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O. R. P. Q. 圓的弦線. A. 圓的弦線. 定理 1 (p.124) 由圓心至弦的垂必平分該弦。 i.e. 如果 OR  PQ 那麼 PR = RQ 參考 : 圓心至弦的垂平分弦. O. R. P. Q. 證明 :. OP = OQ ( 半徑 ).  ORP =  ORQ = 90  ( 已知 ). OR = OR ( 相同邊 ). OPR  OQR (RHS). PR = RQ. O. R. P. =. =. Q. 定理 2 (p.124) 連接圓心和弦的中點的直必垂直於該弦。

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O

R

P

Q

圓的弦線

A.圓的弦線

定理1 (p.124)

由圓心至弦的垂必平分該弦。

i.e. 如果OR  PQ

那麼PR = RQ

參考:圓心至弦的垂平分弦


O

R

P

Q

證明 :

OP = OQ (半徑)

ORP = ORQ = 90 (已知)

OR = OR (相同邊)

OPR OQR(RHS)

PR = RQ


O

R

P

=

=

Q

定理 2 (p.124)

連接圓心和弦的中點的直必垂直於該弦。

i.e. 如果 PR = RQ

那麼OR  PQ

參考:

圓心至弦中點的連垂直弦


O

R

P

=

=

Q

ORP +  ORP = 180 (直上的鄰角)

證明:

OP = OQ (半徑)

PR = RQ (已知)

OR = OR (相同邊)

POR QOR(SSS)

ORP = ORP(同位角s ,  )

  ORP =  ORP = 90

OR  PQ


O

R

P

=

=

Q

一條弦的垂直平分必通過圓心。


D

N

C

O

M

A

B

定理3 (p.127)

等弦與圓心等距

i.e. 如果 AB=CD

那麼 OM = ON

參考:等弦與圓心等距


D

N

C

O

M

A

B

AB = CD (已知)

AM=MB=1/2AB (圓心至弦的垂平分弦)

CN=ND=1/2CD (圓心至弦的垂平分弦)

AM=CN

OA=OC (半徑)

OMA = ONC= 90 (已知)

OAM OCN(RHS)

OM=ON (同位邊, )


D

N

C

O

M

A

B

定理4 (p.127)

與圓心等距的弦等長

i.e. 如果 OM = OM

那麼 AB = CD

參考:與圓心等距的弦等長


D

N

C

O

M

A

B

OM = ON (已知)

OA=OC (半徑)

OMA = ONC= 90 (已知)

OAM OCN(RHS)

AM=CN (同位邊,  )

AM=MB=1/2AB (圓心至弦的垂平分弦)

CN=ND=1/2CD (圓心至弦的垂平分弦)

AB = CD


圓心角和圓周角

R

R

R

Q

Q

O

S

S

O

O

P

P

P

Q

S

弧AQB對向兩個角:

POR -圓心角

PQR -圓周角


R

a

Q

b

O

S

P

定理 5 (p.132)

圓心角兩倍於圓周角

i.e. 2a = b

參考圓心角兩倍於圓周角


x

m

y

n

R

O

Q

P

OQ = OR (半徑)

OQR = ORQ = x (等腰底角)

m =2x(外角)

相似地,

n = 2y

a = n+ m = 2x +2y = 2b

POR = 2PQR


R

Q

S

O

x

m

P

y

n

OQ = OR (半徑)

OQR = ORQ = x (等腰內角)

m =2x(外角)

相似地,

n = 2y

a = n+ m = 2x +2y = 2b

POR = 2PQR


R

O

R

m

P

Q

n

O

x

y

P

Q

OQ = OR (半徑)

OQR = ORQ = x (等腰底角)

m =2x(外角)

相似地,

OQP = OPQ = y (等腰底角)

n = 2y (ext.  of  )

a = n- m = 2y- 2x = 2b

POR = 2PQR


R

Q

P

O

定理 6 (p.134)

半圓上的圓周角是直角

i.e. 如界 PQ是直徑,

那麼 PRQ= 90

參考: 圓周上的圓周角


C

x

y

A

B

陰影部份ACB -主要弓形

x -弓形ACB上的角

x , y -在同一弓形上的角


C

x

D

y

A

B

定理 7 (p.136)

在同一弓形上的圓周角相同

i.e. 如果AB是弦,

那麼x = y

參考: 同弓形內的圓周角


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