Livro Geometria Euclidiana (2)

8/10/2019 Livro Geometria Euclidiana (2)

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A

B

r s

A B r s

P

r

P r P r

P

r

P /r

P

r

A r B /r
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A B

r

r=

AB

r

A

B

A r

A

B

C

r

A

AB

AC

r

AB

AC

AB A

A B r AB

r

A

B

AB

AB

AB

CD

AB > CD
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A

C

D

A

B

1.

C

E

CD

CE= AB

AB = CE CD

AB

CD

EF GH EF = AB +C D GH= 3 AB

B

A D

C

r


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r X

AB

r

EX

EX= AB

CD r X

F

XF= CD

XEF

GH

3 AB = AB+ AB+ AB

A

B

d(A, B) AB

AB

d(A, B) = AB.

O

r > 0

O r

P r O OP =r

O

P

r

O r

O

r

O

r

O

r

P

O r OP < r
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P

O

r

OP > r

O

P

r

O r

(O; r) O

r

O r

O

OA,OB OP

AB

CD

AB

OAB

P

r

C

D


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CD

CD

CP D

CD

O

A

B

AB

l

O

A

l

O O A

A l

B
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R

A, B

R AB R R

A

BA B

R

A, B R AB

R

r

r

A

B

r AB r=

r

A B

OA

OB

O

OA

OB

OA

OB
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AO

B

A

O

B

OA

OB

AOB

O

360 X Y 360

XOY 1 1

X OY = 1.

O

X

Y

O

360

XOY

1
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O

AB

A

B

O

A, B

A B

OA

OB

AB

AB

O

1

360 AOB

O

A

B

OA

OB AOB

AB

A OB AOB 360

O A

B

A

AOB
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AB 16

AOB

A OB =1

6 360 = 60.

AOB =

AOB

O

r

r

O

R

X

Y
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R O Y

r

X

R

O

XY Y

XOY

AOB

OA

OB

A O B

O AB

A OB = 180

AO

B

180

180

180

AOB

AOB

AOB

0 12X Y s X

Y

C

OC

AOB


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XOC Y OC

OX = OY = r XC = Y C = s OC

XOC Y OC

XOC =YOC AOC=BOC

ABC

BC

A AP A A

BC P BC BC

ABC

AC

AB B C

ABC

A

AB

A

B

r >

1

2AB

r

A B X Y

M

XY

AB

AB

AXY

BX Y

AX= BX AY =

BY XY
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AXY BX Y AXY = BXY A

XM = B

XM

AXM

BX M

AX =

BX AXM =BXM XM

AXM BX M

AM = BM

ABC BC A

A

BC

ABC AC AB B

C

ABC

A

r s r s

s r r s r s

90

rs r s

r

A

s rs A s

r

A

A

r B C
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M BC s =

AM

ABM ACM AB = AC

BM = CM AM

ABM ACM AM B = AMC AM B + AM C = 180

AM B =AM C= 90 AMr

rA

A

r

B

C

r > 1

2BC

B

C A

AAr

ABA ACA

AAB = A

AC AAB + AAC= 180 AAB =AAC= 90

A / r

s r A

A

r

A r

s

r

A

A r A / r

A

r

AP

P

A r

d

A r

d = AP

AP
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r

A

P

d

P

A r

A

P r

P =P

d < AP

ABC BC A

A

A

BC

BC

ABC

AC

AB B C

ABC A

ABC

BC

B =C

A

B CM

ABC

B =C
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M BC BM = CM

AB = AC AM AMB AM C

ABM =ACM
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ABC

A

C

B

ABC

A

C

B

ABC

ABC
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ABC

A

r

A / r

A A r

AAr r AA

A

r A

ABC

AB =c BC =a ma A

ABM M

BC

ABC

AB =c AC=b ma BC

ABC

M

BC

A

AM

AM = AM AM C AM B

AAC

M AA B

CM

BM = CM

ABC

AB =c a BC

BAC =

ABP

P

ABC

BC C

BP

AX

BAX =

ABC

BC

BC M

A

A

r

A r

A = A
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BC ABM

ACM

AM

BAC BMA= CM A BMA + CM A=180 AM

ABC

P M

H

BC P H

M H

ABC

BC

P H ABP ACP ALA M H

O AB M

AB

OMAB AM = BM .

