..

..

, 2012

22.151. 2 72

514.133+514.174.5

69 ..

. . 2012. 63 .

.. .

:

1. .., - ,

.. .

2. .., , -

,

.. .

.

1917 (de Sitter, W.) On

the relativity of inertia. Remarks concerning Einstein's latest hypoth-

esis. ,

(H. S.M. Coxeter) 1943

A Geometrical Background for De Sitter's World. 2009

,

.

.

.

,

.

, ,

.

22.151.2 72

c .., 2012

2

H . ,

22

G H . , 10

.

. , ,

,

, ( )

.

H

-

, H,

P2, ( )

[1][2]. ,

,

H. H ,

, ,

H.

- H

[3]

,

G ,

.

H [4][10]. [4], [5]

H, [6][8], [10]

H,

H [9], [10].

H ([6][8], [10])

, ,

, H

. ,

H [3],

H, (past future ).

[11], [12]

.

H

,

[3], [13]

.

2012

The Physics of de Sitter Spacetime (Albert Einstein Institute)

(http://hep.physics.uoc.gr/deSitter/content/schedule.php).

[11][31].

H.

H,

. ,

22 ( 1),

G.

( 48, 1014),

. ,

, ,

( ) .

H,

P2, , , ..

, H .

, H,

.

P2

H. , , .

( 2)

. ,

.

I. H

1. H

H.

R = {A1, A2, A3, E} A1A2A3 : A3

,

, . E

, A1, A2.

R = {A1, A2, A3, E} E , A1A2A3 : A1 , A2 ,A3 A1A2 .

U3 (U3) () .

4

U3 (U3)

x21 + x22 x23 = 0

(x1x2 x23 = 0

). (1.1)

(a1 : a2 : a3) () H A

U3

a21 + a22 a23 > 0

(a21 + a

22 a23 < 0

), (1.2)

U3

a1a2 a23 < 0(a1a2 a23 > 0

). (1.3)

(X1 : X2 : X3)

U3 (U3)

X21 +X22 X32 = 0

(4X1X2 X23 = 0

).

(a1 : a2 : a3) ,

a U3

a21 + a22 a23 < 0, a21 + a22 a23 > 0, a21 + a22 a23 = 0, (1.4) U3

4a1a2 a23 > 0, 4a1a2 a23 < 0, 4a1a2 a23 = 0. (1.5)

2. H

, ,

, H

() [32], [33].

H.

H, ,

, .

H A, B

() , AB

( ) ,

K1,K2. (ABK1K2)

G.

=

2ln(ABK1K2),

H, 2 = 1(2 = 1) () AB. ln(ABK1K2)

w = Lnz

z = (ABK1K2):

ln z = ln |z|+ i arg z, pi < arg z 6 pi.

AB , (ABK1K2) R. A, B (ABK1K2) > 0.

, R. || A,B AB.

AB , (ABK1K2) C. K1, K2 |(ABK1K2)| = 1. , || [0;pi/2]. || A, B. A, B, K1, K2,

. (ABK1K2) = 1,, , = pi/2. ,

pi,

. A, B

||, pi || , A, B.

A, B ()

(a1 : a2 : a3), (b1 : b2 : b3),

U3

chAB

(cos

AB

)= a1b1 + a2b2 a3b3

a21 + a22 a23

b21 + b

22 b23

, (2.1)

U3

chAB

(cos

AB

)= a1b2 + a2b1 2a3b3

2a1a2 a23

b1b2 b23

. (2.2)

3. H

3.1. . H

.

H (),

() .

. ,

() , () H.

.

,

() : , , (, , ).

,

, : , , , , , .

H,

,

. , H

,

, ,

. H

:

6

, , , , , , , , .

, ,

.

1. . a, b D, d

D .

a, b, P2 (H P2), (. 1, ). H

D WD. D

H D WD. [4]

. 1. (WD) (WD

) D ();

() () ( );

() ()

H

,

[4].

H. d WD

D d , D.

2, 3. , . a b

(. 1, ),

(. 1, ) .

a, b P2 ,

H.

a, b.

, a, b,

.

H, H , a, b

a, b.

a, b b ,

, .

4. . a b

(. 2, ). a, b,

( ) ,

( ).

. 2. () ()

(); () ( );

() () ()

5. . a b (. 2,

) H .

H a b,

.

6. . a b

H ,

a, b,

(. 2, ).

7. . a b

H (. 3, ). , ()

a b

() H a b.

. 3. (), () (); ()

( ); (), () ()

8. . a b (. 3, )

H .

a, b.

8

1. H

()

ab

a b

1

2

3

4

5

6

7

8 [0;pi] 9 R+ 10 = pii , R+ 11

12

13 = + pii/2, R 14 R+ 15 = pii , R+

9. . a, b H

(. 3, ). P2 a,

b, ( ) ,

( ) H

a b.

, H

(. . 1).

, ,

, [32], [33].

,

(. [32], [33])

H.

a, b ,

a b = K, k1, k2 K. K l1, l2: (l1l2ab) = 1,(l1l2k1k2) = 1. l1, l2 a, b, , ,

a b.

3.2. . :

;

;

.

I. a b

() K. K

() k1, k2.

(abk1k2),

K, G, H

. , H : ,

, ,

. .

. a,

b 1, 2. k1,

k2, K,

. , (abk1k2) C, |(abk1k2)| = 1.

=

12i ln(abk1k2) . (pi ), [0;pi/2], , , 1 (2). H

pi.

. a b

1, 2. a, b

K

k1, k2. , (abk1k2) R. w = ln z

ln(abk1k2) = pii+ ln |(abk1k2)|, ln(bak1k2) = pii ln |(abk1k2)|.

[pii ln |(abk1k2)|] /2, ln |(abk1k2)| R, , , 1, 2.

a, b k1, k2, .. (abk1k2) = 1, pii/2. a, b

.

().

a, b 1,

2 .

a, b H K

k1, k2, .. (abk1k2) R+.

=

12 ln(abk1k2) , R+, , , 1 (2).

a a , K. a a

10

pii/2 pii.

1

. , ,

, pii . ().

a, b

. K a, b

k1, k2, .. (abk1k2) R+.

=

12 ln(abk1k2) , R+, , , .

a a , K. a a

pii/2

pii,

. , ,

, pii .II. a, b ,

k1, k2. (abk1k2) .

, ,

() , H.

III. a, b

, , K, :

k1 = k2. (abk1k2) = 1, ln(abk1k2) = 0, ..

(),

, . ,

G.

, H ,

(. . 1).

() a, b

a (ai), b (bi), i = 1, 2, 3, ab

U3

ch ab(

cos ab)

= a1b1 + a2b2 a3b3a21 + a

22 a23

b21 + b

22 b23

, (3.1)

U3

ch ab(

cos ab)

= 2a1b2 + 2a2b1 a3b34a1a2 a23

4b1b2 b23

. (3.2)

4. H

H

,

. , , ,

, .

H ,

,

, ,

, .

a, b, c : A = b c, B = a c, C = a b. H,

H .

H

G :

; ;

.

,

H e, h p .

,

:

eee, eeh, ehh, hhh, eep, ehp, hhp, epp, hpp, ppp. (4.1)

, [5]

H.

(4.1)

:

ehh, hhh, hhp, hpp. (4.2)

, (4.2).

1. ehh. b, c

B1, B2 C1, C2. :

B1, B2 C1, C2 ( ) (. 4, ( )).

A = b c , , A, B, C

H. ehh

B1, B2 C1, C2.

2. hhh. a, b, c

(A1, A2), (B1, B2) (C1, C2).

, A, B, C

H. a, b, c ,

hhh

:

1) (A1, A2), (B1, B2), (C1, C2)

, (. 4, );

12

. 4.

ehh (, ), hhh (, )

2) (A1, A2), (B1, B2), (C1, C2) ,

(. 4, ).

()

ABC hhh(I) (hhh(II)).

3. hhp. a, b , a = {A1, A2}, b ={B1, B2}, c = C0. A1, A2 B1,B2, C = a b , , A, B, C . A1, A2 B1, B2 .

A1, A2, B1, B2, C0 :

1) (A1, A2), (B1, B2) ,

C0, (. 5, );

2) (A1, A2), (B1, B2) ,

C0, (. 5, ).

H, ()

, hhp(I) (hhp(II)).

. 5.

hhp (, ), hpp (, )

4. hpp. a ABC

A1, A2, b = B0, c = C0. B0C0 A. :

1) A1, A2 B0, C0,

M = a B0C0 (. 5, );2) A1, A2 B0, C0,

M (. 5, ).

