geometrija 4, slajdovi, 2. deo, 2015
TRANSCRIPT
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http://goforward/http://find/http://goback/
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RP
a
x
+ a
x
x
+ a
x
+ a
x
x
+ a
x
x
+ a
x
= .
http://find/
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RP
a
x
+ a
x
x
+ a
x
+ a
x
x
+ a
x
x
+ a
x
= .
A = (aij )
http://find/
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RP
a
x
+ a
x
x
+ a
x
+ a
x
x
+ a
x
x
+ a
x
= .
A = (aij )
A =
http://find/
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RP
a
x
+ a
x
x
+ a
x
+ a
x
x
+ a
x
x
+ a
x
= .
A = (aij )
A =
x x x a
a
a
a
a
a
a a a x
x
x = ,
http://find/
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RP
a
x
+ a
x
x
+ a
x
+ a
x
x
+ a
x
x
+ a
x
= .
A = (aij )
A =
x x x a
a
a
a
a
a
a a a x
x
x = ,
x T Ax = ,
http://find/
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RP
a
x
+ a
x
x
+ a
x
+ a
x
x
+ a
x
x
+ a
x
= .
A = (aij )
A =
x x x a
a
a
a
a
a
a a a x
x
x = ,
x T Ax = , (AT = A)
http://find/
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RP
a
x
+ a
x
x
+ a
x
+ a
x
x
+ a
x
x
+ a
x
= .
A = (aij )
A =
x x x a
a
a
a
a
a
a a a x
x
x = ,
x T Ax = , (AT = A)
x
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http://find/http://goback/
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x
x
−x
=
http://find/
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x
x
−x
=
λx = Px A
C T
AC , C = P −
.
http://find/
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x
x
−x
=
λx = Px A
C
T
AC , C = P −
.
λx = Cx .
http://find/
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x
x
−x
=
λx = Px A
C
T
AC , C = P −
.
λx = Cx .
= x T Ax =
http://find/
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x
x
−x
=
λx = Px A
C
T
AC , C = P −
.
λx = Cx .
= x T Ax = (Cx )T A(Cx ) =
http://find/
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x
x
−x
=
λx = Px A
C
T
AC , C = P −
.
λx = Cx .
= x T Ax = (Cx )T A(Cx ) = x T C T ACx =
http://find/http://goback/
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x
x
−x
=
λx = Px A
C
T
AC , C = P −
.
λx = Cx .
= x T Ax = (Cx )T A(Cx ) = x T C T ACx = x T (C T AC )x .
http://find/http://goback/
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x
x
−x
=
λx = Px A
C
T
AC , C = P −
.
λx = Cx .
= x T Ax = (Cx )T A(Cx ) = x T C T ACx = x T (C T AC )x .
A = C T AC
http://find/
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x
x
−x
=
λx = Px A
C
T
AC , C = P
−
.
λx = Cx .
= x T Ax = (Cx )T A(Cx ) = x T C T ACx = x T (C T AC )x .
A = C T AC
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x
+ x
+ x
=
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x
+ x
+ x
=
x
+ x
− x
=
http://find/
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x
+ x
+ x
=
x
+ x
− x
=
x
+ x
=
http://find/
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x
+ x
+ x
=
x
+ x
− x
=
x
+ x
=
http://find/
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x
+ x
+ x
=
x
+ x
− x
=
x
+ x
=
x
− x
=
http://find/
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x
+ x
+ x
=
x
+ x
− x
=
x
+ x
=
x
− x
=
http://find/http://goback/
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x
+ x
+ x
=
x
+ x
− x
=
x
+ x
=
x
− x
=
x
=
http://find/http://goback/
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x
+ x
+ x
=
x
+ x
− x
=
x
+ x
=
x
− x
=
x
=
http://find/
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x
+ x
+ x
=
x
+ x
− x
=
x
+ x
=
x
− x
=
x
=
http://find/http://goback/
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x
+ x
+ x
=
x
+ x
− x
=
x
+ x
=
x
− x
=
x
=
A
C ,C − = C T ,
C T AC = C − AC
http://find/
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x
+ x
+ x
=
x
+ x
− x
=
x
+ x
=
x
− x
=
x
=
A
C ,C − = C T ,
C T AC = C − AC = diag (λ
, λ
, λ
),
λi A.
