0 kvadratu

7/22/2019 0 kvadratu

1/78

-

: :

.

,2012.


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3 51 62 a 12

2.1 ................................................................. 122.2 (5 ).................. 192.3 (4 ).................. 222.4 ........................................................... 232.5 .............................. 312.6 ........................................................................................... 342.7 ....................................................................... 38

3 483.1 ?.................................. 483.2 ............................... 503.3 a............................................................................................ 553.4 ................................ 583.5 ......................................................................... 593.6 .................................................................................. 633.7 ............................................................................ 65

73 74


3/78

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, . .

, ,

, . [11]

(). GeoGebra 3.2.,

[14].

,

.

, -

, , - ,

.


4/78

4

,

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, .

, ,

, , .

,j2012.


5/78

5

N-

CBA - B A C

In -

Ex -

Bd -

BAp , - A B

BApp , - A B

batr , - () a b

vl|| - l v

vl - l v

FS - F

Fp - () F

-

-


6/78

6

1

( 1.1.).

().

.

, ,

. . , , .[18]

1.1.

.

.

. .

.

4, 1.2. ) ,

1.2. ).[20]


7/78

7

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1.2.

[12], ,

. :

,

.

,

:

( ),

,

() .

, ( )

( ).

. , .

( ).

, ,

. , ,

.

.

.

1.1. [3]: F

NkFFF k ,...,, 21

1. ki

iFF1

;


8/78

8

2. ji FInFIn kji ,...,2,1, , ji . iF ki ,...,2,1 1k

F .

- [3]

[16].

[3] .

.

1.2.[3]: F G

k F

k kFFF ,...,, 21 , G kGGG ,...,, 21 , ki ,...,2,1 , iF

1 iG . Ka F( ) G .

,

(

).

1 iG iF .


9/78

9

. .

1.

2.

3.

4.

5.

6.

7. 4:3:2

Ta 1.


10/78

10

.

.

8.

9.

10.

11.

2.


11/78

11

. .

12.

13.

14.

15.

3.


12/78

12

2 a

2.1

. ,

.

, .

, , ,

.

,

:

,

.

.

, . o

, ,

,

.

, , ,

, .

, . ( 2.1.)[8]

2.1.


13/78

13

EKDHLK ( ,EDHL 45EDKHLK EKDHKL ) .

:

.

1. 1., ,

.

. .

.

,

.

( 7 )

ABCD ( 2.2.) , 1ABCDS . , .

( x , x3 ) , :

13 xx 13 2 x .

, x 3

1,

3

3 3 .

3 , DC DE ,

, . 2 ( :

1, 2 ). A

EApp , , BC G .

ADE :

222 DEADAE


14/78

14

2. 2.

3212 AE 3AE .

B AE CDp , F . ABFE ( ABEFAEBF ||,|| ), . ,

:

ADABS .

B F BK FH EAp , . HFBK , ABFE ( ADABS ). ABFE, , ABCD .

, HFBK .

ABCD HFBK , .

AD L , BGAL , BF M AGBM , AELN|| EFMP . ABCD BKAG , LN,

, 2.2. 1, 2, 3 4.

HFBK MPEF, BG .

, BKG

(3) .


15/78

15

EFHABK (4)

( EFAB - , K H , A E ).

GL , M DF ( ),

MPGCLD , PMFDLN ( ). :

MPFLDN (1),

GEMFLN .

ALNCG BGEPM , .

ABCD HFBK . , BF BK . , BF

AE , . 3BF , ABCD HFBK : 1BFBK , :

33

3

3

11 BF

BFBK .

( 2.3. ).

2. 3.


16/78

16

ABCD , AG

ASKR , ST , BK . SR , T

, , AG , , 11, SR 1T .

ABCD HFBK .

2.3.

.

.[12]

( 6 )

,

1:3 , 7, 6 .

ABCD ,

, 3DE , EDC . ( 2.4.)

2.4.

A E AE

BC F .2

2, 60DAE .

313

AD

DE

DAEtg , 60DAE .

30AED , DAE .


17/78

17

DE DEGK,

3

3

3

DEDK . LEApGKp ,, .

. :

FCEAKL LGEABF .

:

3

31

13

DKADAK

DCDECE

. (1)

ADEAKL . , :

AD

AK

DE

KL

131

33

31

AD

DEAKKL (2)

(1) (2) KLCE , :

FCEAKL .

