Luận văn thạc sĩ Phương pháp diện tích

E
2
L ⊂ F F\L F
1
,F
2
F

1
F
1
∪ F

2
F
2
∪
F

1
, F

2
F = F


1
F

2
E
2
S : M → R
∗
+
∼
=
⇒
⇒
S(F
0
) = 1 F
0
M
0
x, y ∈ R
∗
+
f(x
1
+x
2
, y) = f(x
1
, y) + f(x
2

, y)
f(x, y) |
y=const
= g(x)
g(x
1
+x
2
) = g(x
1
) + g(x
2
),∀x
1
, x
2
∈ R
∗
+
R
∗
+
f(x, y) |
y=const
= k.x
F
0
S(F
0
) = 1

S
ABC
= S
AFNB
+ S
FNC
= S
BNMA
= EF.AH =
1
2
BC.AH
S
ABCD
= S
ABD
+ S
BCD
=
1
2
AB.DH+
1
2
CD.BH
1
= AB.DH.
ε F
1
= F ∩ B(H, ε)

[HN) ∩ B(H, ε)\F
1
= ∅.
−→
n
1
l
i
−→
n
i
H
i
k

i=1
l
i
−−→
OH
i
.
−→
n
i
.
H
i
−−→
O


H
i
=
−−→
O

O+
−−→
OH
i
k

i=1
l
i
−−→
O

H
i
.
−→
n
i
=
−−→
O

O

k

i=1
l
i
−→
n
i
+
k

i=1
l
i
−−→
OH
i
.
−→
n
i
.
k

i=1
l
i
.
−→
n

i
=
−→
0 .
HP ∈
−→
AB HQ ∈
−→
BC HR ∈
−→
CA
HP

∈ AB.
−→
n
1
HQ

∈ BC.
−→
n
2
HR

∈ CA.
−→
n
3
ϕ = −

π
2
−→
AB+
−→
BC+
−→
CA =
−→
0 ,
H

i
−−→
OH

i
.
−→
n
i
=
−−→
OH
i
.
−→
n
i
−−→

OH

i
.
−→
n
i
=
−−→
OH
i
.
−→
n
i
−−→
H
i
H

i
.
−→
n
i
−−→
H
i
H


i
−→
n
i
S :M → R
∗
+
S(F) =
1
2
k

i=1
l
i
−−→
OH
i
.
−→
n
i
.
M → R
∗
+
F, F

∈ M, F
∼

=
F

∂|∂(F) = ∂(F

)
ϕ
E
2
∂(O) = ∂(O

) ∂(H
i
) = ∂(H

i
) ϕ(
−−→
OH
i
)
−−→
O

H

i
ϕ(
−→
n

i
) =
−→
n

i
F =F
1
+F
2
AB ∈ L S(F
1
)
1
2
AB
−−→
OH.
−→
n H ∈ AB
S(F
2
)
1
2
AB.
−→
OH
−→
n


∈
−→
n
−→
n

S(F
1
) + S(F
2
)
S(F) = S(F
1
) + S(F
2
).
H
i
1
2
l
i
.
−−→
OH
i
.
−→
n

i
=
1
2
, (i = 1, 2, 3, 4)
S

: M → R
∗
+
∆ ∆ ∆
∆ ∆ ∆
S

(F) =
n

j=1
S

(∆j); S(F) =
n

j=1
S(∆j).
F ∈ M
FρF

ρ
FρF


FρF

F ⊂
o
F

o
F

ρ
F ∈ M FρF FρF

⇒ F

ρF
ρ F, F

, F

∈ M
FρF

, F

ρF

F =
n


i=1
F
i
, F

=
n

i=1
F

i
, F
i
∼
=
F

i
; F

=
m

j=1
G

j
, F


=
m

i=1
G

j
, G

j
∼
=
G

j
.
F

i
, G

j
∈ F

F

i
∩G

j

F

ij
F

i
⊂ F

=
m

j=1
G

j
F

i
=
m

j=1
F

ij
F

i
∼
=

F
i
F
i
m

j=1
F

ij
F =
n

i
m

j=1
F

ij
.
G

j
m

i=1
F

ij

G

j
∼
=
F

j
F

j
m

i=1
F

ij
F

=
m

j
n

i=1
F

ij
.

FρF

S(F) = S(F

) FρF

S(F) = S(F

) ⇔ FρF

∀Φ ∈ J∃F, F

∈ M|F ⊂ Φ ⊂ F

.
M ⊂ J Φ ∈ J
S(F) ≤ S(F

).
Φ ∈ J
(F) = {F|F ∈ M&F ⊂ Φ}.
(F

) = {F

|F

∈ M&Φ ⊂ F

}.