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http://find/
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Γ(A) P = C −
= λ
x
+ λ
x
+ λ
x
http://find/
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Γ(A) P = C −
= λ
x
+ λ
x
+ λ
x
= sgn(λ
)(
λ
x
) + sgn(λ
)(
λ
x
) + sgn(λ
)(
λ
x
) .
http://find/
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Γ(A) P = C −
= λ
x
+ λ
x
+ λ
x
= sgn(λ
)(
λ
x
) + sgn(λ
)(
λ
x
) + sgn(λ
)(
λ
x
) .
x i =
√ λi x
i λi = ,x i λi = ,
http://find/
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39/264
Γ(A) P = C −
= λ
x
+ λ
x
+ λ
x
= sgn(λ
)(
λ
x
) + sgn(λ
)(
λ
x
) + sgn(λ
)(
λ
x
) .
x i =
√ λi x
i λi = ,x i λi = ,
http://find/
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Γ(A) P = C −
= λ
x
+ λ
x
+ λ
x
= sgn(λ
)(
λ
x
) + sgn(λ
)(
λ
x
) + sgn(λ
)(
λ
x
) .
x i =
√ λi x
i λi = ,x i λi = ,
http://find/
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Γ(A) P = C −
= λ
x
+ λ
x
+ λ
x
= sgn(λ
)(
λ
x
) + sgn(λ
)(
λ
x
) + sgn(λ
)(
λ
x
) .
x i =
√ λi x
i λi = ,x i λi = ,
−
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http://find/http://goback/
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, , p ∞ : x
= .
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, , p ∞ : x
= .
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http://find/http://goback/
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RP R̄P
http://find/
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RP R̄P
λu = Ax ,
http://find/http://goback/
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RP R̄P
λu = Ax ,
A, A = , x u
http://find/
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RP R̄P
λu = Ax ,
A, A = , x u
http://find/http://goback/
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8/18/2019 Geometrija 4, slajdovi, 2. deo, 2015
51/264
RP R̄P
λu = Ax ,
A, A = , x u
http://find/
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8/18/2019 Geometrija 4, slajdovi, 2. deo, 2015
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RP R̄P
λu = Ax ,
A, A = , x u
http://find/http://goback/
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8/18/2019 Geometrija 4, slajdovi, 2. deo, 2015
53/264
RP R̄P
λu = Ax ,
A, A = , x u
http://find/
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http://find/
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Γ(A) A, AT = A,
λm = AM .
http://find/http://goback/
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Γ(A) A, AT = A,
λm = AM .
m M
http://find/
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Γ(A) A, AT = A,
λm = AM .
m M
M ( : : ) x
x
− x
= .
http://find/
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Γ(A) A, AT = A,
λm = AM .
m M
M ( : : ) x
x
− x
= . A∞( : : ) B ∞( : : ).
http://find/
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http://find/
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M n nT M = M T n =
http://find/
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M n nT M = M T n =
M (m
: m
: m
) n : n
x
+ n
x
+ n
x
= .
http://find/
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M n nT M = M T n =
M (m
: m
: m
) n : n
x
+ n
x
+ n
x
= .
M
∈ n
http://find/
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63/264
M n nT M = M T n =
M (m
: m
: m
) n : n
x
+ n
x
+ n
x
= .
M
∈ n
⇔ = n
m
+n
m
+n
m
http://find/
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8/18/2019 Geometrija 4, slajdovi, 2. deo, 2015
64/264
M n nT M = M T n =
M (m
: m
: m
) n : n
x
+ n
x
+ n
x
= .
M
∈ n
⇔ = n
m
+n
m
+n
m
= (n
n
n
)m
m
m
http://find/
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8/18/2019 Geometrija 4, slajdovi, 2. deo, 2015
65/264
M n nT M = M T n =
M (m
: m
: m
) n : n
x
+ n
x
+ n
x
= .
M
∈ n
⇔ = n
m
+n
m
+n
m
= (n
n
n
)m
m
m = nT M .
http://find/http://goback/
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8/18/2019 Geometrija 4, slajdovi, 2. deo, 2015
66/264
M n nT M = M T n =
M (m
: m
: m
) n : n
x
+ n
x
+ n
x
= .
M
∈ n
⇔ = n
m
+n
m
+n
m
= (n
n
n
)
m
m
m
= n
T M .
http://find/
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8/18/2019 Geometrija 4, slajdovi, 2. deo, 2015
67/264
M n nT M = M T n =
M (m
: m
: m
) n : n
x
+ n
x
+ n
x
= .