:

3

31 AKFC ,

3

3

3

311

FCBCBF

GEBF . (3)

:

1133 KLKGLG


18/78

18

ABLG . (4)

(3) (4) LGEABF . [12]

:

, .

DEGK MN PQ

( 2.5.) .

EPLL 1 LK 1L AKL 3' 4',

3 4. GQBB 1 1B ABF 5 6,

5 6. 1 2 ABCD DMNK MNQN.

.

2.5.

, ,

[5].

.


19/78

19

2.2 (5 )

, 12 3.

ABCD

.

. x

a . 2a . x , h ,

2

2a

hx

. ,

2

3

2

hx ,

22

3

3a

h

. 33

3 22

2

aa

h

4 3ah .

3322 aaah h a 3a . aAB

( 2.6.) D aDE 2 .

DAE : 34 2222 aaaADDEAE .

2.6.

aAB

3aAE . AE . BC F . AF AB AE,

4 3a .


20/78

20

ABCD ( 2.7.). DC K

AK 4 3a ( 2.6.). B

AKBL || DC , L BLBM

AK, M . MLKP|| . ABCD AK, MN,

BM KP .

, MB MBBQ Q

L . BQMB MQBL , MLQ : QLML .

, ABCD .

AKBL|| 4 3aAKBL . MBL

22222 3 MBaMBBLML . (1)

2.7.

AMB ADK ( MABDKA ). :

AK

AD

AB

MB ;

4 3a

a

a

MB ;

4 3

aMB .

(1):

34

33

22

22 aaaML , . 4 32aML


21/78

21

MBML 2 , MQML .

MQQLML , . MLQ .

BMN , 1.

APKBNL ( AKBL , NBLPAK NLBPKA ), 2.

BL MABS SQ . AMBSBQ ( MABS ,

BQMB SBQAMB ) 3.

:

ABQS (2) BAMBSQ . (3)

BSQ QSL , :

SLQBSQSQL . (4)

:

PKDAKPAKDBAM .

AKPBAMPKD

SLQMLBAKP .

,

SLQBAMPKD . (5)

(3) (5):

SLQBSQPKD . (6)

(4) (6):

PKDSLQ . (7)

QS DKQT , DCDK , QSABDC , QSQT .

QTTR . (2) (7) KDPQTR , 4.


22/78

22

MKCN TRLS ( ).

ABCD

QLM . [12]

2.3 (4 )

, ,

.

MQL ( 4.7.) ,

4

32xS ,

x - . a ,

. :

4

322 xa .

4 32

xa

xxx

xxxx

a 34

34

34

32

22

.

a :

x3 x , 4

x.

MQL . E F

ML QL . E a

MQ H. F EHpp , EHpp , A . A a ,

EApp , FApp , ,, D B . D B EApp , FApp , , C.

ABCDSMQLS , ,. .

. L MQ , DC

BC J K. P MQ , PHJK

W EHpp , , P EHpp , .


23/78

23

(1) 2.8.

.

(4) JKCPHW ( PHJK 90JCKPWH , JKCPHW ).

(2) EWDEELME , ,

DEL MEW , .

2.8.

(3) ( ).

. ()

2.4

?

,

.

.

,

.

.


24/78

24

,

25x ,

.

2.9.

2.9. AD BC

10CFAE ,

QVNO DF ABEK|| , K

EK DF. NO QV 2QWNP

VP QOWM|| , M WM VP.

,

.

?

25x .

a b , b ( 2.10.), , ,

, . abCFAE (

a b ).


25/78

25

2.10.

DF ABEK||

. EK

DF, abAE bAD . :

abb

2

:

abb

4

2

,

ab 4 .

ab 4 ( 2.11.) :

ABaabCFAE 22

EF AB . DF, .

DEFC AEFB .

ab 4 ( 2.12.)). ,2

bAE

2

bCF . ABEK||

DF. ABKE ED

EKCD abAEEL , CK abCF


26/78

26

ABLM|| . LM DF M . 2, 3 4

11CLEF ( 2.12.)), abLE ,

aLMDCLMLC 1 .

2.11.

ABCD EK11CEAF ( 2.12.)).

11CEAF , EA 1EC .

abEA , 11 KCEKEC

aLMaLMaLCEKEC 211 (1)

LM . LM ( 2.12.)) BC N

LM MNF MLD :

DL

FN

LM

MN

abbAEbDL 22

DLabDLFCNCFCFN ,


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27

DL

DLab

LM

MN

:

DL

DLDLab

LM

LMMN

;

abb

ab

LM

a

2 ,

:

ab

abbaLM

2 .