, hpp.

()

ABC hpp(I) (hpp(II)).

,

hhh, hhp, hpp.

,

,

.

G. , ,

eee, eeh, ehh, eep, ehp, epp. (4.3)

.

1. eee. a, b, c .

, A, B, C,

a, a, b, b, c, c , a, b, c

, a, b, c

ABC. (a, b,

c), (a, b, c), (a, b, c), (a, b, c) eee(I), eee(II),

eee(III), eee(IV ) (. 6, , , , ).

. 6. eee: eee(I) (), eee(II) ( ),

eee(III) (), eee(IV ) ()

2. eeh. a, b ABC.

CA, CB CA, CB , (0;pi/2). CA, CB () C, ABC eeh(I) (eeh(II)).

3. ehh. a ABC. B, C a ,

b, c,

.

ehh

( )

, ehh(I) (ehh(II)).

4. eep. a, b ABC.

a, c b, c .

14

a, b,

A, B, C, a, a, b, b , a, b

, a, b

. c,

A, B, c. a, b

. ABC : 1) (a, b, c),

(a, b, c); 2) (a, b, c); 3) (a, b, c). ABC

1)3) eep(I), eep(II), eep(III).

5. ehp. a, b, c ABC ,

. B, C

a ,

,

b, c. ehp,

(

) , ehp(I) ehp(II).

6. epp. a ABC.

B, C a ,

, A. ,

A, B, C ,

epp(I), epp(II).

, 15 H (4.3).

ppp 3-

[4], [8]. 3- G- [8], ,

ppp.

G, G ,

.

,

1. H

22

G .

5. H

M H, F ,

F , M

F . F

H F . H,

, .

2. H

, ,

.

. I. ABC H

. H ,

ABC, D. ,

D, ABC . ,

Da = AD a, Db = BD b, Dc = CD c a, b, c . ABDaDb C, D,

E = c DaDb . A, B

Dc, E. , E c. ,

DaDb, ABC

.

II. l,

ABC : La a, Lb b, l = LaLb, a, b ABC.

R0 = {A,B,C,E}, E = ALaBLb, , E ABC, .. ,

E, .

R0: La(0 : 1 : 1), Lb(1 : 0 : 1), l(1 : 1 : 1). l c ABC Lc(1 : 1 : 0). Lc ABC, , Lc c . Lc = CE c (1 : 1 : 0) R0 c, (ABLcL

c) = 1 < 0. , , E ,

. t,

E , R0 (t1 : t2 : t1 + t2), t1t2 6= 0, t1 + t2 6= 0. Ta, Tb,Tc t :

Ta(0 : t1 + t2 : t2), Tb(t1 + t2 : 0 : t1), Tc(t2 : t1 : 0). R0 Ia = (BCLaTa), Ib = (ACLbTb), Ic = (ABLcTc),

Ta, Tb, Tc a, b, c :

Ia =t1 + t2t2

, Ib =t1 + t2t1

, Ia = t2t1.

Ia, Ib, Ic :

(+ + ), (+ +), ( + +). , ABC t .

, , E,

ABC . , E

ABC . ABC .

.

3. F1, F2 H

a, b, c a, b, c, c c

c.

F1, F2 .

. a, b F1, F2

La, Lb, .

16

LaLb c, c, ,

F1, F2 , .

2 F1, F2,

LaLb , .

.

.

1. E ABC ,

E , .

2. , H,

, .

3. , H,

, , .

2, 3 .

II.

1. eee

1.1. eee(I). ABC

eee(I) a, b a,

b, A, B b, c a, c , c c,

C a, b. , a, b, c, a, b, c,

, A, B, A, B, , C, C. ( ) , ().

4. H , R+, :

1) eee(I) ;

2) H

eee(I) ;

3) ABC eee(I) :

cosa

= cos

b

cos

c

+ sin

b

sin

c

chA, (1.1)

chA = chB ch C shB sh C cos a. (1.2)

sin ashA

=sin bshB

=sin csh C

, (1.3)

. ABC

R , A1 A, A2 A (. 7, ). b, c A1A2

R

b(0 : t : 1), c(0 : t : 1), t R. (1.4)

. 7. eee(I)

, R E , c , AA2, AE.

c, AE AA2, AE32:

(c(AE)(AA2)(AE32)) > 0, E32(0 : 1 : 1) (1.1 . I).

(c(AE)(AA2)(AE32)) =2t

t+ 1> 0. (1.5)

(1.4) b, c

(1.4 . I), , t21 < 0. (1.5) :t (0; 1). B, C c, b :

B(b1 : 1 : t), C(c1 : 1 : t), b1, c1 R. (1.6) B, C H, (1.2 . I)

b21 + 1 t2 > 0, c21 + 1 t2 > 0. (1.7) a :

a(2t : t(b1 + c1) : c1 b1), (1.8)18

(1.4 . I)

4t2 + t2(b1 + c1)2 (c1 b1)2 < 0. (1.9)

, b, c ,

AA3, A2A3,

E. b < pi/2(b > pi/2

) ,

B3 = A3B AA2 () E12 = AA2 EA3 A2, A: (B3E12A2A) = b1 > 0, ((B3E12A2A) = b1 < 0).

AA3 , B / AA3, .. B3 6= A. b = pi/2 B3 = A2, .. b1 = 0. ,

b < pi/2 b1 > 0, b = pi/2 b1 = 0. (1.10)

c:

c < pi/2 c1 > 0, c = pi/2 c1 = 0. (1.11)

,

b, c. b (c) (1.4)

(1.1 . I)

H1(t2 1 : 1 : t) , H2 (t2 1 : 1 : t)(

K1(t2 1 : 1 : t) , K2 (t2 1 : 1 : t)) ., S1, S2 (Q1, Q2), b (c),

(S1S2H1H2) = 1, (S1S2AC) = 1((Q1Q2K1K2) = 1, (Q1Q2K1K2) = 1). (1.12)

(1.12) :

S1

(c1 +

c21 + 1 t2 : 1 : t

), S2

(c1

c21 + 1 t2 : 1 : t

),

Q1

(b1 +

b21 + 1 t2 : 1 : t

), Q2

(b1

b21 + 1 t2 : 1 : t

).(1.13)

b, c ,

, E

A1A2, A2A3, b, c

, .. A3

A1A2 E12 A1,

A2. S1, S

2 (Q

1, Q

2), (1.13) A1A2

A3, R

S1(c1 +

c21 + 1 t2 : 1 : 0

), S2

(c1

c21 + 1 t2 : 1 : 0

),

Q1(b1 +

b21 + 1 t2 : 1 : 0

), Q2

(b1

b21 + 1 t2 : 1 : 0

),

(S1E12AA2) > 0, (S2E12AA2) < 0 ((Q

1E12AA2) > 0, (Q

2E12AA2) < 0).

S1 (Q1) b (c) ,

S2 (Q2) .

O = AA2 a R (b1 + c1 : 2 : 0) a.

u,

A3, R (u1 : u2 : 0).

A / u, B / u, C / u. u1 6= 0, b1u1 + u2 6= 0, c1u1 + u2 6= 0. u a, b, c (1.8), (1.4) ABC :

Pa(u2(b1 c1) : u1(c1 b1) : tu1(b1 + c1) + 2tu2),Pb(u2 : u1 : tu1), Pc(u2 : u1 : tu1). Ia = (PaOBC), Ib = (PbS1BC), Ic = (PcQ1BC)

Pa, Pb, Pc a, b, c,

O, S1, Q1. Pj , j = a, b, c,

ABC , Ij > 0.

R Ia, Ib, Ic

Ia = b1u1 + u2c1u1 + u2

, Ib = u1c21 + 1 t2

c1u1 + u2, Ic = u1

b21 + 1 t2

b1u1 + u2,

:

(+ +), (++), (+ +), (). , ,

A3 , ABC

. 2 ABC

eee(I) .

A3 ,

ABC. , A3

. A3 , ,

H

ABC eee(I).

.

(1.6), (1.7), (1.10), (1.11) (2.1 . I)

cosb

=

c1c21 + 1 t2

, cosc

=

b1b21 + 1 t2

, (1.14)

,

sinb

=

1 t2

c21 + 1 t2, sin

c

=

1 t2

b21 + 1 t2. (1.15)

20

B (b1 : 1 : t) a K

(t2(b1 + c1) + b1 c1 : 2t2 + b1c1 b21 : 2t+ tb1(b1 + c1)

).