OA = OB
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1
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r A r

A

r

ABC M AC

BM

B

BM= MB

ABM CBM AM= CM BM= BM AM B =CM B

AM B CM B

BCM=BAM XCA > B CA= B CM=BAM=BAC.

XCA > ABC
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A

B C

M

B

X

s

r

A

A

C X r A C

CAY CAY = ACX X Y

AC

s=

AY r

AY r B

C BX

A

B CX

Y
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B

AC=Y

AC=A

CX;

ACX ABC

BAC < ACX,

AY r

r s r s

r A / r


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r

A / r

s

r

A

r s t t r s A B

r

s

t

B

A
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r s = + = 180.

+ = 180 = + = 180

r s =

= r s

r s

s r B

=

180

ABC

XY

BC

A

B =BAX

C=CAY

A

B C

YX

A +B+C=A + BAX+ CAY= 180.


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60

180

60

ACX= 180 C =A+B

A

B CX

90

ABC A 90 B 90 A +B+C >A +B 90 + 90 = 180,


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A

B

C AB

C

A

BC

AB

AC

ABC

ABC

BC= BCA=AB =B

=AB C ABC,

A A B B C C

C=C, AC= AC AB = AB.

A=A

B =B

C= 180 AB= 180 A B =C.

BC= BC; B =B; C=C.
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B C

A


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ABC BC D AB E AC

DE

BC

F

CD BE BF = CF

ADE DE BD = CE

DBC ECB

DCB =EBC

ABC BC

AB

AC

AB

AC

Mb Mc AC AB

BC Mb CBMc

ABC

A= 90

P AC

B P BC

2

AP E

P

BC

BAP BE P

AB

CD

ABC BC D 3x20 x + 40 x

A

D

B

C


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r s =

r

s

t

r s

+ =

r

s

r s

ABC = 20 BCD = 60 DEF = 25

AB

EF

CDE

A BC

D

= DAB+ ABC+ BCD
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A

D

C

BD

ABD

ACD

A B C D E

A

B

D

X AB CD Y

AE

CD

BC X DEY

n

n 2

n 3

180(n2)

360
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ABC M BC

AM = 12BC

BAC = 90

ABM ACM

ABM = BAM = ACM = CAM =

ABC

I

B C ABC BIC= 90 + 12

BAC

B = 2

C= 2

BIC BAC

ABC

A

B C

A

ABC

Ia

B C BIaC= 90 12BAC

ABC BC D BC

E AC AD = AE BAD = 48 CDE

B =C = CDE=

ADE AED

ADE=AED

ABC

BC

D F

AB E AC BC= CD= DE= EF = F A

BAC A =

AEF DEF

CDE BC D

ABC

ABCDEF

AD BE

CF

M

A+B +C = 180
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1
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ABC

B > C

AC > AB

B > C

BX

ABC

C BX= 1

2( B C)

P

BX

AC

A P B =C BP+ B CP=1

2( B C) + C=

1

2( B+ C).

A BP = B 1

2( B C) = 1

2( B+ C)

ABP

B C

A

P X

BP

AB = AP < AC.

ABC

A 90

BC
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A 90 A ABC

BC

ABC

AB = c

AC=b

BC=a

a < b+ c

D

CA

A CD

AD= AB

B C

A

D

CD= AC+ AD= AC+ AB =b + c,

B DC < D BC

B DA = D BA

B DC=B DA = DBA < D

BA + A

BC=D

BC.

a

b

c

a < b + c, b < a + c, c < a + b.
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a

b

c

P

ABC

P B+ P C < AB+ AC

P A + P B+ P C < AB+ AC+ BC

BP

AC

Q

CP Q

ABQ

P B+ P C < P B+ ( P Q + CQ) = BQ + CQ

< ( AB+ AQ) + CQ= AB+ AC.