2.12.

(1) :

ab

ab

ab

ab

ababbaa

ab

abbaEC

222

21 .

, EAEC 1 11CEAF .


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ABCD

AEEL ABLM|| DF,

AD

ab , . .

ab .

1,2,3 4.

.[12]

a b - ABCD ( 2.13. 2.14.3), h -

a ba ( hb ha ).

ahr a h .

A r, DC , hr . ha :

hah hr , aah ar . aahrh .

2.13.

CD :

1. CD , . 2.13. ( E);2. HD , . 2.14. ( 1E );3. D .

,

h , AB AE ( 1AE ).

3

: 2.13., 2.14. .


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ADE BCF , CBF1

DAE1 . .

2.14.

h a

ahr .

, .

ahBK . , ABEL ( 2.13.) ABLE 1 (

2.14.). AEL ABK ( LAE1

ABK ) , KAB .

:

BK

AB

EL

AE , ELABAE .

ELABELAB

ELAB

AE

ELAB

BK

AEBK (, 2.13.:1

AEBK ).


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30

2.15.

ABFE 1ABFE

( 2.15. 2.16.) a , h ahrAEAE 1 .

, BK B

AE ( 2.16.) ( 2.14.).

ah : ahBKKH ahBKBG .

2.16.


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31

H G BKHG ,

ABFE .

.[12]

2.5

10. 11. 3.

, .

, .

( )

.

.

1. ( 2.17. ))

.

2.17.

2. ( 2.17. ))

.

1. FC( 2.18.) )

BFEC1 .( 2.18.) )

,

. , BF H ,

FH . H BF , BF , P. FP

. F ,

FP 1EC , K.

FK .

FK , .

KSBFFK , KS .


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32

2.18.

B BL FK . FLBFSK ~ :

BL

BF

KS

FK , KSBF

KSBF

KSBF

FK

KSBFBL

,

FKBL .

BLMN

, .

, FENN ||1

BF 1N .

, KFEBNN 1 ( NBNEFKFKBN 1, BNNEKF 1 )

11 BCFENN , TBCQNN 11 .

FQ KT

.

KCTFMQ .

,

. 2.18. .

2. CE ABCDE ( 2.19.),

:

EDC ABCE.

CE EDEF ( EADC ) AF .

AEF

EDC . EABAEF ( CFAB|| AE), EDCAEF .

EDCAEF .


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33

EDC AEF . ABCF

. G , AF ,

BC CF K, AB L .

, FKGALG , LBCK

ABCF , .

2.19.

EDC GK, ab ,

. ,

LBCK BMNP.

, ,

LBCK 1LL .

1LL AF R . Eab bc , LR . CDba de , RT .

. , BMTL,1 EC

. BMNP.

, 2.20.

[14].


34/78

34

2.20.

2.6

. [14]

. , .

. 2.21.

.

) )

)

)

2.21.


35/78

35

,

. .

. .

,

.

: . ( 2.22.)

2.22.

2

2 ,

( 2.23.

180 ).

2.23.

. ,

180 .

180 . 2.24.

.


36/78

36

) )

)

)

2.24.

.

,

.

. ( 2.25.)

2.25.


37/78

37

2

,

, .

. ( 2.26.)

2.26.

1

1 .

.

.

.

. 2.27.

2.27.

2

2 1 .

,

.

. 2.28. ,

2.3.


38/78

38

2.28.

22

22 .

, , .

. 22

. ,

. 2.29.

.

2.29.

2.7

, , .

,

, ?

.

[3]:

2.7.1.: n - 3...21 nPPP n (

, , ).[3]


39/78

39

: iP nPPP ...21

. :

1) 11 ii PP ;

2) 11 iii PPP nPPP ...21 ,

11 iii PPP 11. ii PPp ,

iP n - nPPP ...21 ;

3) n - nPPP ...21 11 ii PP 11 iii PPP ,

iP .[16]

2.7.2.: nPPP ...21

, . ,

nPPP ...21 , ,

2n , nPPP ...21 .[3]

: 2.7.1. nPPP ...21 .

n - nPPP ...21

. .

n - nPPP ...21

.

2n n :

4n .; nk k -

kPPP ...21 2k ;

n n - nPPP ...21

2n . 2.7.1. .