(AK), (AA3), b, c A

((AK)(AA3)bc) =t2 + 1 + b1c1b21 + 1 t2

. (1.16)

((AK)(AA3)bc) > 0 (((AK)(AA3)bc) < 0) ,

AK, AA3 b, c (), , , K

() a. B, K

pi/2. K / a (K a) a < pi/2(a > pi/2). K = C a = pi/2.

B (1.7), (1.16)

a < pi/2 t2 + 1 + b1c1 > 0, a = pi/2 t2 + 1 + b1c1 = 0. (1.17) (1.6), (1.7), (1.17) (3.1 . I)

cosa

=

t2 + 1 + b1c1b21 + 1 t2

c21 + 1 t2

, (1.18)

,

sina

=

(b1 c1)2 4t2 t2(b1 + c1)2

b21 + 1 t2c21 + 1 t2

. (1.19)

a, b

A, B, c C. , chA R+, chB R+, ch C R (. . 1 . I), (3.1 . I) (1.4), (1.8), (1.9)

chA =1 + t2

1 t2 , (1.20)

chB =

b1(t2 + 1) + c1(t2 1)1 t2(b1 c1)2 4t2 t2(b1 + c1)2 , (1.21)

ch C = c1(t2 + 1) + b1(t2 1)

1 t2(b1 c1)2 4t2 t2(b1 + c1)2 . (1.22) (1.21), (1.22)

chB ch C = b1(t2 + 1) + c1(t2 1) c1(t2 + 1) + b1(t2 1)

(1 t2) ((b1 c1)2 4t2 t2(b1 + c1)2) =

= t4(b1 + c1)2 (b1 c1)2

(1 t2) ((b1 c1)2 4t2 t2(b1 + c1)2) .

(1.9)

t4(b1 + c1)2 (b1 c1)2 < 0.

chB ch C =t4(b1 + c1)

2 (b1 c1)2(1 t2) ((b1 c1)2 4t2 t2(b1 + c1)2) . (1.23)

C = pii C, sh C = shC. (1.20)(1.22)

shA =2t

1 t2 , (1.24)

shB =2tb21 + 1 t2

1 t2(b1 c1)2 4t2 t2(b1 + c1)2 , (1.25)sh C =

2tc21 + 1 t2

1 t2(b1 c1)2 4t2 t2(b1 + c1)2 . (1.26) (1.14), (1.15), (1.18)(1.20), (1.23)(1.26)

(1.1)(1.3).

.

1. ABC a, b, c,

A, B, C,

a, c ( . 7,

).

a = pi a, c = pi c, A = pii A, C = pii C,

(1.1), (1.2)

cosa

= cos

b

cos

c

+ sin

b

sin

c

ch A, (1.27)

ch A = chB chC shB shC cos a. (1.28)

ABC, (1.27), (1.28)

cosc

= cos

b

cos

a

+ sin

b

sin

a

ch C, (1.29)

ch C = chB chA shB shA cos c. (1.30)

(1.1), (1.29) (1.2), (1.30) ,

(1.1), (1.2),

eee(I) H, ,

,

. (1.3) .

22

4.

eee(I)

, .

, eee(I) .

t2 + 1 + b1c1 = 0, b1 = 0, c1 = 0 (1.10), (1.11),

(1.17) , ABC (a, b, c

) .

, ABC eee(I)

. ,

,

. H ,

, .

, eee(I)

. ,

eee(I) .

, 4.

1. eee(I) ,

, . a = pi/2 (1.1)(1.3)

shB = sinb

shA, sh C = sin

c

shA, chA = ctg b

ctg

c

, chA = chB ch C.

2. eee(I) ,

, . c = pi/2 (1.1)(1.3)

cosa

= sin

b

chA, shA = sin

a

sh C, shB = sin

b

sh C.

1.2. eee(II).

5. H , R+, :

1) eee(II) ;

2) ABC eee(II) :

cosa

= cos b

cos

c

+ sin

b

sin

c

chA. (1.31)

chA = ch B ch C sh B sh C cos a. (1.32)

sin ashA

=sin bsh B

=sin csh C

. (1.33)

. ABC eee(I)

a, b, c, 4. b

a, c, , b

b .

a, b, c eee(II).

4 3

a, b, c eee(II) .

(1.1)(1.3) b = pi b, B = pii B eee(II) (1.31)(1.33).

. 2. 1 (1.31)(1.33)

eee(II),

, .

1.3. eee(III).

6. H , R+, :

1) eee(III) ;

2) H

eee(III) ;

3) ABC eee(III) :

cosa

= cos

b

cos

c

+ sin

b

sin

c

ch A. (1.34)

ch A = ch B ch C sh B sh C cos a. (1.35)

sin ash A

=sin bsh B

=sin csh C

. (1.36)

. ABC eee(I)

a, b, c, 4. a, b

b, c a, c , ,

a, b a, b

. a, b, c

eee(III).

5 a, b,

c . 3

a, b, c eee(III) .

.

a, b, c a, b, c

a, b.

H

. 4

6.

eee(III)

, .

(1.1)(1.3) a = pi a, b = pi b,A = pii A, B = pii B , eee(III), (1.34)(1.36).

.

24

1.4. eee(IV ).

7. H , R+, :

1) eee(IV ) ;

2) ABC eee(IV ) :

cosa

= cos b

cos

c

+ sin

b

sin

c

chA. (1.37)

chA = chB chC shB shC cos a. (1.38)

sin ashA

=sin bshB

=sin cshC

. (1.39)

. ABC eee(I)

a, b, c, 4. c

a, b, , c

c .

a, b, c (. 7, ) eee(IV ).

4 a,

b, c . 3

a, b, c eee(IV ) .

.

eee(IV )

, ( . 4,

).

(1.1)(1.3) c = pi c, C = pii C ,

eee(IV ), (1.37)(1.39).

.

2. eeh

2.1. eeh(I). ABC

eeh(I) a, b a, b,

A, B b, c a, c , c c,

C a, b. 8. H , R+, :

1) eeh(I) ;

2) ABC eeh(I) c

:

chc

= cos

a

cos

b

+ sin

a

sin

b

chC, (2.1)

cosa

= cos

b

chc

+ i sin

b

shc

chA, (2.2)

chC = chA chB shA shB ch c, (2.3)

chA = chB chC + shB shC cos a, (2.4)

sh cshC

=i sin ashA

=i sin bshB

. (2.5)

. ABC eeh(I)

R , A3

C, A1, A2

l C (. 8, ).

E l2 ACB.

a, b (1.5 . I)

R :

a(t : 1 : 0), b(1 : t : 0), t R. (2.6)

. 8. eeh(I), eeh(II)

A0 = a l, B0 = b l

A0(1 : t : 0), B0(t : 1 : 0). (2.7)

a, b

l1(1 : 1 : 0) ACB. ,

a (b) l1 CA1 (CA2),

.. (A0A1B0A2) > 0. 1 t2 > 0., t R, : t (1; 0). A, B b, a (2.6)

A(t : 1 : a3), B(1 : t : b3), a3, b3 R, (2.8)26

(1.3 . I)

a23 t > 0, b23 t > 0. (2.9)

a, b ABC, B, C

A, C , ,

l, A1C,

E ( . 8, ). B0 l C , |CB0| = pi/2. A1A, A1E () l, A1C

, b < pi/2(b > pi/2

). A1A, A1E,

l, A1C A1 ((A1A)(A1E)l(A1C)) = a3.

,

b < pi/2 a3 > 0, b > pi/2 a3 < 0. (2.10)

B, , t (1; 0) ((A1B)(A1E)l(A1C)) = b3/t,

a < pi/2 b3 < 0, a > pi/2 b3 > 0. (2.11)

c

c(b3 ta3 : a3 tb3 : t2 1), (2.12)

(1.5 . I)

(t2 1)2 4(b3 ta3)(a3 tb3) > 0. (2.13)

a, b.

a (b) (2.6) (1.1 . I)

H1

(1 : t :

t), H2

(1 : t : t

) (K1

(t : 1 :

t), K2

(t : 1 : t

)).

S1, S2 (Q1, Q2) a (b) :

(S1S2H1H2) = 1, (S1S2CB) = 1((Q1Q2K1K2) = 1, (Q1Q2CA) = 1) . (2.14)

R

S1

(1 : t : b3

b23 t

), S2

(1 : t : b3 +

b23 t

),

Q1

(t : 1 : a3

a23 t

), Q2

(t : 1 : a3 +

a23 t

).(2.15)

a, b ,

E l,

CA1. , a, b.