B C

A

QP

P A + P B c3 a3 + b3

a + b > c

ABCD

P

P A+ P B + P C+ P D

ABCD

P AC

P BD

n 3

n

n 1

n 3
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A1A2 . . . AkAk+1 k 3

A1A2 . . . Ak

A1AkAk+1

r

s

B r

C s

AB+ BC+ CD

r

s

A

D

A

D

A

D

r

s

B

C

AD

r s

B r C s

B = B

C = C

AB+ BC+ CD < AB + BC + CD

ABC

B

AB > BC

P

ABC

P A +P B + P C < AB +AC

P

QR

BC

Q AB

R AC

AP < AR

BP+ P C < B Q + QR + CR

A

B

C

B AC > 60
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1


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A B

D C

ABCD

AB

CD

AD

BC

ABCD

AD

BC A B

AB A +B = 180

B+C= 180 A= 180 B =C B =D ABCD A =C B =D

A +B =C+D

A +B+C+D= 360

A +B =C+D= 180 A+D =B +C = 180

A+B = 180

AD

BC B+C= 180
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A B

D C

A =C B =D ABCD

AB

CD

ABCD

ABCD

A=C

AD

BC

A

DB =C

BD ABD CDB

AB = CD

AD= BC

A B

D

ABCD

AB = CD AD = BC

ABCD AB = CD

AD= BC

ABD

CDB

ADB = CBD ABD = CDB

AD

BC

AB

CD
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A B

D C

AB = CD AD = BC ABCD

r A / r

r A

A

r

A

r B C

A

BC

C

D

A

r

ABCD

AD r


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a b

a b

r

AB

b

BAX

AX

D

AD= a

D

AB

B

AD

C

ABCD

ABC a BC

A

B


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a

r BC

a

CBX

BX Y Y

C

BX

C

XY A

BX

ABCD

M

AB

CD B

AM = D

CM

ABM =CDM AB = CD ABM

CDM

AM= CM

BM= DM

ABCD

AC

BD

M

M A= MC M B = M D AM B = CM D

ABM CDM

BC M DAM
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A B

D C

M

ABCD

AM = CM BM = DM

AB = CD

BC= AD

ABCD

ABC

M N

N P

MP

M N A BC

NP M P

ABC

B C AB AC

B C

A

M N

P

MN P ABC

ABC
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ABC

MN

ABC

BC

M N

BC M

AB

BC

AC N

M N=1

2BC.

M

M N

MN= NM

N

AC

AN M=CNM

AM N CMN

M

C = MA

MCN = MAN

MC

AM

BM=

AM=

MC

BM=

AM

MC.

MBCM

BC

MM =

M N BC= M M = 2 M N.