, - [16]

. k 2 kn .

k 2k

2 kn kn , 2n .

2.7.3.:

.[3]

(): ABC 111 CBA

p ( 2.30.). ABC 111 CBA

MN 11NM . ,

111 CBAABC , 111 CNMMNC . MN ( 11NM ) P

(1

P) CPMC ( 1111 PCCM ). , MPC

111 MCP .


40/78

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MN(. 11CM ) Q (. 1Q ) MQPN (. 1111 QMNP ),

CPNAMQ ( .

111111 NPCQMA ),

111111 NPCQMACPNAMQ . AMQ 111 NPC , CPN

111 QMA , .

2.30.

NBCNAQ ( . 111111 BNNCQA ) AQNB (. 1111 AQNB )

, AQNB 1111 AQNB .

.

2.7.4.: P R Q R , P Q .[3]

[12]: R P Q , .

R ( 2.30. R ).

P Q ,

.


41/78

41

2.31.

2.31. ) R P ( ),

1P, 2P , 3P 4P .

2.31. ) R

Q ( ),

1Q , 2Q , 3Q 4Q .

R (

2.31. )). ,

1P, 2P , 3P 4P P ( . 1 2

1P, 3,4 5 2P , . ), P.

,

1Q , 2Q , 3Q 4Q Q ( . 5 8 1Q ,

2, 4 7 2Q , . ), Q .

R ( )

P Q ,

.

2.7.5.: ,

,

.[12]

: DE ABC ( 2.32.)

1BF ( B AC) F .


42/78

42

2.32.

CDE BFE . , ABFD

ABC .2.7.6.: .[12] (): 2.7.5.

,

.

, . 2.7.4. .

2.7.7.: .

, .

2.7.8.: ,

.[12]

[12]: :

.

. , . (

2.33. )

BFGC ACKL , ABC , AEDB , , ,

:

1. AM , AE;2. ABFN|| ;3. AENP|| ;4. BCDY|| ;5. DYES ;6. BR FB ;7. NPDQ DYQT .


43/78

43

2.33.

ABFN ,

. BDAB BDFN . , DBRNFG .

:

BRDFGN

FBBRFG

.

2 ,

.

EDSABC ( DEAB ) ACES

CAMSEY , CAMESY . AMEY .

DQNPAY DEAE , AMEYEQ . ,

4, ( ALACES AMEQ ).

2.33. .

,

.

: ,

, , ,

.

2.7.9.:

.[3]


44/78

44

[3]:

. ABCD ABEF ,

BAp , .

DEF ,, C CD EF

, ABM , M

. , ,...,,210 mmm

BApm ,0 , 1mM im 1i 11, ii mmtr .

kmmm ,...,, 21 EFpBAptr ,,, , im , ki ,

. : ,,,, iiii EEBpmAFApm iiii BCBpmDDApm ,,, , ,,(,1,...,1,0 0011 DAADMEki

)00 EBB .

CBk

, 1

C 1

kk

DB CD , 1

F 1

kk

EA

EF . ABCD 1, CCBABM k ,

121 iiii DDBB , 2,...,1,0 ki , kkk DDCBB 11 ,

ABEF FFAABM k 1, , 121 iiii AEEA ,

2,...,1,0 ki 11 EFEAA kkk .

.

2.7.10.:

, .[12]

[12]: ABCD ,

( 2.29.)

CD , E AE

. B , AE

AF EK A E. CD BK AF L N .

EN AB , A B , n .

ABENn , EN , : ABENn 1 .4

121

...,,, nAAA AF LD

,..., 21 NN nA AF , BL

M5.

4 .

5 ABCDn , nA ( M ) B .


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AN EK ,..., 21 EE

EN , AF ,..., 21 FF

KF 1

M . ,..., 21 EE ...,, 21 AA ,

n .

, 2.29.

, 1 .

ABCD AFKE .

.

2.34.

AFKE ( 2.34. ).

KF B , AB

. ABEC|| , ECAD ECBC . EK AF

E A , AN, ,...,2211 FEFE

AFKE ABCD .

, AD

AFKE( 2.34. ).

EC

E AD A

.

.

2.7.11.: n - 3n .[3]

[3]: n - 2n (2.7.1. 2.7.2.) 21,..., kTT . 2200 nn ABBA

00BA .