(2.15) a3, b3 t (1; 0)

((A1S1)(A1E)l(A1C)) =b3

b23 tt

> 0,

((A1S2)(A1E)l(A1C)) =b3 +

b23 tt

< 0,

((A1Q1)(A1E)l(A1C)) = a3 a23 t < 0,

((A1Q2)(A1E)l(A1C)) = a3 +a23 t > 0,

, S1 (Q2) a(b)

(2.14) . S2 (Q1)

.

S1Q2 (u1 : u2 : u3),

u1 = b3 b23 t ta3 t

a23 t,

u2 = tb3 + tb23 t+ a3 +

a23 t, u3 = t2 1.(2.16)

(v1 : v2 : v3) V = S1Q2 c:

v1 = u2(t2 1) u3(a3 tb3), v2 = u3(b3 ta3) u1(t2 1),

v3 = u1(a3 tb3) u2(b3 ta3). (2.17)

M = l1 c (t 1 : 1 t : a3 b3) c. (2.16), (2.17), (MVAB):

(MVAB) =tv1 v2v1 tv2 =

u1 + tu2 + b3u3tu1 + u2 + a3u3

= b23 ta23 t

.

M , V A, B, (MVAB) < 0.

, V c.

, S1Q2 ABC

. 2 ABC .

.

(2.2 . I) (2.8)(2.11)

cosa

= b3

b23 t, cos

b

=

a3a23 t

, (2.18)

,

sina

=

tb23 t

, sinb

=

ta23 t

. (2.19)

28

a, b pi/2,

pi/2, (2.10), (2.11) a3b3 < 0.

t2 + 1 2a3b3 > 0. (2.20) a, b pi/2,

pi/2, a3b3 > 0 ( . 8,

ABC). T = lc c , , H A, B. c,

,

H. (TApAB) < 0,

Ap = cpA, pA A . . R:

pA(1 : t : 2a3),Ap(t3 t+ 2a23 2ta3b3 : 1 t2 2a3b3 + 2ta23 : a3

(t2 + 1

) 2tb3)

(TApAB) =a3(t2 + 1 2a3b3)

2b3(a23 t)< 0.

(2.9) a3b3 > 0 (2.20).

A, B l, ..

a, b pi/2, a3b3 = 0, (2.20).

, a3, b3, t

(2.20). (2.2 . I),

chc

=

t2 + 1 2a3b32a23 t

b23 t

, shc

=

(t2 1)2 4(b3 ta3)(a3 tb3)

2a23 t

b23 t

. (2.21)

(3.2 . I) C

a, b (2.6)

chC = t2 + 1

2t, shC =

t2 12t

. (2.22)

A, B b, c a, c

(2.6), (2.12) (3.2 . I), (2.13)

chA = i1a3(t

2 + 1) 2tb3t(t2 1)2 4(a3 tb3)(b3 ta3) , 1 = 1, (2.23)chB = i2

2ta3 b3(t2 + 1)t(t2 1)2 4(a3 tb3)(b3 ta3) , 2 = 1. (2.24) 1, 2.

a, b: B a, A b, aa, bb, A = a b, B = b a. R:

a(tb3 : b3 : 2t), b(a3 : ta3 : 2t),

A(2t2 : 2t : b3

(t2 + 1

)), B

(2t : 2t2 : a3

(t2 + 1

)).

I1 = (BBCA0), I2 = (A0BBC), J1 = (AACB0), J2 = (B0AAC).

R:

I1 =a3(t

2 + 1)

2tb3, I2 =

2tb32tb3 a3(t2 + 1) , (2.25)

J1 =b3(t

2 + 1)

2ta3, J2 =

2ta32ta3 b3(t2 + 1) . (2.26)

I1, I2 (J1, J2) B (A)

a (b). .

1. a3 > 0, b3 < 0. (2.10), (2.11) A0, B0 (2.7)

a, b. I1 > 0, J1 > 0, t (1; 0). , B (A) CA0 (CB0),

B (A).

I2 < 0 (I2 > 0), B ( ) a (.

9, ( )), , A ABC

pii/2 + (pii/2), R+. chA = i sh (chA = i sh), sh > 0., a3 > 0, b3 < 0 I2 < 0 (I2 > 0) (2.25)

a3(t2 + 1) 2tb3 > 0

(a3(t

2 + 1) 2tb3 < 0), (2.27)

: 1 = 1.

. 9. A, B a3 > 0, b3 < 0

, J2 < 0 (J2 > 0), A ( )

b (. 9, ()), , B

ABC pii/2 + (pii/2 ), R+. chB = i sh (chB =i sh), sh > 0. a3 > 0, b3 < 0 J2 < 0 (J2 > 0) (2.26)

2ta3 b3(t2 + 1) > 0(2ta3 b3(t2 + 1) < 0

), (2.28)

(2.24) 2 = 1.

2. a3 < 0, b3 > 0. (2.10), (2.11) A0, B0 a, b. (2.23), (2.24) I1 > 0, J1 > 0.

, 1 , B (A)

CA0 (CB0), B (A).

30

I2 > 0 (I2 < 0), B A0B, a

( a = BC, a) (. 10, ( )), , A

ABC pii/2+ (pii/2). chA = i sh(chA = i sh), sh > 0. , a3 < 0, b3 > 0 I2 > 0 (I2 < 0) (2.23) () (2.27), : 1 = 1.

. 10. A, B a3 < 0, b3 > 0

J2 > 0 (J2 < 0), A B0A,

b ( b = AC, b) (. 10, ()), ,

B ABC pii/2 + (pii/2 ). chB =i sh (chB = i sh), sh > 0. a3 > 0, b3 < 0 J2 > 0 (J2 < 0) (2.26) () (2.28), (2.24)

2 = 1.

3. a3 > 0, b3 > 0, A0 (B0) (

) a (b) (. 11, ). (2.25), (2.26)

:

I1 < 0, J1 < 0, I2 > 0, J2 > 0. (2.29)

, (2.27) (2.28) .

B (A) A0C, a ( B0C,

A). A = pii/2 + (B = pii/2 ), chA = i sh(chB = i sh). 1 = 1 (2 = 1).

. 11. A, B a3 > 0, b3 > 0 (), a3 < 0, b3 < 0 ( )

4. a3 < 0, b3 < 0, A0 (B0)

() a(b)(. 11, ). (2.25), (2.26)

(2.29). , (2.27)

(2.28) . B (A) A0C,

B ( B0C, b). A = pii/2 (B =pii/2 + ), chA = i sh (chB = i sh). 1 = 1(2 = 1).

, (2.23), (2.24) 1 = 2 = 1.

, ch Re(A) R+, ch Re(B) R+

shA = sh

(pii

2+ Re(A)

)= i ch Re(A), shB = sh

(pii

2+ Re(B)

)= i ch Re(B),

shA = i(1 t2)

a23 tt(t2 1)2 4(a3 tb3)(b3 ta3) , (2.30)

shB = i(1 t2)

b23 tt(t2 1)2 4(a3 tb3)(b3 ta3) . (2.31) (2.18), (2.19), (2.21)(2.24), (2.30), (2.31)

(2.1)(2.5).

8.

1. ABC a = pi/2, b 6= pi/2, (2.1)(2.5) :

chc

= sin

b

chC, shB = sh

b

shA, shC = i sh c

shA.

2. a = pi/2, b = pi/2, A = B0, B = A0, ,

ABC : ac, bc. A0B0 a, b, A0 a,

B0 b, a, b. (2.3) chC = ch c/. , C = c/.

,

9.

H.

3. ABC , , A =

pii/2, (2.5)

shc

= shC sin

a

, sin

b

= i shB sin a

.

2.2. eeh(II).

10. H , R+, :1) eeh(II) ;

2) ABC eeh(II) a, b

c :

chc

= cos a

cos

b

+ sin

a

sin

b

chC, (2.32)

32

cosa

= cos b

chc

+ i sin

b

shc

ch A, (2.33)

chC = ch A chB sh A shB ch c, (2.34)

ch A = chB chC + shB shC cosa

, (2.35)

sh cshC

=i sin ash A

=i sin bshB

. (2.36)

. 8,

, a, a. a, b, c (.

8, ) ABC eeh(II). 3

8

.

, a, A, A. A = pii A, chA = ch A, shA = sh A. a, a : a = pi a. (2.1)(2.5) eeh(II) (2.32)(2.36).

.

