Τεύχος 18 Δεκέμβριος 2017 - mathematica.gr · Α = 150 +450 = 600 και Γ = 300...

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AM B .

53.

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.

:


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1 ( )

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.

180 = = . , ,

.

33
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.

2 ( )

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= + 90 - = 90 .

.

, .

45= .

= =2+900=1800 .

900 +

.

.

54.

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(O,r).

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PB, PC. PA=PB+PC.

:


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=

(= =600)

= , = =120

=

=, =

, =

=+=+.

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BPC 60 B ( ABD).

D AP

BP=BCP (

) BDP

(BD = BP P - BD=+60 =60 ).

: PA = PD + DA = PB + PC.

34


55.

ABC

a cI ,I

A, C .

AC K

CA L.

AC, KL

.

: (kostas136)


c.php?f=20&t=12826&p=69219#p69219

( )

D, E

aI

BC, AB .

2 , :

AK=AEAC+CK=AB+BE

AC+CD=AB+BD (1)

ABC :

AC+CD+AB+BD=2s (2) ,

s .

(1), (2)

AC+CD=AK=s .

: CL = s,

AC

: AK = CL

AK - AC + CM = CL - AC + AM

KM = LM KL.

56.

ABCD

D, B

EC EDC=EBC.

DEC=AEB.

: Triantafyllos


c.php?f=20&t=12935

( )

35
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EDC=EBC=a , DEC=b, AEB=c.

ADFE

BC = //EF BCFE

.

CE=CFE=CDE=a

CEFD

CED=CFD=b .

AEB, DFC (-

-) .

57.

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E . A B

Z ,

AE

BZ ,

:


c.php?f=20&t=12971

( )

,:

=+=+=

(=)

.

58.

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() BC

D. BF CG AD.

A BC,

EFG .

:


c.php?f=20&t=12976

1 ( )

FG

FG.

36

www.mathematica.gr 18, 2017 FEG.

EFG .

2 ( )

E BC

CG BG M.

FG ME .

BFG FM = MG, ME

FG, FEG

.

59.

ABC , o o A=90 , B=30 .

M BC , K

AB.

BC AB D.

1) CD C

2) 1KD= AB6

.

: KARKAR


c.php?f=20&t=13017

( )

D DC =

DB BDC ,

DCM=30 , CD

C .

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60.

(=900)

. :

1.

.

2. <

3.

, .

:


c.php?f=20&t=13049

( )

1) .

A, E

37

www.mathematica.gr 18, 2017 (

), A E = A E. =

2) EB

E ( A )

B ,

A B. <

1) AE

( )A E,A E = = ,

A,B

B E.

61.

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:


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( )

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ZN// ZN A

AN

AZ N

AK BE

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AKB .

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.

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=.

:


c.php?f=20&t=13436

( )

38


=450

450

.

63.

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.

=.

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A=E.

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.

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B / /EZ ZE Z 60 . = =

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.

=120 =30

39


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=.

64.

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( )

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:

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:

EM=EBM=EAM .

,,,

.

65.

E AB ,

ABCD,

DE.

CZ , AH.

EZ=DE-AH.

: KARKAR

40

www.mathematica.gr 18, 2017 http://www.mathematica.gr/forum/viewtopi

c.php?f=20&t=13722

1 ( )

A CZ

DE G CZ F.

AC.

AFCD

AFC=ADC=90 , AFD=ACD=45 ,

GDF

GF=GD (1) .

EZFG,

AHEG :

EZ = GF (2),AH = GE (3).

(1),(2),(3)

:

EZ = GD = DE - GE = DE - AH.

2 (KARKAR)

DE

AG ,CK , KCD, GDA

( = -

, CD=DA - )

EZ=KC , AH=GE

EZCK , AHEG.

EZ=KC=DG=DE-GE

66.

AB=BC=CD=DE=EF=FG=GH

oAGH=82 . GAH .