20 nAA 31,..., nAA , 20 nBB 31,..., nBB ,

11 iii AAA 11 iii BBB , 3,...,1 ni , iT ,


46/78

46

2,...,1 ni . 2.7.10. 2.7.6. n -

2200 nn ABBA .

2.7.12.:

.[12]

:

,

.

.

( 2.35.) .

2.35.

2.7.1. ,

. , a . , .

,

, , ,

.

2.7.13..: (F. BolyaiGervin, 1832/33)

.

[3]:

, . ,

.

1M 2M , 21 MSMS . 2.7.11.,

1 2 1M 2M , .

2.7.10., 1 2 .

2.7.4. 11,M 2 , 21,M 2M ,

1M 2M , .


47/78

47

(): 1M 2M .

2.7.2. . 2.7.6.

2.7.7. ,

2.7.8.

, 1

Q 2

Q , , .

1Q 1M 2Q 2M

( 2.7.4) 1M 2M .

2.7.13.

.


48/78

48

3

3.1 ?

, , ,

, , . (, , .) .

, , .

, . [1]

, .

:

, ,

. ,

: ?

:

3.1.1.: .

[12]: ABCD BEFG .

a , b c , cb . ABEK . AKFG

KECD , . KDDCFGAG . DCFG , KDAG ;

caab . acb 2 acb 422 , . , , . ,

, .

, , , .

?

:

3.1.2.: ,

.

(): a , ap 4 ,

2aq . , , ,

cbp 22 , b c , bcQ . cbap 224 2

cba

,

Qcb

bccb

q

22

2222

. 02

22

cb

, Qq .

[12]: p q . ,

, p , qQ .

,

1p . 3.1.1.,


49/78

49

pp 1 . , , .

, .

,

. .

, .

, ,

.

1: ( 3.1.),

.

, :

1.

, ;2. ( , ).

3.1.

.

, (

3.2.). . ?

. .

.

:

, , .


50/78

50

3.2.

: ( 3.3.)

.

, ,

. ,

.

.

? .

CBA ,, D 459045 PCQNBPMAN ,, QDM - , . ,

QMPQNPMN , , MNPQ .

, ,

.

3.2

.

,

,

.

, . ABC ( 3.3.). BC

AB , BC AB , BD ( 3.4.) ABC .


51/78

51

3.3. 3.4.

.

3.5.

, 1XX ( 3.5.),

, . ,

1YY 1XX , B .

ABCD .


52/78

52

AB ABCD . ,

AB BC ( 3.6.). EDCF .

AEFB .

5.7.

.

2:

.

, , X

AB , BX

AX

AX

AB BXABAX 2 .

3.7.

[12]: ABCD , BC

E( 3.7.). AE. K ,


53/78

53

EBEK , F AB

BEK . AK AB , AKAX ( AS)

X .

, XB XA ( XH) XH

EA L . , LX AB . FK . K ,

B , AKFAXL (

a AKAX ).

FKXL , BFFK , :

XLBF (1)

AXLABE ~

2

ABBE ; :

2

AXXL . (2)

(1) (2) :

2

AXBF . (3)

AB Y , :

FYBF (4)

( BF FA ). FYAFFBAFAB . , FYAFAY .

:

22 FYAFFYAFFYAFAYAB .

(4) (1) : FKFY , 22 FKAFAYAB . 222 AKFKAF (

AKF ); AXAK ( ) :

2222 AXAKFKAFAYAB ,

AYABAX 2 . (5)

BYFYBFBFBFBFXLAX 412

22 .

, BYAX , :


54/78

54

BXAY . (6)

(5) (6) :

BXABAX 2 . (7)

:

AYABBY 2 . (8)

. Y AB :

AY

BY

BY

AB .

AX Y BY X .

, AX BY AXRM BYNP

( 3.7.). PM . (8)

BYNP AY AD (6)

BXHC.

,

BXRP XYNP CPRH , .

2PRXRXY AXXR AYBXPR , 2AYAXXY ,

. .

3: ,

15'11,'3022,45 . 36 ,

9,18,72 .?

, 1.

: Y , AB

( 3.8.). AYGD , EF . YB Y B EF ,

K A . KBYK, KA .

BAK BKA KAB

ABK .


55/78

55

3.8.

, YKBY AEEY . AKYK AKYKBY . ,KYABAK , YKBKBY ; KBYKYA 2 (

BYK )

KBABAK 2 . (1)

,

,2

22

222222222

AYABBY

YEBEABBYAEBEAEBEBY

AEBEYKBEAEAKBEEKBK

YBABBY 2 ; 22 ABABABBYYAABBYABYAABBK ABBK . :

BKABAK . (2)

(1) (2) : KBAAKBBAK 2 .