M ∈ n ⇔ = n
m
+n
m
+n
m
= (n
n
n
)
m
m
m
= n
T M .
m
M
n
N
Γ(A).
M ∈ n ⇔ m N .
http://find/
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68/264
M ∈ n ⇔ = nT M =
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M ∈ n ⇔ = n M =
M ∈ n ⇔ = nT M = (AN)T (A− m)
http://find/http://goback/
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70/264
M ∈ n ⇔ = n M = (AN ) (A m)= N T AT A− m = N T AA− m
M ∈ n ⇔ = nT M = (AN)T (A− m)
http://find/
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8/18/2019 Geometrija 4, slajdovi, 2. deo, 2015
71/264
M ∈ n ⇔ = n M = (AN ) (A m)= N T AT A− m = N T AA− m = N T m
http://find/http://goback/
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8/18/2019 Geometrija 4, slajdovi, 2. deo, 2015
72/264
M ∈ n ⇔ = nT M = (AN)T (A− m)
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M ∈ n ⇔ n M (AN ) (A m)= N T AT A− m = N T AA− m = N T m
⇔ m
N .
M m Γ M .
M ∈ n ⇔ = nT M = (AN)T (A− m)
http://find/http://goback/
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M ∈ n ⇔ n M (AN ) (A m)= N T AT A− m = N T AA− m = N T m
⇔ m
N .
M m Γ M .
M
M ∈ Γ(A) ⇔ = M T AM
M ∈ n ⇔ = nT M = (AN )T (A− m)
http://find/http://goback/
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∈ ⇔ ( ) ( )= N T AT A− m = N T AA− m = N T m
⇔ m
N .
M m Γ M .
M
M ∈ Γ(A) ⇔ = M T AM = M T (AM ) = M T m
M ∈ n ⇔ = nT M = (AN )T (A− m)
http://find/
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( ) ( )
= N T AT A− m = N T AA− m = N T m
⇔ m
N .
M m Γ M .
M
M ∈ Γ(A) ⇔ = M T AM = M T (AM ) = M T m ⇔ M ∈ m.
M ∈ n ⇔ = nT M = (AN )T (A− m)
http://find/
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( ) ( )
= N T AT A− m = N T AA− m = N T m
⇔ m
N .
M m Γ M .
M
M ∈ Γ(A) ⇔ = M T AM = M T (AM ) = M T m ⇔ M ∈ m.
M ∈ n ⇔ = nT M = (AN )T (A− m)
http://find/
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8/18/2019 Geometrija 4, slajdovi, 2. deo, 2015
78/264
( ) ( )
= N T AT A− m = N T AA− m = N T m
⇔ m
N .
M m Γ M .
M
M ∈ Γ(A) ⇔ = M T AM = M T (AM ) = M T m ⇔ M ∈ m.
m M .
M ∈ n ⇔ = nT M = (AN )T (A− m)
http://find/
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8/18/2019 Geometrija 4, slajdovi, 2. deo, 2015
79/264
= N T AT A− m = N T AA− m = N T m
⇔ m
N .
M m Γ M .
M
M ∈ Γ(A) ⇔ = M T AM = M T (AM ) = M T m ⇔ M ∈ m.
m M .
m ∩ Γ = {M ,N }, N = M .
M ∈ n ⇔ = nT M = (AN )T (A− m)
http://find/
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8/18/2019 Geometrija 4, slajdovi, 2. deo, 2015
80/264
= N T AT A− m = N T AA− m = N T m
⇔ m
N .
M m Γ M .
M
M ∈ Γ(A) ⇔ = M T AM = M T (AM ) = M T m ⇔ M ∈ m.
m M .
m ∩ Γ = {M ,N }, N = M . N ∈ m n M .
M ∈ n ⇔ = nT M = (AN )T (A− m)
http://find/http://goback/
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8/18/2019 Geometrija 4, slajdovi, 2. deo, 2015
81/264
= N T AT A− m = N T AA− m = N T m
⇔ m
N .
M m Γ M .
M
M ∈ Γ(A) ⇔ = M T AM = M T (AM ) = M T m ⇔ M ∈ m.
m M .
m ∩ Γ = {M ,N }, N = M . N ∈ m n M . n N ,
M ∈ n ⇔ = nT M = (AN )T (A− m)
http://find/
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8/18/2019 Geometrija 4, slajdovi, 2. deo, 2015
82/264
= N T AT A− m = N T AA− m = N T m
⇔ m
N .
