0smo0
00smo
k P p
p
Pk 7⇐7 \h sht tfh p mo tfh hquktmbo sb wh fkvh obw9
1
0smo4
0smo;9
4
0smo4
4
0smo;9
4
0smo0
4
0smo6
6
0smo4
4
0
000
4
044
0
4
4
40
:
:
0
00
7+∐+∐++∐k
P P K
k
K P P
k
P
k
P
k
P sP
k
sP
k
P s
⇐
\h guetmpe' wmtf0
smo4 4 k sb wh fkvh 9
⇐7+∐+∐++∐ 10
smo0
smo40
smo40
smo60
smo40
smo4 0400444044:::000 k
P K K Pk
Pk
P Pk
sPk
sPk
P s
10
smo0
smo40
smo40
smo60
smo40
smo4 0000040000:00 7+∐+∐++∐⇐ k
P K K Pk
Pk
P Pk
sPk
sPk
s (4)
\h .obw +rbg trmlbobghtr' 90
:smo
0smo:
0smo4
0smo4
0smo:
0
:smo :: kkkkkk
∐7⇐∐7
\h sht @k7
0
:smo kod x
k 70
smo sb wh fkvh 9
7/29/2019 Angle Trisection - FINAL English
http://slidepdf.com/reader/full/angle-trisection-final-english 8/13
@ x x ∐7⇐ :4 : ()
0rbg (4),()
⇐7+∐∐+∐+∐+∐⇐ 1):(4):(044 00000000000 x P K K P @ x x P x P P @ x sP sPx x s
⇐7∐+∐∐+∐+++∐⇐ 10).94().:44( 0000000000
K P P sP@ x @ P sP sP x P K P P s⇐7∐+∐∐+∐∐⇐ 1);(0).0()W.;(4^ 000000 K P sP@ x P@ s P x K P s
\h rhgmod tfkt0
:jbs
k K 7 kod @
k7
0
:smo
⇐7+∐∐+∐⇐ 10
:smo0).0().
0
:smo4( 000000 k
P sP@ x P@ s P xk
P s
⇐7∐∐∐++∐⇐ 1)0
:smo.0.(
0
:smo.).
0
:smo.0.().
0
:smo0)(
0
:smo0( 0 k
P sk
P xk
P s P xk
P sk
P s
10
:smo..).
0
:smo.0( 0 7∐++⇐ k
P x P xk
P s (;)
wfhrh0:jbs.. ;;kBS d s Kvd s ∐7⇐∐7
0rbg hquktmbo (;) tfh' 'mhed tfh rbbts;;
x kod;0
x /fh rbbts dbo2t lmvh tfh
sbeutmbo b+0
smo k
\h tk.h tfh hquktmbo (4) kod wh krh tr'mol tb od tfhmr rbbts9
⇐7+∐+∐++∐ 10
smo0
smo40
smo40
smo60
smo40
smo4 0000040000:00 k P K K P
k P
k P P
k sP
k sP
k s
⇐ ⇐7∐+∐+∐++ 1);(0
smo.40
smo)44(0
smo60
smo4 0000000:40 K Pk
P sk
P K P sk
sPk
P
⇐ ⇐7+∐+∐++ 10
:smo.
0smo.4
0smo)44(
0smo6
0smo4 0000000:40 k
Pk
P sk
P K P sk
sPk
P
\h sht r k 70
smo
7smol gktfhgktmjk prblrkg +br gktfhgktmjs wh fkvh b`shrvhd tfkt :0; r r r ∐7+ \h tr' tfms trkos+brgktmbo motb tfh k`bvh hquktmbo kod usmol Amhtk hquktmbos whwmee shh m+ tfms trkos+brgktmbo sktms+' kee tfh rbbts
Amhtk hquktmbos (\h wmee ush tfh ohjhsskr' hquktmbos boe') 9
;r # 0
r # :r # ;4 kr 7∐7
λ
ΰ ;4 kr 7⇐ (<)
7/29/2019 Angle Trisection - FINAL English
http://slidepdf.com/reader/full/angle-trisection-final-english 9/13
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
44:0; kr r r r 77λ θ ()
8/@9 * fkvh jfhj.hd kod kee tfh rbbts sktms+' kod tfh 4 hquktmbos b+ Amhtk
() ⇐7∐∐∐⇐7∐∐⇐ 1..0
smo..0
smo.).(0
smo 4;:
0
;
0
:4;:;: kkr k
kr k
kkr r r k
⇐
∐»∐
7⇐7++⇐0
smo00
smo4
0
smo
0
smo.