: (nonlinear)


c.php?f=20&t=13908&p=74632#p74632

1 ( )

GH=x

: 5 180 12 82 .x x + =

x=GH=14

2 ( )

:

41


=2 =+

=+ =+

=+

, , , , ,

. :

=+4=5

=+5=6

0AGH=82

( )00 0

AGH=+FGH= + 180 -2 =

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: 0 0180 -7=82 0=14

67.

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D AC ,

oABD=18 DBC ,

. BDC .

( , , )A C .

: KARKAR


c.php?f=20&t=14012&p=74991#p74991

( )

1 :

: B=C=

CBD=BCD

=-18

CBD=BDC=-18 , BCD :

2-36 +=1803=216

=72

( ) ( ) A, , C = 36 , 54 ,72 BDC=C= , BCD :

2+-18 =1803=198

=66

( ) ( )A, , 48 , 66 ,66C = 2 :

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B=180 -2

CBD=162 -2

BDC=C

BDC>A=

CBD=C= ,

42


162 -2==54

( ) ( )A, , 54 , 72 ,54C = CBD=BDC=162 -2 , BCD :

BDC+CBD+C=+324 -4=1803=144

=48

( ) ( )A, , 48 , 66 ,48C = 3 :

A=B=

CBD=-18 , BDC=+18

CBD=BDC ,

: -18 =+18

CBD=C=-18 , BCD :

BDC+CBD+C=2-36 ++18 =1803=198

=66

( ) ( ) A, , = 66 , 84 ,48C BDC=C=+18 , BCD :

BDC+CBD+C=2-36 ++18 =1803=162

=54

( ) ( ) A, , C = 54 , 72 ,72

68.

CD = 2 ,

C = 1 , CD = 3 , D = 1 D = 60

.

:


c.php?f=20&t=14123

(KARKAR)

CD EF=ED=1/2,

AB=FC=2, BC=AF=2 ABCF

.

69.

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.

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:


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1 ( )

43


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++ 2 -- =2 2

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( )

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= .

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+ + =

= =

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71.

.

=+.

:


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1 ( )

Z 90

( AK ).

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:

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44


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A , D, E .

,

, c .

: (KARKAR)


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.

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)

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), M B x = (

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E A = .

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1) A=B=B B

BM .

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E. BE

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BM=ME .

74.

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:

i) ,

R.

ii) , ,

R

:


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( )

46


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c.php?f=20&t=15015

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. oNB=BNA=90

NA=180 .

76.

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47

www.mathematica.gr 18, 2017 ) T .

) , , ,

(:).

: .


c.php?f=20&t=15014&p=78850#p78850

( .)

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).

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. B

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AC E

CD Z.

DZ=2KE.

:


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( .)

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= (

.

KIE=EZC ( )

KEI=ZEC (). To . ECZ

CL (L Z

ACD)

. CEL=CZL .

KIE=KEI = .

78.

48
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www.mathematica.gr 18, 2017 , ,

, ,

.

( , 7, 153,

, 1)

: dimplak


c.php?f=20&t=15057

( )

:

79.

BC

ABC , BCCD=2

,

CDE , E ( BD)

.

=AEB =ADB ,

.

: KARKAR


c.php?f=20&t=15103&p=79273#p79273

1 ( .)

BD DZ = CD. .

ABZ: C EC//AB .

2C = BZ oBAZ=90 .

CEZ 2ED = CZ oCEZ=90 .

:

o o o o

o o o o

=180 -90 -180 +120 +EBC=30+EBC

=180 -60 -60 -CAD=60 -CAD.,

49


EBC=CAD .

BC CD -- (

C = BC, ACD = BCE , CE = CD)

EBC=CAD .

2 ( )

F BD,AE AG

ABC .

ACG, ACE (--),

AEC=90 F=30 .

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.

AED,EGB (--:

- GEC=30 CGE).

E AD = B EG = x,

+x=60 (1)

=30 +x (2) .

( ) ( )1 , 2 .