M m Γ M .
M
M ∈ Γ(A) ⇔ = M T AM = M T (AM ) = M T m ⇔ M ∈ m.
m M .
m ∩ Γ = {M ,N }, N = M . N ∈ m n M . n N , n = MN = m.
M ∈ n ⇔ = nT M = (AN )T (A− m)
http://find/http://goback/
-
8/18/2019 Geometrija 4, slajdovi, 2. deo, 2015
83/264
= N T AT A− m = N T AA− m = N T m
⇔ m
N .
M m Γ M .
M
M ∈ Γ(A) ⇔ = M T AM = M T (AM ) = M T m ⇔ M ∈ m.
m M .
m ∩ Γ = {M ,N }, N = M . N ∈ m n M . n N , n = MN = m. n = m N = M .
M ∈ n ⇔ = nT M = (AN )T (A− m)
http://find/http://goback/
-
8/18/2019 Geometrija 4, slajdovi, 2. deo, 2015
84/264
= N T AT A− m = N T AA− m = N T m
⇔ m
N .
M m Γ M .
M
M ∈ Γ(A) ⇔ = M T AM = M T (AM ) = M T m ⇔ M ∈ m.
m M .
m ∩ Γ = {M ,N }, N = M . N ∈ m n M . n N , n = MN = m. n = m N = M .
m
M
http://find/
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8/18/2019 Geometrija 4, slajdovi, 2. deo, 2015
85/264
http://find/http://goback/
-
8/18/2019 Geometrija 4, slajdovi, 2. deo, 2015
86/264
M ∈ Γ m ∩ Γ = {M }
http://find/
-
8/18/2019 Geometrija 4, slajdovi, 2. deo, 2015
87/264
M ∈ Γ m ∩ Γ = {M } M m ∩ Γ = {T T }
http://find/
-
8/18/2019 Geometrija 4, slajdovi, 2. deo, 2015
88/264
M m ∩ Γ = {T
,T
}
M ∈ Γ m ∩ Γ = {M } M m ∩ Γ = {T ,T }
http://find/
-
8/18/2019 Geometrija 4, slajdovi, 2. deo, 2015
89/264
M m ∩ Γ {T
,T
}
M
m ∩ Γ = ∅
M ∈ Γ m ∩ Γ = {M } M m ∩ Γ = {T ,T }
http://find/http://goback/
-
8/18/2019 Geometrija 4, slajdovi, 2. deo, 2015
90/264
M m ∩ Γ {T
,T
}
M
m ∩ Γ = ∅
x
x
− x
=
M ∞(− : : ).
M ∈ Γ m ∩ Γ = {M } M m ∩ Γ = {T ,T }
http://find/
-
8/18/2019 Geometrija 4, slajdovi, 2. deo, 2015
91/264
{
,
}
M
m ∩ Γ = ∅
x
x
− x
=
M ∞(− : : ).
,
M ∈ Γ m ∩ Γ = {M } M m ∩ Γ = {T ,T }
http://find/http://goback/
-
8/18/2019 Geometrija 4, slajdovi, 2. deo, 2015
92/264
{
,
}
M
m ∩ Γ = ∅
x
x
− x
=
M ∞(− : : ).
,
CP
M ∈ Γ m ∩ Γ = {M } M m ∩ Γ = {T
,T
}
http://find/http://goback/
-
8/18/2019 Geometrija 4, slajdovi, 2. deo, 2015
93/264
{
,
}
M
m ∩ Γ = ∅
x
x
− x
=
M ∞(− : : ).
,
CP
Γ M ∈ Γ. p M p ∩ Γ = {P
,P
} ∈ CP .
M ∈ Γ m ∩ Γ = {M } M m ∩ Γ = {T
,T
}
http://find/
-
8/18/2019 Geometrija 4, slajdovi, 2. deo, 2015
94/264
{ }
M
m ∩ Γ = ∅
x
x
− x
=
M ∞(− : : ).
,
CP
Γ M ∈ Γ. p M p ∩ Γ = {P
,P
} ∈ CP .
N
H(M ,N ,P
,P
) M Γ.
http://find/
-
8/18/2019 Geometrija 4, slajdovi, 2. deo, 2015
95/264
M
Γ m M.
http://find/http://goback/
-
8/18/2019 Geometrija 4, slajdovi, 2. deo, 2015
96/264
Γ m M .