3. ehh

3.1. ehh(I). ABC

ehh(I) b, c b, c,

BAC, a a, BAC.

11. H , R+, :1) ehh(I) ;

2) ABC ehh(I) a

:

cosa

= ch

b

chc

sh b

shc

chA, (3.1)

chc

= cos

a

ch

b

i sin a

shb

chC, (3.2)

chA = chB chC shB shC cos a, (3.3)

chC = chA chB + shA shB ch c, (3.4)

sin ashA

=i sh bshB

=i sh cshC

. (3.5)

. ABC ehh(I)

R , A3

A, A1, A2

l A (. 12).

E l1 BAC. ,

b (c) l1,

AA1 (l1, AA2), .. ((AA1)bl1(AA2)) > 0 (((AA2)cl1(AA1)) > 0).

b, c (1.5 . I)

R :

b(t : 1 : 0), c(1 : t : 0), t (0; 1). (3.6)

. 12. ehh(I) (), ehh(II) ( )

B, C c, b (3.6)

B(t : 1 : b3), C(1 : t : c3), b3, c3 R. (3.7) AE BAC, B1, C1,

E23, B, C, E AA2 A1

AA2, .. (B1E23AA2) > 0, (C1E23AA2) > 0.

t

b3 > 0, c3 > 0. (3.8)

B, C H, (1.3 . I)

b23 t > 0, c23 t > 0. (3.9) a

a(c3 tb3 : b3 tc3 : t2 1) (3.10)34

, (1.5 . I)

(t2 1)2 4(b3 tc3)(c3 tb3) < 0. (3.11)

B0 = b l, C0 = c l, R: B0(1 : t : 0), C0(t : 1 : 0). b, c B(1 : t : 2c3), C (t : 1 : 2b3).

(ACBB0) < 0, (ABC C0) < 0,

B, C b, c . BC

R (2(tb3 c3) : 2(tc3 b3) : 1 t2) a (3.10) Q(b3 tc3 : tb3 c3 : 0) l. a A, ,

A, Q / a. , BC ABC . 2

ABC .

.

(2.2 . I) (3.7)(3.9)

chb

=

c3c23 t

, chc

=

b3b23 t

. (3.12)

,

shb

=

t

c23 t, sh

c

=

t

b23 t. (3.13)

(2.2 . I)

cosa

=

t2 + 1 2b3c32b23 t

c23 t

, = 1. (3.14)

(3.11)

sina

=

4(b3 t3)(c3 tb3) (t2 1)2

2b23 t

c23 t

. (3.15)

(3.14).

P = a l1 R (t+ 1 : t+ 1 : b3 + c3) a H,

(b3 + c3)2 (t+ 1)2 > 0. (3.16)

pB (pC) B (C) Bp =

pB a (Cp = pC a). a pi/2 , C = Bp (B = Cp). a pi/2 ,

C, P (B, P ) B, Bp (C, Cp). ,

cos a (CPBBp), (BPCCp).

R: pB(1 : t : 2b3), pC(t : 1 : 2c3),

Bp(bp : 2b3(c3 tb3) + t2 1 : 2tc3 b3(t2 + 1)),Cp(2c3(b3 tc3) + t2 1 : t t3 2c3(c3 tb3) : cp).,

(CPBBp) =t2 + 1 2b3c3 2(b23 t)

t2 + 1 2b3c3 , (BPCCp) =t2 + 1 2b3c3 2(c23 t)

t2 + 1 2b3c3 .

(CPBBp), (BPCCp) , ,

(CPBBp) + (BPCCp) = 2(b3 + c3)2 (t+ 1)2

t2 + 1 2b3c3 . (3.17)

(3.16), (3.17) (CPBBp), (BPCCp), , ,

cos a t2 + 1 2b3c3.

= 1, (3.14) :

cosa

=

2b3c3 t2 12b23 t

c23 t

. (3.18)

(3.2 . I) t (0; 1) A b, c

chA =1 + t2

2t, shA =

1 t22t

, (3.19)

B, C a, c a, b

(3.11)

chB = i1b3(1 + t

2) 2tc3t

4(b3 tc3)(c3 tb3) (t2 1)2, 1 = 1, (3.20)

chC = i2c3(1 + t

2) 2tb3t

4(b3 tc3)(c3 tb3) (t2 1)2, 2 = 1. (3.21)

1, 2 b, c, C, B b, c .

b, b c, c pii/2. B (C), ABC, ,

a = c (a = b). c, c () a, BB0, .. (cc

a(BB0)) > 0 ((cca(BB0)) < 0), B ABC : B = pii/2 + 1 (B = pii/2 1), 1 a, c

, 1 R+. ,

chB = ch

(pii

2 1

)= i sh1, sh1 R+,

36

: chB

(cca(BB0)). , chC (bba(CC0)). R : b(t : 1 : b3), c

(1 : t : c3), B0(1 : t : 0), C0(t : 1 : 0),BB0(tb3 : b3 : t2 1), CC0(c3 : tc3 : t2 1). ,

(bba(CC0)) =c3(1 t2)

c3(1 + t2) 2tb3 , (cca(BB0)) =

b3(1 t2)b3(1 + t2) 2tc3 .

(3.20), (3.21) ,

b3 > 0, c3 > 0, t (0; 1), : 1 = 2 = 1. ,

chB = ib3(1 + t

2) 2tc3t

4(b3 tc3)(c3 tb3) (t2 1)2, (3.22)

chC = ic3(1 + t

2) 2tb3t

4(b3 tc3)(c3 tb3) (t2 1)2, (3.23)

, ,

shB = i(1 t2)

b23 t

t

4(b3 tc3)(c3 tb3) (t2 1)2, (3.24)

shC = i(1 t2)

c23 t

t

4(b3 tc3)(c3 tb3) (t2 1)2. (3.25)

(3.12), (3.13), (3.15), (3.18), (3.19), (3.22)(3.25),

(3.1)(3.5).

. 11.

1. a ABC

, .. a = pi/2, (3.1)(3.5)

chA = cthb

cth

c

, cth

b

= th

c

chA,

chA = chB chC, shB = i sh b

shA, chc

= i sh b

shC.

2. ABC , , C =

pii/2, (3.1)(3.5)

chc

= cos

a

ch

b

, ch

c

= cthA cthB,

chA = i shB cos a, sin

a

= sh

c

shA, sh

b

= i sh c

shB.

, ABC .

b, c a. (t : 1 : b3),

(1 : t : c3) b, c . , , ,

t2 = 1. , t (0; 1). , ehh(I) .

, a = pi/2 C = pii/2. (3.14)

1 + t2 = 2b3c3. (3.26)

C = pii/2 b a, , . ,

c3 tb3t

=b3 tc3

1. (3.27)

(3.26), (3.27) c23 = t,

(3.9). , a = pi/2 C = pii/2 .

11 .

3.2. ehh(II).

12. H , R+, :1) ehh(II) ;

2) ABC ehh(II) b, c

a :

cosa

= ch b

chc

sh b

shc

ch A, (3.28)

chc

= cos a

ch

b

i sin a

shb

chC, (3.29)

ch A = chB chC shB shC cos a, (3.30)

chC = ch A chB + sh A shB chc

, (3.31)

sin ash A

=i sh bshB

=i sh cshC

. (3.32)

. a , a

ABC ehh(II),

11. a A, A. , a, b, c ehh(II). 3

11 .

(3.1)(3.5) a = pi a, A = pii A (3.28)(3.32).

.

38

4. hhh

4.1. hhh(I).

13. H , R+, :1) ABC hhh(I)

;

2) hhh(I) ;

3) ABC hhh(I) a,

BAC, :

cha

= ch

b

chc

+ sh

b

shc

chA, (4.1)

chc

= ch

a

ch

b

sh a

shb

chC, (4.2)

chA = chB chC + shB shC cha

, (4.3)

chC = chA chB shA shB ch c, (4.4)

sh ashA

=sh bshB

=sh cshC

. (4.5)

. a, b, c

(K1, K2), (B1, B2), (C1, C2) ,

(K1, K2) ,

(B1, B2), (C1, C2) (. 13).

ABC, A = b c, B = a c, C = a b hhh(I). a, b, c ABC,

A, B, C. ABC

R . A3 A,

E c. b, c R

:

b(1 : t : 0), c(1 : 1 : 0), (4.6) (1.5 . I) t R+, b . l = A1A2

A , , b, c,

A, H B0(t : 1 : 0), E12(1 : 1 : 0). A1,

A2 l , A1, B0 A2,

E12: (A1B0A2E12) > 0. t > 1.

