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Замечательные. Замечательные. точки и линии. точки и линии. треугольников. треугольников. учитель: Шевелева Ольга Ивановна; г.Москва. учитель: Шевелева Ольга Ивановна; г.Москва. СОДЕРЖАНИЕ. Ортоцентр. Точка Торричелли. Точка пересечения медиан. Прямая Симпсона.

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6269271

: ; .

: ; .


6269271

,


6269271

B

H

C1

A1

C1

B

H

A1

A

C

B1

B1

C

A

A1, B1, C1 ABC;

H ABC


6269271

C

B

A1

C

C1

B1

1

1

A

A

1

B

A1, B1, C1 ABC;

, ABC


6269271

B

P

V

M

A

C

N

M, N, P ABC;

V , ABC


6269271

AG : GA1 = 2 : 1

C

BG : GB1 = 2 : 1

CG : GC1 = 2 : 1

1

1

G

A

B

1

A1, B1, C1 ABC;

.G ABC.


6269271

,

C2

A1, B1, C1 ;

H ABC

B

C1

A2, B2, C2 , . ABC

H

A1

2

A

, 2, , 2, , 2

?

B1

C

B2


6269271

,

, .2 , ABC

:

B

  • 2.

2. A1HB = A1A2;

3. A1HB ~ B1;

4. 2. 3.: A1A2 ~B1;

H

5. 4. : LA1A2 = LB1;

2

A1

6. ;

A

7. -, LA1A2 LB1 , .2 , ABC.

B1

C

H ABC

...

A2 , . BC


6269271

,

1 = 12

H

B1 = B1B2

C1 = C1C2

C1

B

A1

C2

2

C

B1

, 2, , 2, , 2

?

A

2


6269271

,

H ABC

H

1 = 12

:

B1 = B1B2

, . 2 , ABC.

C1

C1 = C1C2

B

1. 2.

A1

2. AHB! = A21;

C2

2

3. AHB1 ~ 1 (.. 1 ~B1,

, LA1 = L1);

C

4. 2. 3. : A21~B1;

B1

5. 4. : LA2 = L ;

6. ;

A

7. -,LA2 L , , .2 , B.

...

2


6269271

.

C1

,

B

, AB1C, AC1 B1C,

A

M

?

A1

B1

C


6269271

C1

:

B

1. 1 1.

A

M

?

2. LAMC=1200

A1

3 LBMC=1200.

4.,LAMB=1200.

B1

5. LAMB + LAB =1800., . , 1.

C

...


6269271

:

B

A

LAMC=1200

M

?

120

  • 1 . , 180.

  • 1 . , 180.

A1

60

B1

C

2. .. L1 + L1 = 180

  • L1 = 60

  • -, L1 = 180 - 60 = 120.


6269271

,

C

,

F

P

E

?

D

A

B


6269271

:

C

1...LCFP =LCEP = 90,

CFP .

, LCEF = LCPF , .

F

?

P

2. LCPF=90 - LPCF= =90 - LDBP = LBPD.

3. .. LBEP = LBDP = 90,

BEPD .

E

A

B

D

LBPD = LBED.

4. -, LCEF = LBED.

5. , D, E, F .

...


6269271

?

2. LCPF=90 - LPCF= =90 - LDBP = LBPD.

C

:

LPC LBP.

) AP, AP.

F

P

,

LPC =180 -LBP.

) LBP LDBP .

E

,

LDBP = 180 - LBP

D

A

B

) , LPC = LDBP, ..

LPCF=LDBP

, LCPF=90 -LPCF=90 - LDBP = LBPD.


6269271

,

B

D

:

- 1,1 , 1;

Y

A1

C1

- D, E, F;

E

H

- AH,BH,CH X,Y,Z

X

A

Z

F

B1

C

?


6269271

?

:

B

1. .. 1=1 =, 1 II BF.

2. .. 1=1 1=1, 11IIAC.

D

C1

3. .. BF AC, 1 11.

Y

4. , 1 11.

A1

H

E

X

5. 1, 1, 1, .

A

Z

6. .. XD DA1, X, D, A1, B1 .

?

B1

F

C

7. , X D , 111.

8. , Y, E Z, F .

...


6269271

?

B

1. .. 1=1 =, 1 II BF

D

1.- .

C1

Y

, 1 II BF .

A1

H

E

X

A

Z

B1

F

C


6269271

B

1, 1, 1, .

?

D

,

1 11 1 11

C1

A1

H

E

, 111. 180

.. L11 + L11 = 180

X

A

B1

, 111. .

F

C

1, 1, 1, .


6269271

,

B

A1, B1, C1

D

:

Y

A1

- H,

C1

O

N

G

- G,

H

E

X

- .O

A

Z

F

B1

C

?


6269271

:

. - . OG BF.

.O- -

G .

1, 1

BF

B

D

,

.

C1

G

N

H

:

?

1... BF II OB1,

BGH ~ B1GO.

F

2. - HG:GO=BG:GB1=2:1,

B1

C

A

3. CG:GC1= 2:1. , CG:GC1=HG:GO. -, GH ~ 1GO.

?

4. LGH =LGO1, IIOC1, 1 .

5.C- , .. CD B.

6. . .

...


6269271

:

. - . OG BF.

.O- -

G .

1

BF

B

D

:

1... BF II OB1,

BGH ~ B1GO.

C1

G

N

?

H

  • , 1 . -, 1 .

F

B1

C

2. 1 , BF . -, BF II OB1

A

3. .. BF II OB1, L BGH L B1GO , BGH B1GO .

- BGH ~ B1GO.


6269271

:

. - . OG BF.

.O- -

G .

1, 1

BF

B

D

2.HG:GO=BG:GB1=2:1, CG:GC1=HG:GO.

?

C1

G

N

BG:GB1=1:2, ..

. G 1 1 B, 2:1, .

H

F

B1

C

A


6269271

- . K, 1 , 2 , 3

B

D

K3

K

Y

K1

A1

C1

E

X

Z

A

B1

F

C

K2

- 1,1, 1, - D, E, F,- AH,BH,CH X,Y,Z


6269271

,

C2

B

C1

H

A1

2

A

B1

B2

C


6269271

H

,

C1

A1

B

C2

2

C

B1

A

2


6269271

D

A2

A1

C

B

A3


6269271

C

F

P

E

D

A

B


6269271

B

F

C1

Y

A1

H

X

D

A

Z

B1

E

C


6269271

B

D

Y

A1

C1

O

G

N

H

E

X

Z

F

A

B1

C


6269271

B

D

K3

Y

K

Y

K1

A1

C1

E

X

Z

X

Z

B1

F

A

C

K2


6269271

OB = 2.NA1

B

R . = 2R.

D

A1

C1

O

N

E

H

F

A

C

B1


6269271

,

:

B

1,1, 1 -

D

D, E, F -

Y

A1

C1

X,Y,Z - AH,BH,CH

H

E

X

A

Z

F

B1

C

H

H

H


6269271

C1

2.

B

AA1

=

BB1

=

CC1

A1

3.

,

M+M+M -

M

A

C

B1


6269271

F1, E1, D1- .

C

F1, E1, D1- F1D1.

F1

F1D1 .

F

P

!

H

E

D

A

E1

B

=

D1


6269271

.- -

G .

OG : GH = 1 : 2

B

D

. - . OG BF.

.N

C1

.N OH .

G

N

O

H

ON = NH

F

B1

C

A


6269271

  • :

  • .. , .. , , , 2009 .

  • 2. .. , .. . , 7 -11, , , 2004 .

  • 3. , , , 1989.


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