M
Γ m M .
http://find/
-
8/18/2019 Geometrija 4, slajdovi, 2. deo, 2015
97/264
Γ = Γ(A) M ∈ Γ N
M
Γ m M .
http://find/
-
8/18/2019 Geometrija 4, slajdovi, 2. deo, 2015
98/264
Γ = Γ(A) M ∈ Γ N p = MN M
p : X = λM + N , λ ∈R.
M
Γ m M .
http://find/
-
8/18/2019 Geometrija 4, slajdovi, 2. deo, 2015
99/264
Γ = Γ(A) M ∈ Γ N p = MN M
p : X = λM + N , λ ∈R.
p Γ λ ∈ R :
= X T AX
M
Γ m M .
http://find/http://goback/
-
8/18/2019 Geometrija 4, slajdovi, 2. deo, 2015
100/264
Γ = Γ(A) M ∈ Γ N p = MN M
p : X = λM + N , λ ∈R.
p Γ λ ∈ R :
= X T AX = (λM + N )T A(λM + N )
M
Γ m M .
http://find/
-
8/18/2019 Geometrija 4, slajdovi, 2. deo, 2015
101/264
Γ = Γ(A) M ∈ Γ N p = MN M
p : X = λM + N , λ ∈R.
p Γ λ ∈ R :
= X T AX = (λM + N )T A(λM + N )
= λ
M
T
AM A
+λ (M
T
AN + N
T
AM )
B +N
T
AN C
.
M
Γ m M .
http://find/
-
8/18/2019 Geometrija 4, slajdovi, 2. deo, 2015
102/264
Γ = Γ(A) M ∈ Γ N p = MN M
p : X = λM + N , λ ∈R.
p Γ λ ∈ R :
= X T AX = (λM + N )T A(λM + N )
= λ
M
T
AM A
+λ (M
T
AN + N
T
AM )
B +N
T
AN C
.
M T AN = (M T AN )T
M
Γ m M .
http://find/
-
8/18/2019 Geometrija 4, slajdovi, 2. deo, 2015
103/264
Γ = Γ(A) M ∈ Γ N p = MN M
p : X = λM + N , λ ∈R.
p Γ λ ∈ R :
= X T AX = (λM + N )T A(λM + N )
= λ
M
T
AM A
+λ (M
T
AN + N
T
AM )
B +N
T
AN C
.
M T AN = (M T AN )T = N T AT (M T )T = N T AM
M
Γ m M .
http://find/http://goback/
-
8/18/2019 Geometrija 4, slajdovi, 2. deo, 2015
104/264
Γ = Γ(A) M ∈ Γ N p = MN M
p : X = λM + N , λ ∈R.
p Γ λ ∈ R :
= X T AX = (λM + N )T A(λM + N )
= λ
M
T
AM A
+λ (M
T
AN + N
T
AM )
B +N
T
AN C
.
M T AN = (M T AN )T = N T AT (M T )T = N T AM ⇒ B = N T AM .
http://find/
-
8/18/2019 Geometrija 4, slajdovi, 2. deo, 2015
105/264
λ
, λ
∈ C
http://find/
-
8/18/2019 Geometrija 4, slajdovi, 2. deo, 2015
106/264
P = λ M + N , P = λ M + N ,
p
Γ.
λ
, λ
∈ C
http://find/
-
8/18/2019 Geometrija 4, slajdovi, 2. deo, 2015
107/264
P = λ M + N , P = λ M + N ,
p
Γ.
H(M ,N ,P ,P ) ⇔ − =
λ
:
λ
= λ
λ
λ
, λ
∈ C
http://find/
-
8/18/2019 Geometrija 4, slajdovi, 2. deo, 2015
108/264
P = λ M + N , P = λ M + N ,
p
Γ.
H(M ,N ,P ,P ) ⇔ − =
λ
:
λ
= λ
λ
⇔ λ
+ λ
=
λ
, λ
∈ C
http://find/http://goback/
-
8/18/2019 Geometrija 4, slajdovi, 2. deo, 2015
109/264
P = λ M + N , P = λ M + N ,
p
Γ.
H(M ,N ,P ,P ) ⇔ − =
λ
:
λ
= λ
λ
⇔ λ
+ λ
= ⇔
B =
λ
, λ
∈ C
http://find/
-
8/18/2019 Geometrija 4, slajdovi, 2. deo, 2015
110/264
P = λ M + N , P = λ M + N ,
p
Γ.