B C

A

M N M

r M AB

BC

M N

M

BC r

MN

Nr
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ABC

M

N P BC CA AB

M

P

M

r

NP

N s

MP

P t

MN

s t= {A} r t= {B}

r s= {C}

2 : 1


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N P AC AB

BN CP = {G1} S T

BG1 CG1 N P

ABC

BC

ST

BC G1

BC NP ST

BC

BC

NP = ST

N P

ST NP ST

P G1= G1T N G1= G1S

BS = SG1 CT = T G1 BS = SG1 = G1N

CT = T G1 = G1P BG1 = 2 G1N

CG1= 2 G1P

A

B C

P N

S T

G1

M

BC

G2

AM BN G2 AM BN

2 : 1

G1 G2

BG1 = 2 G1N BG2 = 2 G2N

G1G2 G G1G2 AM BN

CP G G 2 : 1

ABC G
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A B

CD

ABCD

AB

CD

AB CD

BC

AD

ABCD

ABCD

E

F

AB

ADCE

BDCF

BC E BE= AB CD CE= AD BCE=

AD BC

ACF AF = AB+ CD CF = BD A

CF =

AC

BD


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A B

CD

E F

ABCD

ABCD

AB

CD

AC BD 135

AB =a

AC=d1 BD = d2

ad1

d2

ACF

AC=d1

CF=d2 ACF= 135

B

AE

AB =a

B

r

CF

C

s

AF

D

r

s
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ABCD

AB CD

AD

BC M N AD

BC P Q AC BD

(a) M N P Q

MN

AB,

CD

(b) M N= 12( AB+ CD) P Q= 12 | AB CD|

A B

CD

M NQP

a

b

M P

DAC

M P

CD M P = b2

MQ

ADB

MQ

AB M Q= a2

AB

CD

MP=

M Q M P Q

P Q= M Q MP=a

2

b

2=

a b

2 .

N Q

N P

CBD ABC P Q N


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N Q= b2

M N= M Q + NQ=a

2

+b

2

=a + b

2

.

360

90

A B

CD

E

F

G

ABCD

EFGH

r s

t

r

s

r

s

r

s

P Q Pr Q s

P Qr, s
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P, P r Q, Q

s

P Q,

PQr P QQP

90

P Q= PQ

r

s

P

Q

P

Q

r

r

r

d

r

d r

A r A tr

t

B

B

AB = AB

=d

B B s s r

s

s
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ABCD

AC

BD

DAB =ADC= 90 ABCD AB = DC

DAB ADC

AD

AC= BD

ABCD

AC=

BD AB = DC DAB

A B

D

ABCD

AC = BD

ADC

AD

DAB = ADC ABCD

DAB +ADC = 180

DAB = ADC = 90

ABC=DCB = 90 ABCD

ABC A B

AC

C

AB

D

BAC+ ABD = 180 BAC= 90 ABD = 90

ACD = 90

ABDC 360

BDC= 90 ABDC
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A B

C D

M

AD = BC

M

AD

BC

BC= AD= 2 AM

ABC

m

h BC

m h

BC = 2m

BC

M

r

BC

h

r

A

r M m
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EFGH EG F H

EF = EH

GF = GH

EF G

EH G

M

EG

F H

FEM=FEG= HEG= HEM. EM F EH EF H

F H EM

F H

F H

EM=

EG

EFGH

EG

F H

EG

F H

M EFGH EHG

HM

EG

EH = GH

EH= F G EF = GH

E

F

GM

EGFH EFGH
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A B

CD

ABCD

45

T

P

R

L

Q

R L P T

R L= Q,


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AB CD

ABCD

AC BD = {M}

ABM CDM

ABCD

ABCD

r

G

ABC

A B C

B

C

r

A

r

M BC P Q

A M r

AP= 2 M Q

R

S

R

AG

S

R

r

RSG M QG

ABC

a

BC

ma mb

BC AC

M BC G ABC

BGM C

M G

A

G AM AG= 2 GM

3

2

3

4

G

Ma Mb BC

AC AB = c AC=b BC=a


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MaGMb

2

3(ma+ mb)> c2

2

3(mb+ mc)>

a

2

2

3(ma+ mc)>

b

2

O AB

AP

Q

P

AQ OQ BP = {R}

RQ

RO

R

AQB

ABCD

AB = 7

CD= 3

AD BC A = 43 B = 47 M N

CD AB M

P Q

AB

MP

AD

APMD

BCMQ

MMQ= 90

P Q

M N

P M Q

ABCD AC BD

r

ABCD

A B C r 2 3 6

D r M

ABCD

A C r

B

D

r

AB CD a b

a > b

AD

BC

BC D= 2DAB BC=ab CEAD


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EAB AECD

BC E

CE

ABCD

AB

CD

BC

A

70

C E AB BE= BC

AECD

a b c d

ABCD AB CD BC

AD AB =a BC=b BC=c AD =d C

AD

AB

E

EBC

ABC

A

BC= 2 AB

M

BC

ABM

ABC

M

BC

Hb

Hc AC AB

MHbHc

BC Hc BC Hb

ABCD AC BD E

CD

AE= AB+ CE

F

CD

EAB = 2 FAD G H G

BC

H

EG

AB

BGH CGE

AEH

EH

ABG

ADF
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P

P L

(a)