B, C :

B(1 : 1 : b3), C(t : 1 : c3), t > 0, b3, c3 R. (4.7)

. 13. hhh(I)

a R :

(b3 c3 : c3 tb3 : t 1). (4.8)

B, C H, a ,

(1.3, 1.5 . I)

b23 1 > 0, c23 t > 0, (t 1)2 4(b3 c3)(c3 tb3) > 0. (4.9)

R , E

C1, C2. , E

A, C ( . 13

- ). C23, E23 C E

A2A3 A1 AA2:

(C23E23A2A3) > 0, ..

c3 > 0. (4.10)

a ,

B1, B2 C1, C2, A0 a l

B0E12 a: (A1A0B0E12) < 0.

, b3/c3 < 0, (4.9)

b3 < 0. (4.11)

()

() , ,

, () .

A, B, C ABC A1. R: A

(t 1 : 0 : c3 b3),40

B(t : 1 : b3), C (1 : 1 : c3). B, C b, c, (4.10), (4.11)

(ACBB0) =c3

c3 b3 > 0, (ABCE12) =

b3 1b3 c3 > 0.

A a, (BCAA0) = c3/b3 < 0. , b, c

ABC, a

b, c.

.

E12A R (b3 c3 : c3 b3 : t 1) c H E12, a

A a, b P (t : 1 : c3 b3). P b, (ACPB0) = c3/b3 < 0. , E12A

ABC . 2 ABC

.

.

(2.2, 3.2 . I), (4.6), (4.7), (4.8)

(4.9),

cha

=

t+ 1 2b3c32b23 1

c23 t

, chb

=

c3c23 t

, chc

=

b3b23 1

,

chA =t+ 1

2t, chB =

2c3 b3(t+ 1)(t 1)2 4(b3 c3)(c3 tb3)

,

chC =c3(t+ 1) 2tb3

t

(t 1)2 4(b3 c3)(c3 tb3).

sha

=

(t 1)2 4(b3 c3)(c3 tb3)

2b23 1

c23 t

,

shb

=

t

c23 t, sh

c

=

1b23 1

,

shA =t 12t, shB =

(t 1)b23 1

(t 1)2 4(b3 c3)(c3 tb3),

shC =(t 1)

c23 t

t

(t 1)2 4(b3 c3)(c3 tb3).

(4.1)(4.5).

.

4.2. hhh(II).

. ,

H .

14. H , R+, :1) hhh(II)

;

2) hhh(II) ;

3) ABC hhh(II) a, b, c

:

cha

= ch b

chc

sh b

shc

chA, (4.12)

chA = chB chC shB shC ch a, (4.13)

sh ashA

=sh bshB

=sh cshC

. (4.14)

. a, b, c

(K1, K2), (B1, B2), (C1, C2)

(. 14). ABC, A = b c, B = a c,C = a b hhh(II). a, b, c ABC, A, B, C.

ABC R . A3 A, E c, A1,

A2 l A ,

A1, B0 = l b A2, E12 = l c: (A1B0A2E12) > 0. b, c R :

b(1 : t : 0), c(1 : 1 : 0), t > 1. (4.15)

B, C :

B(1 : 1 : b3), C(t : 1 : c3), b3, c3 R. (4.16)

a R :

(b3 c3 : c3 tb3 : t 1). (4.17)

B, C H, a ,

(1.3, 1.5 . I)

b23 1 > 0, c23 t > 0, (t 1)2 4(b3 c3)(c3 tb3) > 0. (4.18)

b, c , K1, K2

(B1, B2), (C1, C2), B, C

42

. 14. hhh(II)

A. R

E ( . 14

). B23, C23, E23 B, C E

AA2 A1 AA2:

(B23E23AA2) > 0, (C23E23AA2) > 0,

,

b3 > 0, c3 > 0. (4.19)

b, c , l A

b c H, ,

l b, c ABC. , l

a.

AA1 a

L(t 1 : 0 : c3 b3) a. V = a l (c3 tb3 :c3 b3 : 0) (V LBC) = b3/c3 > 0, , V a. , l a.

ABC hhh(II) ,

.

l ABC .

2 ABC .

.

N = c pC , pC C , c.

, (ABNE) < 0. .

R: pC(1 : t : 2c3), N(2c3 : 2c3 : t+ 1).

(ABNE) =2c3(1 b3)t+ 1 2b3c3 < 0.

(4.18), (4.19) : b3 > 1. ,

t+ 1 2b3c3 > 0. (4.20) t > 1 (4.19), (4.20),

t+ 1 > 2b3c3,

c3(t+ 1) < 2b3c23, b3(t+ 1) < 2b23c3,2tb3 c3(t+ 1) < 2tb3 2b3c23 = 2b3(t c23) < 0,2c3 b3(t+ 1) < 2c3 2b23c3 = 2c3(1 b23) < 0. ,

c3(t+ 1) 2tb3 > 0, b3(t+ 1) 2c3 > 0. (4.21) hhh(II)

. , R+, , R+, , R+, (b, c),(a, c) (a, b) .

ABC A = pii , B = pii ,C = pii . ,

chA = ch, chB = ch, chC = ch,shA = sh, shB = sh, shC = sh.

(2.2, 3.2 . I), ,

(4.15)(4.17) (4.18)(4.21),

cha

=

t+ 1 2b3c32b23 1

c23 t

, chb

=

c3c23 t

, chc

=

b3b23 1

.

chA = t+ 12t, chB =

2c3 b3(t+ 1)(t 1)2 4(b3 c3)(c3 tb3)

,

chC =2tb3 c3(t+ 1)

t

(t 1)2 4(b3 c3)(c3 tb3).

sha

=

(t 1)2 4(b3 c3)(c3 tb3)

2b23 1

c23 t

,

shb

=

t

c23 t, sh

c

=

1b23 1

.

shA =t 12t, shB =

(t 1)b23 1

(t 1)2 4(b3 c3)(c3 tb3),

shC =(t 1)

c23 t

t

(t 1)2 4(b3 c3)(c3 tb3).

(4.12)(4.14).

.

44

III.

H,

(. . 4 . I).

H ,

G.

G,

.

.

, .

ABC A a A0, A

6= B, A 6= C. (BC,A),(CB,A) , A BC, CB, B, C.

(BC,A), (CB,A) (., , [32], [33]): (BC,A) = (BCAA0), (CB,A) = (CBAA0). , (BC,A), (CB,A) G.

: (BCAA0) = 1/(CBAA0). (BC,A)(CB,A) = 1. (BC,A), (CB,A) a ABC ma., ppp [4], [8].

, G [8]

2 1 [4] .

1. eep

1.1. eep(I).

15. H , R+, :1) eep(I) ;

2) ABC eep(I) b, c,

, ma(ma = (BC,A)) :

cosb

cos

c

+ sin

b

sin

c

chA = 1, (1.1)

cosb

cos c

= sin

b

sin

c

shA, (1.2)

cosc

= ma cos

b

. (1.3)

. ABC eep(I) (.

15, ) b, c,

R . A3 A, E

a. R a b, c

(1.5 . I) :

a(1 : 1 : 2), b(b1 : 1 : 0), c(c1 : 1 : 0), b1 R+, c1 R+, (1.4)

:

A(0 : 0 : 1), B(2 : 2c1 : 1 c1), C(2 : 2b1 : 1 b1). (1.5)

. 15. : eep(I) a, b, c ();

eep(II) a, b, c, eep(III) a, b, c ( )

A1, A2 , AA1, c

AA2, b: ((AA1)c(AA2)b) > 0. :

b1 > c1. (1.6)

Sb, Sc A1E b, c

: Sb(1 : b1 : b1), Sc(1 : c1 : c1). , ABC.

BA1(0 : 1 c1 : 2c1) (CA1(0 : 1 b1 : 2b1)) a, c (a, b)

b (c) Mb(2c1 : 2b1c1 : b1(c1 1)) (Mc(2b1 : 2b1c1 : c1(b1 1))). (1.4) (1.6)

(ACSbMb) =c1 b1c1(b1 + 1)

< 0, (ABScMc) =b1 c1b1(c1 + 1)

> 0. (1.7)

Mb (Mc) a, c (a,

b), () (1.7) Sb(Sc) ()

46

. () a, c

(a, b) b (c). , Sb (Sc)

.

a A1E E,

, A1E Sb, Sc

ABC. 2 ABC eep(I)

.

.