KBA , 2 AKBBAK , 18022 . . 36 .

36KBA , .

3.3 a

.

ABCD ( 3.9.) .

EFKL ,

ABCD .


56/78

56

3.9.

,

EFKL . , ,

ABCD . ABCD ,

2

1.

EFKL , , 1111

DCBA (

3.9.)) , .

2

1 EFKL

4

1

ABCD . ( 3.9.)) ,

8

1 ABCD .

,

.

: ,...16

1,

8

1,

4

1,

2

1.6

, , .

12

1...

16

1

8

1

4

1

2

1lim

nx.

3.9.),

.

6 21 .


57/78

57

3.10.

ABCD , A ,

:1111

,,, DDCCBBAA 1111 ,,, DCBA

11111111 ,,C, ADDCBBA .

, 1111 DCBA . 111 BABADA ( 11 BBAA ,

11 BAAD 901111 BBAAAD ). :

1111 BADA ;

1111 ABBDAA ;

90111 BBADAA ;

111 DAB .

1111

DCBA .

S 1111 DCBA xAA 1 ?

ABCD a , xaBA 1 xBB 1 .

:

22211

xaxBA .

211

BAS ,


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58

22 xaxS .

:

22

2

22

2

22

2222

22

242

222

2222

xaaxaxaaxaxaaxaxaxax

.

22

22

2

x

aaS .

:

1. 0x ABCD ;2.

2,0 a

x 2

a 0 ,

S 2a 2

2a.

3.

a

ax ,

2

2

2

x

a ,

2

2a 2a .

.

xAA 1 , , a ( .

3.10. x ABAE ), EFKL

ABCD .

3.4

.

.

.

A , .

: ?

, : ?

. ,

.

, ABp ( 3.11.).

. 2

p , ABCD .


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59

3.11.

m n . AB a m n .

ABCD BacbDCAd abcd , ( ).

22 nm

Aa Ba Aa Ad , . : abcd . ,

ABCD .

abcd ABCD

a AB .

, :

, .

: 222 nmp mnnmnm 2222 .

2pnm , .

3.5

,

.

.

.

.


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60

,

.

. 12.

:

1: ,

.

: ABCD ( 3.12),

. , , ,

.

3.12.

1111 DCBA ( ),

ABCD , ,

ABCD . 1111 DCBA 8 .

EFKL ABCD E CD .

EFKL , MN

1111 DCBA , EFKL , . 1.


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61

3.12. M N 11DC , 1111 DCBA

EF EL EFKL . LED

MNEP . :

tgctgEPtgEPctgEPPNMPMN .

2

1EP ,:

2sin

1

cossin2

1

cos

sin

sin

cos

2

1

MN .

12sin , 12sin

1

, 1MN .

3.13.

MN EF KL

EFKL ( 3.13.). MNLQ || . MNQL . ,

LQMN , LELQ ; , LEMN 1MN .

2

DCDE 45 , 3.14.

MN : PNPDMDMN 11 .

ED 11DA Q . :

tgDEtgDEQDQDQMQDMD

2

1

2

1111

,

DEQEPD 21

1 tgtgPEPN

21 .


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62

, :

tgDEtgDEtgDEtgNDMD 112

1

2

1

2

111

.

45 , :

01 tg 111 NDMD .

0DE , 111 NDMD , 0 1111 DENDMD .

3.14.

ABCD EFKL .

1111

DCBA EFKL ,

EFKL .

, . EFKL 1111

LKFE ( 3.14.).

11111 NBBM .

1111

|| LKQN , 11BN 11LE 1P.

:

111 BNQ aNB 11 .

1a , .

1a .

111 PQN :

cos

1111

QNPN


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63

111 QN ,

cos

111 PN ;

aNBPNPB cos

1111111

.

111111 BNQBMP

111 PBM :

sin

cos1

cos

11111

actgactgPBBM .

1sin

cos11111

aa

NBBM

,

( 0sin ):

0sinsincos1 aa .

, :

0sincossin11 a .

1a sin1sincos , . ,

, .

3.15.

, EFKL ABCD ,

1111 DCBA , ABCD .

1111 DCBA . ,

, 3.15. ,

.

3.6

, .