L

P

(b)

P

L

L P L

P

r O

r

O

O

r

AO= r A (O; r).

d r s s r

d

r


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O

A

r

A B

AB

AB

AB

A

B

r > 1

2AB

r

A B X Y

XY

AB

M

XY

AB

M AB

XAB

AB

XM

XM XAB

XY AB AB

AB


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A

B

AB

A B

M

m

AB

Pm

P AB P M

P AB

AB

P A= P B

m

A

B

P

M

P(

AB) PA = PB

P P A= P B

P AB AB

AB P AB

AB

P M

P MAB

P M AB

A

B

M

PA = PB P

AB
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AOB

P

d(P,

AO) =d(P,

BO)P ( AOB).

P AOB

M N P

AO

BO

OM P ON P

M OP =N OP O MP =O NP = 90 OP

P M= P N

d(P,

AO) =d(P,

BO)

A

B

O P

N

P

AOB d(P,

AO) =d(P,

BO)

P AOB

P M = P N

M N

P

AO

BO M OP NOP

OP

P M = P N

M OP =N OP

P AOB

r s O

P

r

s

P

r

s
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r

A B

r A

AB

AB = 2r

AB P O = R QP O = 90 OP Q

Q /

P

t

t

O

P

Q

r

P
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O

P

OP

P r

OP

P P

P

O

O

OA

OB

AOB

AOB

AOB

AB

A B C D

AOB COD AB = CD

AOB = COD < 180 AOB =COD >180

AO= CO

BO =

DO AOB = COD AOB COD

AB = CD


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A

B

CD

O

AB AC O

BAC

BOC

BAC O

OAC

OAB

AC

AB

OAC = OCA = OAB = OBA = BAC= +

COA = 2 BOA = 2

BOC=BOA + COA = 2(+ ) = 2BAC.

BAC

O

OAC OAB AC AB OAC=OCA= OAB = OBA = BAC =

COA = 2 BOA = 2

BOC=BOA COA = 2( ) = 2BAC.
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A

B

C

OA

A

B

C

OA

O

BAC

O AB

AB

P A B

AP B = 12

180 = 90.
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A B

P

O

B

AO

BAC

AOB

BAC= AC AO ABO = BAO= 90 B

OA = 180 2(90 ) = 2= 2B

AC.
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A

BC

D

O

E

AB

CD

E

(a) E

AEC

AC

BD

(b) E

AEC

BD

AC

AEC=ADC+BAD= 12

AC+1

2

BD.
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A

B

C

D

AC

BD

M N P Q

AB

C

BC

D

CD A

AD B

M P

NQ

AB = 2

BC= 2

CD = 2

AD = 2 + ++ = 180

M N = +

P Q= +

E

M P NQ

M

EN=

1

2(

MN+

P Q) =1

2((+) + (+)) = 90.

A

BC

D

M

P

Q

E

AB

0 <


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AB P R P / R

P R

APB = AP B+ PAP > AP B =,

P

= 90

AB

AB

A B

P

O

90

AB

P

AP B =

90 O AB

P O= 12AB = AO P

O

AB

90 <


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O

A

B

90 AB

90 <


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A B

O

AB

O

O

OA = OB

P

P

O A B

OP

90 OP OAP = OBP = 90 OA

AP OB

BP

AP

BP

r

P

X OX

XP OXP = 90

X

90

OP

X

OP X

OP

X=A

X=B
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O

P

A

B

P

O

P

M

OP

M

OM= MP

A

B

AP

BP


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O

P

A, B

P A

P B

(a) P A= P B

(b)

P O

AB

(c)

P O

AOB AP B

(d)