B23(0 : 2c1 : c1 1), C23(0 : 2b1 : b1 1), E23(0 : 1 : 1) B, C, E AA2 A1. E23 Sb, Sc b, c.

b (c) , C23 (B23)

E23 AA2, .. ,

(E23C23AA2) > 0 ((E23B23AA2) > 0).

(E23C23AA2) =b1 1

2b1> 0, (E23B23AA2) =

c1 12c1

> 0.

(1.4) b1, c1

b < pi/2 b1 > 1, b > pi/2 b1 < 1,c < pi/2 c1 > 1, c > pi/2 c1 < 1. (1.8)

(2.2 . I), (1.5) (1.4), (1.8),

cosb

=b1 1b1 + 1

, sinb

=

2b1

b1 + 1, cos

c

=c1 1c1 + 1

, sinc

=

2c1

c1 + 1. (1.9)

(3.2 . I), (1.4) (1.4), (1.6),

chA =b1 + c1

2b1c1

, shA =b1 c12b1c1

. (1.10)

A = a pA, pA = A1A2, a A(1 : 1 : 0). A 6= B, A 6= C, c1 6= 1, b1 6= 1, ma a

ma = (BC,A) = (BCAE) =(b1 + 1)(c1 1)(b1 1)(c1 + 1) . (1.11)

(1.9)(1.11) (1.1)(1.3).

. eep(I).

1. , b ABC,

a, c,

: b = pi/2. ,

a, b,

, .. c = pi/2, BC A

, , pA.

, eep(I) BC

. , b = pi/2 c = pi/2 .

b = pi/2 b1 = 1, (1.9), (1.10)

shA = ctg c.

2. , ABC

eep(I), c ,

, :

shA = ctgb

.

1, 2 .

1.2. eep(II) eep(III). ABC

eep(I) a, b, c, . 1.1.

, b, c, b, c. ,

b (c) ()

a, c (a, b) ABC. ABC

(. 15, ) a, b, c(a, b, c

)

(. . 4 . I) eep(II) (eep(III)).

3 15

.

b = pi b, c = pi c (1.1)(1.3) eep(II) eep(III) :

eep(II) eep(III)

sin b sinc chA cos b cos c = 1, sin b sin c chA cos b cos c = 1

cos b + cosc = sin

b sin

c shA, cos

b + cos

c = sin b sin c shA,

cos c = ma cos b . cos c = ma cos b .

2. ehp

2.1. ehp(I). ABC

ehp(I) a, b, c ,

. C (A),

() , ().

ehp(I) .

.

pA A, A = pA a 48

, c = {C1, C2}, C1

A. P1 = AC1 b (P2 = AC2 b) () b. 1 = (AC,P1), 2 = (AC,P2)

,

ABC.

16. H , R+, :1) ehp(I) ;

2) ABC ehp(I) a,

c, mb (mb = (CA,B)) 1, 2 b :

cosa

= mb ch

c

, (2.1)

cosa

ch c

= i sin

a

shc

shB, (2.2)

cosa

chc

+ i sin

a

shc

chB = 1. (2.3)

12 = ch2 c

. (2.4)

. ABC ehp(I) (.

16) R . A3

A , A1

b. E

c. b, c :

b(0 : 1 : 0), c(1 : 1 : 0). (2.5)

. 16. ehp(I)

B, C c, b :

B(1 : 1 : b3), C(1 : 0 : c3), b3, c3 R. (2.6)

a (1.5 . I)

a(c3 : b3 c3 : 1), (2.7)

4c3(b3 c3) 1 > 0. (2.8) pA A (1.1 . I)

A1A2 R c E12(1 : 1 : 0).

R, , E

c, ,

B. (AE12BE) > 0, .. b3 > 0. B

H, (1.3 . I) (2.8)

b3 > 1, c3 > 0. (2.9)

S AC (ACSA1) = 1 R S(1 : 0 : 2c3). a SE12(2c3 : 2c3 : 1) S0(b3 +c3 : c3 : 2b3c3), AA2 D(0 : 1 : b3c3). C,D a ,

, A.

ABC a A,

D a. S0 a,

(2.9) (BCS0D) = b3/c3 < 0. SE12 a b ABC c, E12 c H . 2

ABC .

.

pB B (1 : 1 : 2b3) b B(2b3 : 0 : 1). mb b

mb = (CA,B) = (CABA1) =2b3c3 1

2b3c3. (2.10)

(2.6) (2.9), (2.2

. I)

cosa

=

2b3c3 12c3b23 1

, = 1, ch c

=b3b23 1

. (2.11)

.

B, Bp,

Bp(2b23 1 2b3c3 : 1 2b3c3 : b3 2c3) = pB a,50

a pi/2. Bp ( ) a, a > pi/2 (a < pi/2). D

a,

(BCBpD) = 2b23 1

2b3c3 1 .

(2.9) (BCBpD), 2b3c3 1 ,

a > pi/2 2b3c3 1 < 0, a < pi/2 2b3c3 1 > 0. (2.12)

(2.12) (2.11) = 1.

(3.2 . I) B a(2.7), c (2.5)

chB = i2c3 b3

4c3(b3 c3) 1, = 1. (2.13)

.

B a, a, :A = a AA2. A AA2 B: a B, A AD.,

B = pii2 + (ADAA2) > 0, B = pii2 (ADAA2) = 0,B = pii2 (ADAA2) < 0, R+.(2.14)

, (ADAA2) , H A2 A.

R: a(1 2b3c3 : 2b23 2b3c3 1 : 2(2c3 b3)), A(0 : 2(b3 2c3) :2b23 2b3c3 1)

(ADAA2) =2(b3 c3)(2c3 b3)

4c3(b3 c3) 1 .

(2.8) (ADAA2) (b3 c3)(2c3 b3). , b3 > c3. A(c3 b3 : c3 : 0) a H, E12 . A

, E12 A1, A2 :

(AE12A1A2) =c3

b3 c3 < 0.

(2.9) : b3 > c3. ,

(ADAA2) 2c3 b3. (2.14) (2.13) = 1.

, (2.10), (2.11), (2.13)

= = 1, (2.1)(2.3).

.

(2.4)

b ABC R.

c E, E(1 : 1 : 1), E c AB. P1 = A

E b(P2 = A

E b) () b. R: P1(b3 : 0 : c3), P2(b3 : 0 : c3). ,

1 = (AC,P1) = b3b3 1 , 2 = (AC,P2) =

b3b3 + 1

.

1, 2

(2.11) (2.4).

. 16 .

1. a ABC pi/2,

mb , (2.2), (2.3)

shB = i cthc

, i sh

c

chB = 1.

2. ABC , ..

B = pii/2, (2.1)(2.4)

chc

cos a

= sin

a

shc

, cos

a

chc

= 1, mb12 = 1.

a = pi/2 B = pii/2 ,

(2.11), (2.13) (2.9) :

b23 = 1. 16 .

2.2. ehp(II). ABC

ehp(I) c a,

. 2.1, a, a. a

b, c, ,

a

. . I

ABC a, b, c ehp(II).

16 3

. (2.1)(2.3) ehp(II)

a = pi a

cosa

= mb ch

c

,

cosa

+ ch

c

= i sin a

shc

shB, i sin

a

shc

chB cos a

chc

= 1.

(2.4) 16

B, C, ehp(II) (2.4)

.

52

3. hhp

3.1. hhp(I). ABC

hhp(I) c ,

a,

b. c c ABC.

17. H , R+, :1) hhp(I) ;

2) ABC hhp(I) ma(ma = (BC,A)) c :

chb

chc

sh b

shc

chA = 1, (3.1)

chc

ch b

= sh

b

shc

shA, (3.2)

chc

= ma ch

b

. (3.3)

. ABC hhp(I) (.

17) a c R

. A3 A, E

a. (1.5 . I)

:

a(1 : 1 : 2), b(b1 : 1 : 0), c(c1 : 1 : 0), b1 R+, c1 R+,A(0 : 0 : 1), B(2 : 2c1 : 1 + c1), C(2 : 2b1 : 1 + b1).(3.4)

. 17. hhp(I)

R A1,

A2 pA A . ,

AA1, b AA2, c: ((AA1)b(AA2)c)) > 0.

b1, c1

c1 > b1. (3.5)

c ,

, E b:

(c(AA2)b(AE)) < 0. (3.5)

b1 (0; 1), c1 (0; 1). (3.6)

B0 = pA b, C0 = pA c H R : B0(1 : b1 : 0), C0(1 : c1 : 0). b, c

Sb, Sc,

(ACSbB0) = 1, (ABScC0) = 1. (3.7)

R: Sb(1 : b1 : 1+b1), Sc(1 : c1 : 1+c1). (3.7) ,

Sb, Sc b, c ABC.

