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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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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.