P O

AB

O P

A

B

OA = OB

PAO = PBO = 90

P OA

P OB

P A = P B

AP O = BP O

A

OP =B

OP

P

O

A

B

P O AB

P O

AB
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P

P

s

P t

O

s

t

Q s O

t

POQ = 2

Q

r

A r

r

A

O

AOr

A

O

s

r

A

r s P r

r

s

P

r

O

O r P

O

r

s

R r

R r

O

O

r R

r s R > 0

R

r

s O

O

R r

a b c a b c

a b

a

b

c O
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O a b

a

c

1(O1; R1) 2(O2; R2) 1 2

O1O2> R1+ R2

O1O2= R1+R2

|R1 R2|< O1O2< R1+ R2

O1O2= |R1 R2|

O1O2< |R1 R2|

P1 2 P O1R1

P O2 R2

|R1 R2| O1O2 R1 +R2

O

A

A

O

R O

AO\ {A} R = AO

A

O

R

r
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r

R

r

O r

OO =R r

A P Q P, Q

AP

AQ

5

B AP C AQ BC

ABC

R

BC BR = BP CR = CQ

ABC

AP+ AQ

ABCD a A

a

M N

BC

CD

MN MAN

P

MAP= 12

BAP NAP= 12

DAP

r s

A

O P r Q s

P Q A

O

PAQ = 30 POQ B C R

AP

AQ

P Q

POR = 12

BOR

QOR = 12

COR

ABOC

360

A B

X

Y

A X Y

XBY

XY

XBY = 180 BXY BY X

AXB AY B

XY

AB CD E

AC

E

BD F F


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BD

DEF

BE F

A

B

C

AB

AC

BC

120

P

BC P A= P B+ P C O

AP

P Q= BP

BP Q

AQB = 120

ABQ CBP

AB

90 <


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AB AD AC

MAP = NAP = 45

A

45 MP NP

B

AM D

AN

AB = AD= l

ABC B C

BAC

BC A A

BC

I ABC

r

O

O

R

R

T T r

R > R s r O

S OT s OO S S

OS=R R

S

T

OS

r

s

T

1(O1; R1)

2(O2; R2)

A

B

l

A

1 2 X

Y X, Y = A XY = l

M

N

AX AY


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M N= l2

O1O2P P

O1P M N O1O2

l

2

1(O1; R1) 2(O2; R2)

A B

A

ABC

a

MN P a A N P B M P C M N

120

ABC

ABC

M NP

A NP

B MP

C M N

ABC

ABC B C

A l

AB

AC

A A

BA

BA = l

BAC = 2

A

B

l

2 BC A AB

A

C

A

B

C

AB > AC M

BC A
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N M

AB

BN = AN+ AC

A

BA \ AB AA = AC

BAC = 1

2BAC

M M B = M C

A

M N

AB

BN= AN
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ABC O O

A B

C

O

s

t

r

OA = OB = OC

R O

R

A,B,C

ABC

ABC

ABC

O

O

R
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B C

A

O

X

O

ABC

B C

A

M

O

X

O

ABC

ABC

A

ABC

O

M

AB A

OM=B

OM=A

CB

(a)

AOM=BOM=12

AOB = ACB.


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I ABC I

ABC

I

ABC r

I r ABC

A

B C

I

ABC

A

C

I ABC
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r I BC

M r BC

I

IM

O

ABC

O OA

ABC BC

AB

AC

BC ABC BC

A

r s B C ABC

Ia

r s BAC

BC

r

s

Ia r r

d(Ia,

BC) =d(Ia,

AB).

Ia s d(Ia,

BC) =d(Ia,

AC)

ra Ia

Ia ra BC AB

AC

ABC

O R

I

r

BC

Ia ra


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A C

B

Iar

s

BC

ABC

ABC

AC

AB

Ib, Ic rb, rc

ABC AB = c BC = a CA = b

p D E F

ABC

BC

CA

AB

BC M

AC

AB

N

P

(a) BD = BF=p b CD= CE=p c AF = AE=p a

(b) AN= AP=p

(c) BM= BP =p c CM= CN=p b

(d) EN= F P=a

(e)

BC

DM
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EN=a F P=a

EN= AN AE=p (p a) =a.