( ) ,

. .


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64

( )

, ,

( ), ( ),

().

.

, ,

.

3.16. , 26 . (

.)

28 .

3.16.

, 26

.


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65

: 9

1, 4, 7, 8, 9, 10, 14, 15 18. ( 3.17.)

3.17.

9 ,

.

,

, . ( )

.

3.7

, ( 3.18.)

(. abca , bcdb , abcda.).

.

i , ( 1i , 2i , 3i .).


66/78

66

3.18.

dcba ,,, . ,

. ,

. .

0 .

. 3.18.: a : 0321 iii ,

b : 0743 iii ,

c : 06542 iiii , .

,

, . , .

, . :

, 0 .

. 3.18.: abca : 0243

iii ,

bdcb : 0467

iii ,

abdca : 02673

iiii , .


67/78

67

, ,

. .

n , ()

n .

, , n , n , ( ).

,

.

,

. .

, , , . ?

.

:

, ,

. , ;

. . ,

, . .

3.19., a f .

3.19.

( )

.


68/78

68

b : 0431 iii ,

c : 0652

iii ,

d : 08754 iiii ,

e : 0973

iii .

acdba : 01452 iiii ,

bdeb : 0374

iii ,

cfdc : 0586

iii ,

dfed : 0798

iii .

. ,

. , . acdeba ,

,

.

.

, . 1i , 832 ,...,, iii . , .

. 9i ;

8i .

1i :

1215

18ii , 13

15

8ii , 14

15

7ii , 15

15

4ii ,

1615

14ii , 17

15

1ii , 18

15

10ii , 19

15

9ii .

151 i , : 182 i , 83 i , 74 i ,

45 i , 146 i , 17 i , 108 i 99 i .

.


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69

5.20.

, 151 i : 15 ( 5.20.),

83 i 74 i .

151 i .

151 i 182 i ,

45 i 146 i .

d 17 i 108 i , : 1 10 ,

7 4 . 9

.

. 3.21.

1i . :

1225

16ii , 13

25

28ii , 14

25

9ii , 15

25

7ii ,

16255 ii , 17 25

2 ii , 18 2533 ii , 19 25

36 ii .


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70

3.21.

6i : d c ,

c d . , , 3.22. 3.23.

3.22. 3.23.

, .

.


71/78

71

,

, ,

.

,

,

.

, . 3.22. 90 . c d .

.

7,6,5,4 8 ,

.

.

.

.

3.24.

, ,

. d , , 76 ii ( 3.24).

:


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72

b 0641 iii , acba 0142 iii ,

c 06542 iiii , aeca 0253 iii ,

d 076

ii , bcfdb 06784

iiii ,

e 011953 iiii , cefc 0895 iii ,

f 010987 iiii , egfe 091011 iii .

:

123

4ii , 13 2ii , 14

3

1ii , 15

3

2ii , 16

3

2ii ,

1732 ii , 18 ii , 19

31 ii , 110 2ii , 111

37 ii .

3.25.

31 i : 42 i , 63 i , 14 i ,

25 i , 26 i , 27 i , 38 i , 19 i , 610 i 711 i .

( 3.25.) 1313x ,

.


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73

[1] Andreescu T., Mushkarov O., Stoyanov L. (2006): Geometric problem on maxima andminima, Birkhauser, Boston

[2] . . (1947): , , [3] .(2000), , ,

,

[4] . ., . . (1965), ,. . .,

[5] http://axel.nm.ru/tangram/about/history.html[6] . ., . .(1951): , , [7] Boltyanskii V. G., Yaglom I. M.(1961): Convex Figures, Holt, Rinehart and Winston, New

York

[8] (1883): , , . 1.,

[9] . ., . ., (1960): ,

[10] . . (1933): . . , [11] . ., (1951):

, ,

[12] . ., . . (1952): ,. ,

[13] . . (1956): , , [14] http://home.btconnect.com/GavinTheobald/Index.html[15] Pach J., Agarwai P. K. (1995): Combinatorial Geometry, John Wiley & Sons Inc., New

York[16] . (2006): , ,

, ,

[17] http://school14-v.ucoz.ru/publ/1-1-0-3[18] www.kotraza-dragacevo.com/Dokument/TANGRAM.ppt[19] The Dissection of Rectangle into squares (1940): Duke Mathematical Journal[20] http://origami-paper.ru/origami/serbian/istoriya_origami/origami_istoriya_poyavleniya.html[21] . ., . ., . . (1974):