3 (KARKAR)

ABC, BCE

(AC=BC, CD=CE 1200)

ABE, AMD (AB=MD,

BE=AD, BAE=AMD=900) AC=2CE.

C=600, =, .

80.

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, , ,

.

.

: (miltos)


c.php?f=20&t=15185

( )

/ AB A 90= > .

AE HZ

:

A 90= > ()

E ZH = /

EZH

1 1E H=

.

AE H=

AE / / H , AE H

/, A EH

O A EH .

50

www.mathematica.gr 18, 2017 / EZH , EH

Z O EH

AB

O A .

/ AB EZH

O .

81.

A

BM,

E.

E AB

Z.

Z//A.

:


c.php?f=20&t=15271

: (badoun)

K//AB . 1 2Z= +

.

Z

AZ 1 1 2 1Z=A + =A +

.

2 2A =

2 1A = ,

1 2 = Z==A

' , Z//A.

82.

O

AB , ,

B, .

A , BK , N

, :

A=BK+N .

: (KARKAR)


c.php?f=20&t=15489

: ( )

MM

( )BK N 2MM : 1+ =

( )OA 2 : 2OM 1

= ( )

2

(

) : ( )

( )

2

1

A OA 2 MM OM 1

A 2MMA BK N

= =

= = +

2 .

51


83.

O

AB AO.

A ,

C , D.

OC E .

:

1) AC=CD

2) BD

OC .

3) AC, AE

: KARKAR


c.php?f=20&t=15169

( )

) oACO=90 .

OD. ACO COD

. AC=CD.

) DAB DOB

.

BD=2OC

) C

AC .

AE

. .

) AOC

AC .

AE

. .

84.

=.

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,

.

: vzf


c.php?f=20&t=16695

(irakleios)

To (

) .

,

52


.

( ,

,

)

.(

,

)

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85.

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) .

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c.php?f=20&t=16697

(irakleios):

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.

//

= 2

(

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.

86.

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.

:


c.php?f=20&t=16734

( )

53


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1) Z HK= = =M2 2

2) O N= = =N2 2

3) KN=NM

KN=NM.

KN=NM=M=K

KNM .

0

KNM=KN+NM= MN+NM=N=90

KNM

87.

60

45 .

,

15 .

.

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,.

) .

) , ,

, ,

.

) 30 .

:


c.php?f=20&t=16993

( )

) A 60=

45 = , B 75= .

E B E A 45 15 30 = = =

B 75= ,

BE 75 . = .

BE B 75 = = .

,

= (1).

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, :

1 P E A 75 BEP= + = = ,

,

= (2).

) 1E 90 BEP 15 = =

ZBE 90 60 150 = + =

.

54


1Z 15=

1 1E Z 15= = ,

BZ BE= (3)

) . A 60= ,

1B 30=

. 1B 30= ,

BEEK2

= (4)

(1)

,

,

EBMB2

= (5)

1 15 = (6)

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= (7)

. =

( (5)), 1Z 15= = ,

.

= (8)

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=, .

, :

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1Z Z 45+ =

15 Z 45 + =

Z 30 =

88.

060 .

,

.


c.php?f=20&t=16968

:

1 ( )

*k -

( )k < ,

3 k = + .

To -

, 60o

:

( 2)180 3 60( 3) 180 180 , 0

o o

o o

Sk k

= =

= .

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, 0 180oS k< < ,

.

0k = , 3 = ,

, .

: -

180o ,

.

2 ( )

- 4

.

n-

3(1800-600)=3600. n=3.

89.

55

www.mathematica.gr 18, 2017 AB , CD ,

OA . S

.

E, CDS

: KARKAR


c.php?f=20&t=17143

( )

1

S

( ).

( CDS) CD :

o

o

o

CSDCED=90 + =2

o(CAD)290 + =2

o(CAD)90 + =ct (1)4

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( ), ..

'( CDS')

(2).

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)

:

CD ( )

.

CD

..

.