H(M ,N ,P ,P ) ⇔ − =
λ
:
λ
= λ
λ
⇔ λ
+ λ
= ⇔
B =
⇔ N T AM =
λ
, λ
∈ C
http://find/http://goback/
-
8/18/2019 Geometrija 4, slajdovi, 2. deo, 2015
111/264
P = λ M + N , P = λ M + N ,
p
Γ.
H(M ,N ,P ,P ) ⇔ − =
λ
:
λ
= λ
λ
⇔ λ
+ λ
= ⇔
B =
⇔ N T AM = ⇔ N T m =
λ
, λ
∈ C
http://find/
-
8/18/2019 Geometrija 4, slajdovi, 2. deo, 2015
112/264
P = λ M + N , P = λ M + N ,
p
Γ.
H(M ,N ,P ,P ) ⇔ − =
λ
:
λ
= λ
λ
⇔ λ
+ λ
= ⇔
B =
⇔ N T AM = ⇔ N T m = ⇔ N ∈ m.
λ
, λ
∈ C
http://find/
-
8/18/2019 Geometrija 4, slajdovi, 2. deo, 2015
113/264
P = λ M + N , P = λ M + N ,
p
Γ.
H(M ,N ,P ,P ) ⇔ − =
λ
:
λ
= λ
λ
⇔ λ
+ λ
= ⇔
B =
⇔ N T AM = ⇔ N T m = ⇔ N ∈ m.
http://find/
-
8/18/2019 Geometrija 4, slajdovi, 2. deo, 2015
114/264
http://find/
-
8/18/2019 Geometrija 4, slajdovi, 2. deo, 2015
115/264
http://find/
-
8/18/2019 Geometrija 4, slajdovi, 2. deo, 2015
116/264
xy − x + y − = .
http://find/
-
8/18/2019 Geometrija 4, slajdovi, 2. deo, 2015
117/264
P , λX = PX
P −T = C T .
http://find/http://goback/
-
8/18/2019 Geometrija 4, slajdovi, 2. deo, 2015
118/264
P , λX = PX
P −T = C T .
http://find/
-
8/18/2019 Geometrija 4, slajdovi, 2. deo, 2015
119/264
n : nT X = n : nT X = .
P , λX = PX
P −T = C T .
http://find/
-
8/18/2019 Geometrija 4, slajdovi, 2. deo, 2015
120/264
n : nT X = n : nT X = .
P , λX = PX
P −T = C T .
http://find/
-
8/18/2019 Geometrija 4, slajdovi, 2. deo, 2015
121/264
n : nT X = n : nT X = .
nT X =
P , λX = PX
P −T = C T .
http://find/
-
8/18/2019 Geometrija 4, slajdovi, 2. deo, 2015
122/264
n : nT X = n : nT X = .
nT X = ⇔ = nT P − X =
P , λX = PX
P −T = C T .
http://find/http://goback/
-
8/18/2019 Geometrija 4, slajdovi, 2. deo, 2015
123/264
n : nT X = n : nT X = .
nT X = ⇔ = nT P − X = ((P − )T n n
)T X
P , λX = PX
P −T = C T .
http://find/
-
8/18/2019 Geometrija 4, slajdovi, 2. deo, 2015
124/264
n : nT X = n : nT X = .
nT X = ⇔ = nT P − X = ((P − )T n n
)T X
P , λX = PX
P −T = C T .
http://find/
-
8/18/2019 Geometrija 4, slajdovi, 2. deo, 2015
125/264
n : nT X = n : nT X = .
nT X = ⇔ = nT P − X = ((P − )T n n
)T X
M m
Γ,
M m
f
Γ
http://find/http://goback/
-
8/18/2019 Geometrija 4, slajdovi, 2. deo, 2015
126/264
f λX = PX C = P − .
http://find/http://goback/
-
8/18/2019 Geometrija 4, slajdovi, 2. deo, 2015
127/264
f λX = PX C = P − . f M m Γ(A)?
http://find/
-
8/18/2019 Geometrija 4, slajdovi, 2. deo, 2015
128/264
f λX = PX C = P − . f M m Γ(A)?
http://find/
-
8/18/2019 Geometrija 4, slajdovi, 2. deo, 2015
129/264
f : λM = PM ,
f λX = PX C = P − . f M m Γ(A)?
http://find/
-
8/18/2019 Geometrija 4, slajdovi, 2. deo, 2015
130/264
f : λM = PM , λm = P T −
m,
f λX = PX C = P − . f M m Γ(A)?
http://find/http://goback/
-
8/18/2019 Geometrija 4, slajdovi, 2. deo, 2015
131/264
f : λM = PM , λm = P T −
m, λA = C T AC = P T −
AP − .
f λX = PX C = P − . f M m Γ(A)?