SbSc(1 : 1 : 1) a A(1 : 1 : 0) . A a, (3.6)

(BCAE) =(1 b1)(1 + c1)(1 + b1)(1 c1) > 0.

, SbSc ABC

. 2 hhp(I) .

.

(3.4) (3.6), (2.2 . I)

chb

=

1 + b11 b1 , sh

b

=

2b1

1 b1 , chc

=

1 + c11 c1 , sh

c

=

2c1

1 c1 . (3.8)

, a

b, c.

K1 = AA1 a, K2 = AA2 a R : K1(2 : 0 : 1),K2(0 : 2 : 1). a

(BCK1E) > 0, (BCK2E) > 0.

, K1, K2 a. , a

BAC. (3.4) (3.5), (3.6), (3.2 . I)

A BAC:

chA =b1 + c1

2b1c1

, shA =c1 b12b1c1

. (3.9)

54

ma a

ma = (BC,A) = (BCAE) =(1 b1)(1 + c1)(1 + b1)(1 c1) . (3.10)

(3.8)(3.10) (3.1)(3.3).

.

3.2. hhp(II).

18. H , R+, :1) hhp(II) ;

2) ABC hhp(II) ma a (ma = (BC,A)) :

chb

chc

+ sh

b

shc

ch A = 1, (3.11)

chb

+ ch

c

= sh

b

shc

sh A, (3.12)

chc

= ma ch

b

. (3.13)

.

17. ABC hhp(II) (. 18)

a R .

A3 A, E

a. (1.5 . I)

(3.4).

. 18. hhp(II)

A1, A2 R pA A

, hhp(I):

((AA1)b(AA2)c)) > 0. b1, c1 (3.5).

hhp(II) b, c

. AA1, AE b, c:

((AA1)(AE)bc) < 0. (3.5)

b1 (0; 1), c1 > 1. (3.14)

B0 = pA b, C0 = pA c H R : B0(1 : b1 : 0), C0(1 : c1 : 0). b,

c Sb, Sc, (3.7),

, , b, c. R:

Sb(1 : b1 : 1 + b1), Sc(1 : c1 : 1 + c1). SbSc(1 : 1 : 1) a A(1 : 1 : 0) . A a, (3.14)

(BCAE) =(1 b1)(1 + c1)(1 + b1)(1 c1) < 0. (3.15)

, SbSc ABC

. 2 hhp(II) .

.

(3.4) (3.6), (2.2 . I)

chb

=

1 + b11 b1 , sh

b

=

2b1

1 b1 , chc

=c1 + 1

c1 1 , shc

=

2c1

c1 1 . (3.16)

a

b, c. ,

AA(1 : 1 : 0) (1.5 . I) , (3.15) A a. a , A, ,

b, c.

(3.4) (3.5), (3.14), (3.2 . I)

A BAC:

chA =b1 + c1

2b1c1

, shA =c1 b12b1c1

.

A = pii A ABC

ch A = chA = b1 + c12b1c1

, sh A = shA =c1 b12b1c1

. (3.17)

ma a (3.10). (3.10), (3.16), (3.17)

(3.11)(3.13).

.

56

4. epp

4.1. epp(I).

19. H , R+, :1) epp(I) ;

2) ABC epp(I) mb(mb = (AC,B)), mc (mc = (AB,C)) :

mb = mc , (4.1)

cosa

mb = 1. (4.2)

. ABC epp(I) (.

19) a,

(. I) A, R . A3 A, E21(1 : 1 : 0) A a. A1 (A2) c (b), E

A, a.

: (E13BAA1) > 0, E13 = A2E AA1. ABC R :

A(0 : 0 : 1), B(1 : 0 : t), C(0 : 1 : t), t R+,a(t : t : 1), b(1 : 0 : 0), c(0 : 1 : 0). (4.3)

. 19. : epp(I) a, b, c;

epp(II) a, b, c

Sb, Sc b, c

(ACSbA2) = 1, (ABScA1) = 1

R : Sb(0 : 1 : 2t), Sc(1 : 0 : 2t). SbSc(2t :

2t : 1) a A(1 : 1 : 0). A a, AA(1 : 1 : 0) ,

.. A, a

A. , SbSc

Sb, Sc. 2 ABC epp(I)

. .

pB , pC B, C (4.3)

R : pB(0 : 1 : 2t), pC(1 : 0 : 2t). b,c (4.3) Bp = a pB : B(0 : 2t : 1), C(2t : 0 : 1),Bp(1 2t2 : 2t2 : t).

mb = (ACBA2) =

2t2

2t2 1 , mc = (ABC

A1) =2t2

2t2 1 . (4.4)

(4.4) (4.1).

(2.2 . I) a B, C

a

cosa

=

2t2 12t2

, = 1. (4.5)

, A, C, B, A2 A1 AA2 B, C, Bp, A

. ,

(BCBpA) = (ACBA2). (4.6)

Bp B ,

, BBp = pi/2. Bp ()

a, .. (BCBpA) > 0 ((BCBpA) < 0), a < pi/2 (a > pi/2). (4.4), (4.6) (4.5) = 1,

(4.2).

.

4.2. epp(II). ABC

epp(I) a, b, c, 19, a

a a ( . 19 a ).

a A,

(. I) ABC a, b, a epp(II).

19 3 ABC

epp(II) .

b, c

B, C. a a a = pi a, epp(II) :

mb = mc , cos

a

mb = 1.

5. hpp

5.1. hpp(I).

58

20. H , R+, :1) hpp(I) ;

2) ABC hpp(I) mb(mb = (AC,B)), mc (mc = (AB,C)) :

mb = mc , (5.1)

cha

mb = 1. (5.2)

. (. 4 . I)

hpp(I) a

. A a hpp(I) H.

R , A3 A, E12 = A2E AA1 A (. 20, ). A1, A2

. ABC R

:

A(0 : 0 : 1), B(1 : 0 : t), C(0 : 1 : t), t R,a(t : t : 1), b(1 : 0 : 0), c(0 : 1 : 0). (5.3)

. 20. : hpp(I) (); hpp(II) ( )

Sc(1 : 0 : 2t), (ABScA1) = 1, c. ScA

b

Sb(0 : 1 : 2t), (ACSbA2) = 1. , ScA

ABC : Sb,

Sc. 2 hpp(I) .

.

pB(0 : 1 : 2t), pC(1 : 0 : 2t) B, C b, c (5.3) B(0 : 2t : 1),C(2t : 0 : 1) .

mb = (AC,B) =2t2

2t2 + 1, mc = (AB,C) =

2t2

2t2 + 1. (5.4)

(2.2 . I) a B, C

a

cha

=

2t2 + 1

2t2. (5.5)

(5.4), (5.5) (5.1), (5.2).

. 5.2. hpp(II).

21. H , R+, :1) hpp(II) ;

2) ABC hpp(II) mb(mb = (AC,B)), mc (mc = (AB,C)) :

mb = mc , (5.6)

cha

mb = 1. (5.7). (. 4 . I)

hpp(II)

, ..

.

ABC (. 20, )

a R

epp(I) (. . 4.1).

ABC (4.3).

E

A, AB, AC. (BE13A1A) = t > 0, E13 = A2E AA1. (1.5 . I) (t : t : 1) a : 4t2 1 < 0. , t (0; 1/2). A(1 : 1 : 0) a,

A, a.

SbSc, (.

19), ABC . ,

2 hpp(II) .

.

ABC

(4.4), , (5.6).

(2.2 . I) a B, C

a t (0; 1/2)

cha

=

1 2t22t2

. (5.8)

60

(4.4), (5.8) (5.7).

.

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I. 1 2 3 3.1 3.2

4 5

II. 1 eee1.1 eee(I)1.2 eee(II)1.3 eee(III)1.4 eee(IV)

2 eeh2.1 eeh(I)2.2 eeh(II)

3 ehh3.1 ehh(I)3.2 ehh(II)

4 hhh4.1 hhh(I)4.2 hhh(II)

III. 1 eep1.1 eep(I)1.2 eep(II) eep(III)

2 ehp2.1 ehp(I)2.2 ehp(II)

3 hhp3.1 hhp(I)3.2 hhp(II)

4 epp4.1 epp(I)4.2 epp(II)

5 hpp5.1 hpp(I)5.2 hpp(II)