CD :

o

o

oo o

o o o

CSDCED=90 + =2

o(CAD)290 + =2

o(CAD) 12090 + =90 + =4 4

90 +30 =120

: 150oCE D =

90.

ABC.

K ,

D , E , Z

BC , CA , AB ,

.

56

www.mathematica.gr 18, 2017 AZ+BD+CE = ZB+DC+EA.

:


c.php?f=20&t=15130

( )

KD

AC F. F

AB, M FG

BC.

AS BC

FG N. () MZ=EF ( FK

AC AB)

AF=2NF=2SD,

( ) ( )( ) ( )

( )

AZ+BD+CE=AM+MZ + BS+SD +CE=

NF+EF + BS+NF +CE=

2NF+ CE+EF +BS=

2NF+CF+BS=3AF+CF+BS= a=2

(

).

AZ + BD + CE = ZB + DC + EA

91.

ABCD

CDEZ BCKL.

KZDB

.

:

http://wwwmathematica.gr/forum/viewtopic

.php?f=20&t=17316

( K)

57
http://www.mathematica.gr/forum/viewtopic.php?f=20&t=15130http://www.mathematica.gr/forum/viewtopic.php?f=20&t=15130http://wwwmathematica.gr/forum/viewtopic.php?f=20&t=17316http://wwwmathematica.gr/forum/viewtopic.php?f=20&t=17316


BK = DZ (1)

BLKC

DECZ

KBC=CKB=45

BCD ,

CBD=CDB= BCD=180 -2

KZC 360 2 90 180 2 = +

KZC=2

KCZ

180 -2CKZ=CZK= =90 -

2

KBD=45 + BKZ=135 -

KBD+BKZ=180

BD//KZ (2)

(1) (2)

KZDB .

92.

(=90).

.

,

, :

1) , , , .

2) =

3) =

: erxmer


c.php?f=20&t=17513

( )

i) ABDZ D=A=90

ii) DZAB DZB DAB=

AD DAB ZAD=

DZAB ZAD ZBD=

DZB ZBD= ZD=DB.

iii) EA ED 90= = D

D DA DE=

AD DAB AD=

D DAB ED=

ED DE= DE=D

58


93.

AD BE ABC ,

H .

HDCE.

D ,

AB.

: KARKAR


c.php?f=20&t=17536

( )

C, H

.

. DM

DBA, :

MDA=MAD=90 -B=HCD,

DM

HDCE (

).

94.

AB, B , , A

AB

K, , M, N :

AK+AN+ + M=2

KM

.

:


c.php?f=20&t=17770

1 ( )

I, KT AB, AD

.

AN+AK+M+=2=DI+NI+AN+AK+KB

AN+AK+T+MT+DI=DI+NI+AN+AK+KB

AN+AK+KB+MT+DI=DI+NI+AN+AK+KB

MT=NI (1)

MTK,

==900, == =

K= =s, N= M=w 0M K 90+ = 0w+s 90=

0K RZK 90+ = 0KRZ 90= (

RZK=OZ=IN ( -- )

59


2. (. )

A AE//KM AZ / /N .

:

2=AK+AN++M=Z+EM++M=B+BZ+-E=

2+BZ-E

.

BZ=E , AE

ABZ , = ,

0EAZ=90 , KM N .

KM=N .

95.

ABC AB=AC

C=640.

By, Cz ABy,

yBC zCB 34, 30 47

.

D By,Cz .

DAB=x .

:


c.php?f=20&t=18373

( )

( ) ABE 112 ,34 ,34 BD E .

( ) ABC 52 ,64 ,64 CAE=112 -52 =60 ACE

AC = AB = AE.

DEC=60 -34 =26 , EDC=77

EDC, EAD .

AD=34

180 -34ED= =732

,

DC=73 -60 =13 x=52 -13 =39 .

60

. : http://mathematica.gr/forum/app.php/portal
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Τεύχος 18 Δεκέμβριος 2017 - mathematica.gr · Α = 150 +450 = 600 και Γ = 300...

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