T T T
http://find/
-
8/18/2019 Geometrija 4, slajdovi, 2. deo, 2015
132/264
f : λM = PM , λm = P T −
m, λA = C T AC = P T −
AP − .
λM = P −
M
, λm = P T
m
, λA = P T
A
P .
f λX = PX C = P − . f M m Γ(A)?
T− T T−
http://find/
-
8/18/2019 Geometrija 4, slajdovi, 2. deo, 2015
133/264
f : λM = PM , λm = P T −
m, λA = C T AC = P T −
AP − .
λM = P −
M
, λm = P T
m
, λA = P T
A
P .
M m Γ(A)
m = AM
f λX = PX C = P − . f M m Γ(A)?
T− T T−
http://find/
-
8/18/2019 Geometrija 4, slajdovi, 2. deo, 2015
134/264
f : λM = PM , λm = P T
m, λA = C T AC = P T
AP − .
λM = P −
M
, λm = P T
m
, λA = P T
A
P .
M m Γ(A)
m = AM ⇔ P T m = (P T AP )P − M = P T AM
f λX = PX C = P − . f M m Γ(A)?
f λM PM λ PT−
λA CTAC PT−
AP
http://find/
-
8/18/2019 Geometrija 4, slajdovi, 2. deo, 2015
135/264
f : λM = PM , λm = P T
m, λA = C T AC = P T
AP − .
λM = P −
M
, λm = P T
m
, λA = P T
A
P .
M m Γ(A)
m = AM ⇔ P T m = (P T AP )P − M = P T AM ⇔ m = AM ,
f λX = PX C = P − . f M m Γ(A)?
f λM PM λ PT−
λA CTAC PT−
AP−
http://find/
-
8/18/2019 Geometrija 4, slajdovi, 2. deo, 2015
136/264
f : λM = PM , λm = P T m, λA = C T AC = P T AP − .
λM = P −
M
, λm = P T
m
, λA = P T
A
P .
M m Γ(A)
m = AM ⇔ P T m = (P T AP )P − M = P T AM ⇔ m = AM ,
M
m
Γ(A).
http://find/
-
8/18/2019 Geometrija 4, slajdovi, 2. deo, 2015
137/264
aij ,
http://find/
-
8/18/2019 Geometrija 4, slajdovi, 2. deo, 2015
138/264
aij ,
M
aij .
http://find/
-
8/18/2019 Geometrija 4, slajdovi, 2. deo, 2015
139/264
aij
aij ,
M
aij .
http://find/
-
8/18/2019 Geometrija 4, slajdovi, 2. deo, 2015
140/264
ij
aij ,
M
aij .
http://find/
-
8/18/2019 Geometrija 4, slajdovi, 2. deo, 2015
141/264
ij
aij ,
M
aij .
http://find/
-
8/18/2019 Geometrija 4, slajdovi, 2. deo, 2015
142/264
j
Γ(A,B ,C ,D ,E )
aij ,
M
aij .
http://find/
-
8/18/2019 Geometrija 4, slajdovi, 2. deo, 2015
143/264
j
Γ(A,B ,C ,D ,E )
aij ,
M
aij .
http://find/
-
8/18/2019 Geometrija 4, slajdovi, 2. deo, 2015
144/264
Γ(A,B ,C ,D ,E )
aij ,
M
aij .
http://find/
-
8/18/2019 Geometrija 4, slajdovi, 2. deo, 2015
145/264
Γ(A,B ,C ,D ,E )
http://find/http://goback/
-
8/18/2019 Geometrija 4, slajdovi, 2. deo, 2015
146/264
a, b , c , d
O
(a, b , c , d ) = ∠aOc
∠cOb : ∠aOd
∠dOb.
http://find/
-
8/18/2019 Geometrija 4, slajdovi, 2. deo, 2015
147/264
∠cOb ∠dOb
a, b , c , d
O
(a, b , c , d ) = ∠aOc
∠cOb : ∠aOd
∠dOb.