CM= BD

ABC I Ia

BC M ABC

IIa M BC

A

M B = M C= M I= MIa.

A

C

B

I

M

Ia

MAB =MAC= 1

2A

M B

M C A M

M B = M C

BM I=BM A= BCA=C IBM = IBC+ CBM=1

2B+ CAM

= 1

2B+1

2A.
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B

IM = 180 I

BM B

M I

= 180 12B 1

2A C

= A +B+C 12B 1

2A C

= 1

2B+1

2A= IBM,

IBM

BM

IM= BM=

CM

BM = MIa


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ABC R

a

b

BC

AC

ABC M N

A

ABC

M N

ABC

M

N

P

ABC A B

C

M = A

N = B

P =C ABC MN P

a, b c

a, b c

ABC

H Ha

A

HHa

BC

P

AH ABC HHa =

HaP

BHaP BHaH

ABC B C

BAC

BC A A

BC

H ABC

ABC

B

C

BAC

BC

A

A

BC


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Ia BC

ABC

A

H

BC I1 I2 ABH

ACH

A

I1HI2

I1I2

ABCD

M , N , P , Q

AB BC

CD

DA

ABC

Q

R

BC ABC

ABC

BC

P

B ABC A QP R=90

M

BC

P M AB =c

AC=b

QR

a

b

ABC p

a

BC ra BC

ANIa

N

AN =p N Ia =ra

BC

Ia ra

A AN

E

ABC

AC

AE = a I ABC

AIa AN E

ABC

I

IE

B

C
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1


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http://-/?-


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ABD

C

ABD

A

D

C

B

ABCD

AB BC

CD

DA

(a) D AB + B CD= 180

(b) B AC= B DC

ABCD

B AC= B DC

D AB + B CD=1

2

BC D +1

2

BAD = 180.


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A

B

E

D

C

B AC+ B DC= 180 ABCD

ABC

ABC

A

BC

Hc

Hb

Ha

H

HaHbHc ABC

H HaB +H HcB = 90

+90 = 180

HHaBHc

H HaHc= H BHc= Hb BA = 90 A.
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H HaC+ H HbC= 180

HHaCHb

H HaHb= HCHb = Hc CA = 90

A.

H HaHc = H HaHb HHa

HcHaHb HHb HHc

H ABC HaHbHc

ABC

P

ABC D E F

P

BC

CA

AB

DEF

P

ABC

D

E

F

ABC

P

P

ABC

P ABC

P

ABC

P

BAC

ABC BC A

ABC

P

ABC

P

ABC

P

ABC

ABC

D

E

F

P

BC

AC

AB

D E BC AC
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A

B C

PF

E

D

F

AB

P F A = P EA =

90

PFAE

PEDC

A

P C D

P F= D

P C F

P A = D

EC F

EA,

A P C= D P F D EC= F EA D, E F .

D P F= 180 A BC

A P C= D P F A P C+ A BC= 180 ABCP

.

P

ABC

D

E

F

P ABC


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ABCD AB BC CD DA

AB + CD= AD + BC .

ABCD

M , N , P , Q

AB

BC

CD

DA

ABCD

A

B C

DM

N

P

Q

ABCD

AB + CD = ( AM+ M B) + ( CP+ P D)

= AQ + BN+ CN+ DQ

= ( AQ + DQ) + ( BN+ CN) = AD + BC .

AB+ CD = AD + BC

ABCD

AD AB BC ABCD

CD

E

AD

CE

E
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A

B C

ED

ABCD

A

D

AB

+ CE

= AE

+ BC

AB

+ CD

= AD

+ BC

CD CE= AD AE= DE

CD= CE+ ED

CDE


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ABCD

E

M

N

P

Q

E

AB

BC

CD

DA

M N P Q

EPCN

ABCD

PEQD

N P E= Q P E

M N P Q

ABC

ABC

ABC O Ha Hb

Hc BC CA

AB

A HbHc= A BC A HcHb = A CB

OA

HbHc

ABCD

BD

M

AMBD

N

A BD

AC N

CD P

BC

Q

C P M Q

B CD = 90

M BQ = M N Q

MNBQ

M N B = 90

M DP =M N P

MDNP

M P D= 90

ABC

P

AC B D

P

BC

Q = P

DP r


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P

ABC

r

AQ

P QA = P CA

ABC P P

AC B r

r

P P

ABC

r

r

P P A

ABCD

ABC

ACD

AC

A1A2A3 . . . A6

A1A2+ A3A4+ A5A6= A2A3+ A4A5+ A6A1.