, ,

[22] http://srb.imomath.com/index.php?options=mat_c%7Cdodatne_c
http://axel.nm.ru/tangram/about/history.htmlhttp://axel.nm.ru/tangram/about/history.htmlhttp://home.btconnect.com/GavinTheobald/Index.htmlhttp://home.btconnect.com/GavinTheobald/Index.htmlhttp://school14-v.ucoz.ru/publ/1-1-0-3http://school14-v.ucoz.ru/publ/1-1-0-3http://www.kotraza-dragacevo.com/Dokument/TANGRAM.ppthttp://www.kotraza-dragacevo.com/Dokument/TANGRAM.ppthttp://www.kotraza-dragacevo.com/Dokument/TANGRAM.ppthttp://www.kotraza-dragacevo.com/Dokument/TANGRAM.ppthttp://origami-paper.ru/origami/serbian/istoriya_origami/origami_istoriya_poyavleniya.htmlhttp://origami-paper.ru/origami/serbian/istoriya_origami/origami_istoriya_poyavleniya.htmlhttp://srb.imomath.com/index.php?options=mat_c%7Cdodatne_chttp://srb.imomath.com/index.php?options=mat_c%7Cdodatne_chttp://srb.imomath.com/index.php?options=mat_c%7Cdodatne_chttp://origami-paper.ru/origami/serbian/istoriya_origami/origami_istoriya_poyavleniya.htmlhttp://www.kotraza-dragacevo.com/Dokument/TANGRAM.ppthttp://school14-v.ucoz.ru/publ/1-1-0-3http://home.btconnect.com/GavinTheobald/Index.htmlhttp://axel.nm.ru/tangram/about/history.html


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74

2. 1987. . oj

5,00 .

, , 2006.

.

, ,

, 2006. 11. 2010.

8,97.

,

.

.

, 2012.


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75

-

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:

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:

:

:

:

: ()

:

:

:

: 2012.

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: , , - , 3

: (3, 74, 22, 3, 63, 0, 0)( , , , , , , )

:

:


76/78

76

/ : , , ,

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: , ,

:

:

. . Bolay Gervin . .

: 27. 02. 2012.

:Maj 2012.

:

: - , , - ,

: , , - ,

: , , - ,


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77

UNIVERSITY OF NOVI SAD

FACULTY OF SCIENCEDEPARTAMENT OF MATHEMATICS AND INFORMATICS

KEY WORDS DOCUMENTATION

Accession number:ANO

Identification number:INO

Document type: Monograph typeDT

Type of record: Printed textTR

Contents Code: Masters thesi

CC

Author: Draginja ProkiAU

Mentor:Nevena Pui, Full ProfessorMN

Title: SquareXI

Language of text: Serbian (Cyrillic)LT

Language of abstract: Serbian and EnglishLA

Country of publication: SerbiaCP

Locality of publication: VojvodinaLP

Publication year: 2012.PY

Publisher: Author's reprintPU

Publ. place: Novi Sad, Department of mathematics and informatics, Faculty of Science, Trg Dositeja Obradovia 3 PP

Physical description: (3, 74, 22, 3, 63, 0, 0)(chapters, pages, references, tables, pictures, charts, supplements)PD

Scientific field: MathematicSF

Scientific discipline: Combinatorial geometrySD


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Subjec/Key words: the square division, the divisible equality of two plane figures, the methods of geometric dissections,

analogous of two figuresSKW UC:

Holding data: The Library of the Department of Mathematics and Informatics, Faculty of Science and Mathematics,

University of Novi Sad

HD

Note:N

Abstract:

ABThis Master thesis deals with the divisible equality of a square and some other plane figure. Apart from basic definitions,

the thesis will provide an evidence of the divisible equality of a square and a polygon or union of a polygons. The Bolay

Gervin theorem refers to the divisible equality of two polygons. Some of the applications of the square division have been

presented.

Accepted by the Scientific Board on: 27.02.2012.ASB

Defended: May 2012.DE

Thesis defense board:DB

President: Dr. Olga Bodroa - Panti, Full Professor, Faculty of Science and Mathematics, University of Novi Sad

Member: Dr. Nevena Pui, Full Professor, Faculty of Science and Mathematics, University of Novi Sad

Member: Dr. Gradimir Vojvodi, FullProfessor, Faculty of Science and Mathematics, University of Novi Sad

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