1.0smo.0smo
;
4;
40
;
0
;
:4:0;0:; kk
kkk
kk
kk
r kr k
kr k
k
0smo
4
0smo
0
0smo
;
4
0
: kk
k
kk
r ∐»∐7⇐
b tfh rbbts b+ hquktmbo (4) krh9
0smo;
k
r 7 ;4 kr 7
0smo
4
0smo
0
0smo
;
4
0
0 kk
k
kk
r ∐+∐7
0smo
4
0smo
0
0smo
;
4
0
: kk
kkk
r ∐∐∐7
\h jko wrmth tfh hquktmbo (4) ks 9
10
:smo...4).44(64 0000000:40 7+∐+∐++
k Pr P sr P K P s sPr r P
0rbg hquktmbo (4) wh fkvh k`uthd tb hquktmbo (;) usmol tfh trkos+brgktmbo +rbg
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
7/29/2019 Angle Trisection - FINAL English
http://slidepdf.com/reader/full/angle-trisection-final-english 10/13
;d B@ 7
0
:jbs.;;
kBS d vKd s ∐7∐7
;d B@ 7
0
:smo.
0
0
0
:smo
k P s
P
sk7⇐7
10
:smo.0
:smo010
:smo..0
:smo0)9( 00 7∐+⇐7∐+⇐ k x xkk P Px xk P ($-)
0br r%x (ks wh fkvh ghotmbohd `h+brh)
;0;0 xr 7⇐ wfhrh ;0 x tfh pbsmtmvh rbbt b+ hquktmbo ($-)
⇐7∐+∐⇐7 ;0
;
4
0
;0;0
0
smo4
0smo
0
0smo
xk
k
k
kk
xr
⇐++7∐⇐+7∐4
0smo
0smo
0smo
4
0smo
0
0smo
0smo
4
0smo 0
;0
0
;0
;
4
0
;0
;
4
0 k
k x x
kk
k
kk
xk
k
k
k
7/29/2019 Angle Trisection - FINAL English
http://slidepdf.com/reader/full/angle-trisection-final-english 11/13
10
smo0
smo 4
0
;0;
0
;0; 7++⇐ kk
xkk
xk
5ut
0
:smo0
:smo.
0.0
0
4
60;
k
P
k P
P
s
P
sPk ∐7∐7∐7∐7∐7
λ
ΰ kod
4
0
:smo
0
:smo
4
0
0
0
0
4
k
k
P
Pk 777
λ
θ
⇐7+∐∐ 14
0
:smo
0smo..
0
:smo
0smo..
0
:smo
0
0
;0
0
;0
k
k x
kk x
k
⇐7+∐∐ 14
0
:smo
0smo..0
:
smo0smo..0
:
smo
0
0
;0
0
;0
kk
x
kk
x
k
⇐7∐+ 10
:smo
0smo...4
0smo..4 0
;0
0
;0
kk x
k x
;0
;0
4
;0
0
;0
6
0
:smo;9;94
0smo
x
k x x x
k +»∐
7;0
0
;0;0
4
0
:smo
400smo
x
k
x xk+»∐7⇐
0
:smo4
0
:smo6;;
);0(
0
k
k
x
+»∐7⇐
\h sht y
d k 70
:smo
b d
d y y
y
d
y
d y y
k
k
x .4
6
.4
6
0
:smo4
0
:smo6;; 00
00
0
;0
++∐
7
++∐
7
++∐
7
I x
k
x7+
;0
0
;0
4
0
:smo
4
7/29/2019 Angle Trisection - FINAL English
http://slidepdf.com/reader/full/angle-trisection-final-english 12/13
\h sht gn
d
d
d y y
y
d
y
d y y
x smo4
.4
6
.4
600
00
;0 77++∐
7
++∐
7
g
kg
g
k
g I smo
)0
:smo(smo
4
;
smo0
:smo
smo4
;:
0
+7
+7
4
:smosmo:smosmo4smo:
4:smo ::
1
ggggg
n
d g
∐7⇐∐77
7
+∐7
+∐7
+∐7
ynn
nn ny yn
d
n d
n
d
y
d
n
d
n
d
g
kgg
I 1
111 :
4.4.4
.4
4.4
444.:
4
;
smo4
0
:smo4:smosmo:
4
;
+∐7
yn
nn ny yn
.
:
;9
;
1
11
/fhrh krh gko' jmrjehs wfhrh :1 .. n yn n 7
7/29/2019 Angle Trisection - FINAL English
http://slidepdf.com/reader/full/angle-trisection-final-english 13/13
011
:1
;9
;
.
:
;9 n
i
yn
n n n y7
+∐7 wfhrh :1: n n n i +∐7
/fhrh krh gko' jmrjehs wfhrh 1
00
1
.n
p i pn i
7⇐7
b1;0
0
;0;0
4
;
0
4
4
0
:smo
400smo
n
p
n
d
x
k
x xk»∐7+»∐7
14
;0
n
p
n
d »∐7