http://find/http://goback/
-
8/18/2019 Geometrija 4, slajdovi, 2. deo, 2015
148/264
∠cOb ∠dOb
p O A,B ,C ,D ,
a, b , c , d
O
(a, b , c , d ) = ∠aOc
∠cOb : ∠aOd
∠dOb.
http://find/
-
8/18/2019 Geometrija 4, slajdovi, 2. deo, 2015
149/264
∠cOb ∠dOb
p O A,B ,C ,D ,
(a, b , c , d ) = (A,B ,C ,D ) =
a, b , c , d
O
(a, b , c , d ) = ∠aOc
∠cOb : ∠aOd
∠dOb .
http://find/
-
8/18/2019 Geometrija 4, slajdovi, 2. deo, 2015
150/264
p O A,B ,C ,D ,
(a, b , c , d ) = (A,B ,C ,D ) =−→AC −→CB
:−→AD −→DB
=
a, b , c , d
O
(a, b , c , d ) = ∠aOc
∠cOb : ∠aOd
∠dOb .
http://find/http://goback/
-
8/18/2019 Geometrija 4, slajdovi, 2. deo, 2015
151/264
p O A,B ,C ,D ,
(a, b , c , d ) = (A,B ,C ,D ) =−→AC −→CB
:−→AD −→DB
= P (ACO )P (CBO ) :
P (ADO )P (DBO )
=
a, b , c , d
O
(a, b , c , d ) = ∠aOc
∠cOb : ∠aOd
∠dOb .
http://find/
-
8/18/2019 Geometrija 4, slajdovi, 2. deo, 2015
152/264
p O A,B ,C ,D ,
(a, b , c , d ) = (A,B ,C ,D ) =−→AC −→CB
:−→AD −→DB
= P (ACO )P (CBO ) :
P (ADO )P (DBO )
= AO · CO ∠aOc CO · BO ∠cOb :
AO · DO ∠aOd DO · BO ∠dOb
=
a, b , c , d
O
(a, b , c , d ) = ∠aOc
∠cOb : ∠aOd
∠dOb .
http://find/
-
8/18/2019 Geometrija 4, slajdovi, 2. deo, 2015
153/264
p O A,B ,C ,D ,
(a, b , c , d ) = (A,B ,C ,D ) =−→AC −→CB
:−→AD −→DB
= P (ACO )P (CBO ) :
P (ADO )P (DBO )
= AO · CO ∠aOc CO · BO ∠cOb :
AO · DO ∠aOd DO · BO ∠dOb
= ∠aOc
∠cOb : ∠aOd
∠dOb .
http://find/
-
8/18/2019 Geometrija 4, slajdovi, 2. deo, 2015
154/264
O Γ = Γ(A,B ,C ,D ,O )
O
∈ Γ (OA,OB ,OC ,OD ) = (O
A,O
B ,O
C ,O
D ).
http://find/
-
8/18/2019 Geometrija 4, slajdovi, 2. deo, 2015
155/264
O Γ = Γ(A,B ,C ,D ,O )
O
∈ Γ (OA,OB ,OC ,OD ) = (O
A,O
B ,O
C ,O
D ).
http://find/
-
8/18/2019 Geometrija 4, slajdovi, 2. deo, 2015
156/264
=⇒
O Γ = Γ(A,B ,C ,D ,O )
O
∈ Γ (OA,OB ,OC ,OD ) = (O
A,O
B ,O
C ,O
D ).
http://find/
-
8/18/2019 Geometrija 4, slajdovi, 2. deo, 2015
157/264
=⇒
O Γ = Γ(A,B ,C ,D ,O )
O
∈ Γ (OA,OB ,OC ,OD ) = (O
A,O
B ,O
C ,O
D ).
http://find/
-
8/18/2019 Geometrija 4, slajdovi, 2. deo, 2015
158/264
=⇒
O Γ = Γ(A,B ,C ,D ,O )
O
∈ Γ (OA,OB ,OC ,OD ) = (O
A,O
B ,O
C ,O
D ).
http://find/http://goback/
-
8/18/2019 Geometrija 4, slajdovi, 2. deo, 2015
159/264
=⇒
⇐= (OA,OB ,OC ,OD ) = (O A,O B ,O C ,O D ).
O Γ = Γ(A,B ,C ,D ,O )
O
∈ Γ (OA,OB ,OC ,OD ) = (O
A,O
B ,O
C ,O
D ).
⇒
http://find/http://goback/
-
8/18/2