S
∗
= sup(S(F)) Φ ∈ J
S
∗
(Φ)
S
∗
= inf(S(F

))
Φ ∈ J S
∗
(Φ) Φ ∈ J
S
∗
(Φ) ≤ S
∗
(Φ).
Φ
S(Φ) =S
∗
(Φ) = S
∗
(Φ)
Φ Φ

∼
=
Φ Φ


S(Φ

) = S(Φ)
Φ=Φ

+ Φ

Φ

, Φ

Φ S(Φ)= S(Φ

) + S(Φ

)
S(Φ)
S(F)
S(Φ) Φ
Φ
ε > 0
F
0
, F

0
|F
0
⊂ Φ ⊂ F


0
&S(F

0
) − S(F
0
) < ε.
Φ sup S(F)) = inf S(F

) = S(Φ)
ε > 0 ε
1
=
ε
2
F
0
, F

0
S(F
0
) > S(Φ)−ε
1
S(F

0
) S(Φ)+ε
1

S(F

0
) S(F
0
) 2ε
1
= ε
S(F
0
) ≤ S
∗
(Φ) S
∗
(Φ) ≤ S(F

0
)
S
∗
(Φ) S
∗
(Φ) ≤ S(F

0
) S(F
0
) ε.
0 ≤ S
∗

(Φ)−S
∗
(Φ) < ε ε S
∗
(Φ)−S
∗
(Φ) = 0
Φ
F
0
⊂ F

0
F

0
= F
0
+F

0
F

0
F

0
F

0

F

0
Φ
Φ
Φ
(X
n
)
n∈N
; (Y
n
)
n∈N
X
n
⊂ Φ ⊂ Y
n
lim
n→+∞
S(X
n
) = lim
n→+∞
S(Y
n
)
S(Φ) Φ
Φ
Φ X

n
Y
n
n ≥ 3
S(X
n
) =
n
2
R
2
sin
2π
n
, S(Y
n
) = nR
2
tg
π
n
.
lim
n→+∞
S(X
n
) = R
2
lim
n→+∞

sin
2π
n
2
n
= πR
2
; lim
n→+∞
S(Y
n
) = R
2
lim
n→+∞
tg
π
n
1
n
= πR
2
.
πR
2
n
0
 n
0
S =

πR
2
n
360
=
R
2
.
Φ ∈ J
J
0
M ⊂ J
0
⊂ J M = J
0
= J
p =
a + b + c
2
S =
1
2
ah.
S = pr
S =
1
2
absinC.
S =
abc

4R
.
S =
a
2
sinBsinC
2sinA
.
S = 2R
2
sinAsinBsinC.
S =

p(p − a)(p − b)(p − c).
(1.10) → (1.11)
S
ABC
= S
AOB
+ S
BOC
+ S
COA
=
1
2
r. (c + a + b) = p.r.
(1.10) → (1.12)
S
ABC

=
1
2
a.AH =
1
2
a.b.sinC.
a
sinA
=
b
sinB
=
c
sinC
= 2R.

A = 90
0
sinA = 1 =
a
2R

A 90
0
sin

BAC= sin

BDC=

BC
BD
=
a
2R
.

A 90
0

BAC= 180
0
−

BDC sin

BAC=
BC
BD
=
a
2R
.
b
sinB
=
c
sinC
= 2R.
(1.12) → (1.13) S

ABC
=
1
2
ab sin C =
1
2
ab
c
2R
=
abc
4R
.
(1.12) → (1.14) S
ABC
=
1
2
absinC =
1
2
asinC.
asinB
sinA
=
a
2
sinBsinC
sinA

.
(1.12) → (1.15) S
ABC
=
1
2
.2RsinA.2RsinB.sinC = 2R
2
sinA.sinB.sinC.
(1.10) → (1.16) S
ABC
=

p(p − a)(p − b)(p − c).
S =
1
2
a.AH 4S
2
= a
2
.AH
2
= a
2

b
2
− CH
2


CH
2
AB ≥ AC BH
2
= AB
2
− AC
2
+ CH
2
.
BH
2
= (BC ± CH)
2
= BC
2
+ CH
2
± 2BC.CH.
AB
2
− AC
2
= BC
2
± BC.CH, c
2
− b

2
= a
2
± 2a.CH,
4a
2
.CH
2
=

a
2
+ b
2
− c
2

2
.
16S
2
= 4a
2
b
2
− 4a
2
.CH
2
= (2ab)

2
−

a
2
+ b
2
− c
2

2
.
16S
2
=

2ab + a
2
+ b
2
− c
2

2ab − a
2
− b
2
+ c
2


= (a + b + c) (a + b − c) (c + a − b) (c − a + b)
= 2p.2 (p − c) 2. (p − b) 2. (p − a) .
S
2
= p (p − a) (p − b) (p − c).
h =
2S
a
; r =
S
p
; R =
abc
4R
.
S = a
2
.
S = ab.
α
S = ah = absinα.
α
S = a
2
sinα = mn.
S = (a + b)h.
α
S
ABCD
=

1
2
AC.BD.sinα.
S = pr.
S = R
2
π.
n
0
 n
0
S =
πR
2
n
360
=
R
2
.
∆
S
ABC
= S
ABF
= S
BDF
= S
BDC
= S

DCE
= S
CEA
= S
AEF
= S.
S
DEF
= 7S.
∆ABC, M ∈ BC, E ∈ AB, F ∈ AC
  S
BME
= a
2
, S
CMF
= b
2
S
ABC
MB
MC
=
ME
AC
;
MC
BC
=
MF

AB
⇒
ME
AC
+
MF
AB
=
MB + MC
BC
= 1.
S
BME
S
ABC
=
a
2
S
2
=
BM
2
BC
2
⇒
a
√
S
=

BM
BC
.
b
√
S
=
CM
BC
.
a
√
S
+
b
√
S
= 1 ⇒ S = (a + b)
2
.
S
ABQ
S
ARD
S
CRQ
S
AQR
S
BSR

S
DPQ
S
CSP
8 − 2(x + (1 − x)) − 2(y + (1 − y)) − (x + (1 − x))(y + (1 − y)) = 3.


EH =
2
5
AI.
S
EFGH
=
2
5
S
AICM
S
EFGH
=
2
5
S
ABCD
S
AICM
=
1
2

S.
∆BMN