A B C

D

ABCD


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AB = a BC = b D

AC

B

CD= x

AD= y

a=b a < b

xy = ba > 0

E CD

CE= ba

E

C

b a

ADE

ABCD

E

180 12

A BC AC


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1


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r s t

u

u

r

s

r

s

t

u u

A

B

C

X

Y

Z

A

B

C

X

Y

Z

t

A

B

C

r

s

t A B C

AB = BC

AB = BC

AB

BC= 1

AB

BC = 1.
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AB

BC

2

3

AB

BC

X Y Z u

AX= XB = BY = Y Z= ZC

X

Y

Z

r

s

t

u X Y Z

AX = XB = BY = YZ = ZC

ABBC

=23 A

B

BC =2

3.

AB

BC = m

n

m, n N

AB

BC

m

n

AB

BC=

m

n

AB

BC =

m

n.

AB

BC= A

B

BC

AB

BC=x,

x

(an)n1

x < an< x +1

n

n N Cn u

AB

BCn=an.
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r

s

t

tn

u u

A

B

Cn

C

A

B

Cn

C

AB

BC

tn r s t Cn Cn tn

u

an Q

AB

BCn=an.

x < AB

BCn< x +

1

nx 90

D

B

AC

D

AC

AD Cx

c ah

b


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AD= x = 180 A

A

BDC

BDA

a2 =h2 + (b + x)2 =h2 + b2x2 + 2bx

h2 =c2 x2

h2

a2 =b2 + c2 + 2bx

cos = xc cos A =xc x= c cos A

a2 =b2 + c2 + 2b(

c cos A)

a2 =b2 + c2 2bc cos A

A

a

A

a2 = b2 + c2 2bc cos A

cos A= 0

a2 =b2 + c2

b2 =a2 + c2 2ac cos Bc2 =a2 + b2 2ab cos C


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7

5 6

72 = 52 + 62 2.5.6. cos

cos = 1

5

=78, 5

cos = 1

5

sen = 26

5

ABC

S=

1

2 AB.AC. sen A

S= 1

2.5.6.2

6

5 = 6

6

A 90 a2 > b2 + c2

a

b

c

a

a2 b2 + c2


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ABC

a

b

c

R

B

A

C

D

a

2R

BAC

A

BD

BC D

BAC

BDC

BC

BDC

BCBD

= a2R

sen A= a2R

asenA

= 2R

ABC

a

sen A=

b

sen B =

c

sen C = 2R

R

ABC


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cos = 15

sen =26

5

7

26/5

= 2R

R=35

6

24=3, 57 .

A

B

C

BAC= 119 ABC= 52 A B

A

C

ABC

C

180

C= 9

AC

sen52 =

1

sen9

AC= 5, 04km


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ABC

AB

BC

CA L M N

LA

LB.

M B

M C.N C

N A= 1 .

B C

A

P

s

t

L

N

M

ABC

t

s

t

A

P

s

BC

s

t

BA

BP

LAMP =

LBMB

CA

CP

MPNA =

MCNC

M P

LANA

= LBMB

MCNC

LA

LB

M B

M C

N C

N A= 1 .

ABC

P

AC

Q

BC

AP= 1

3AC

BQ = 2

3BC

AQ

BP

J

JAJQ


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B Q C

A

2a a

b

2bJ

P

AQC

BJ P

B Q C

A

2a a

b

2bJ

P

J A

JQ.

Livro Geometria Euclidiana (2)

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Transcript of Livro Geometria Euclidiana (2)