. . . (4- . — ! !), 2002

13. | | 14.

13.1. )
a=
BC


,
b=
CA



c=
AB


; AA, BB CC - ABC.

AA

=(cb)/2

,

BB

=(ac)/2



CC

=(ba)/2

.

AA

+
BB

+
CC

=
0


.

)
a1=
AA


,
b1=
BB



c1=
CC


. (c1b1)/2=(baa+c)/4=3a/4 - A2B2C2.

13.2. a, b c - T. (ba)/2, (ac)/2 (cb)/2 - . , a, b c - , T1 . ba, ac cb - .

13.3. ,
2
M1M2

=
A1A2

+
A2A3

=
A1A3


,
2
M3M4

=
A3A5



2
M5M6

=
A5A1


.

M1M2

+
M3M4

+
M5M6

=
0


.

13.4. 90 a1,,an n-.

13.5. . 1, - ; .

13.6.
a=
AE


,
b=
DF



v=
AD


.
2
AK

=b+v


2
AL

=a+v+2b

,

KL

=
AL


AK

=(a+b)/2

.

NM

=(a+b)/2

.

13.7. , , : a1,,an. A1An,

AiAi+1

=ai

. , A1An - . , Ox a1. a2,,ak Ox, ak+1,,an - ( , a1, ). Oy , - . A2,A3,,Ak+1, Ak+1,,An,A1 : d, - d . , A1A2. , , .

13.8. 13.7 . .13.1.

.13.1

13.9. 13.8,) ; . : a , . , . , , (.9.14).

13.10. BE AD AC F G. AFE BCD , AF:FE=BC:CD. , AD:BE=(AF+BC):(EF+CD)=BC:CD. AE:BD=DE:AC. BED EGA AE:DB=EG:BE = CD:BE. ,
BC

AD
= CD

BE
= AE

BD
= DE

AC
=l

. ,

BC

+
CD

+
DE

+
EA

+
AB

=
0


,

AD

+
BE

+
CA

+
DB

+
EC

=
0




BC

=l
AD


,

CD

=l
BE


,

DE

=l
CA


,

EA

=l
DB


. ,

0

=l(
AD

+
BE

+
CA

+
DB

)+
AB

= l
EC

+
AB


, ..

AB

=l
EC


. AB||EC.

13.11.
a=
AB


,
b=
BC


,
c=
CD



d=
DA


. , AC^BD , a2+c2=b2+d2. , d2=|a+b+c|2=a2+b2+c2+2[(a,b)+(b,c)+(c,a)]. AC^BD, .. 0=(a+b,b+c)=b2+(b,c)+(a,c)+(a,b), , d2=a2+b2+c22b2.

13.12. )

AB


,

BC




CD

,br /> ..

AD

=
AB

+
BC

+
CD


,

CA

=
AB


BC




BD

=
BC

+
CD


. .

) D - , A C ABC. ) , , .. BD^AC.

13.13. , AH^BC.

AH

=
AO

+
OH

=
AO

+
OA

+
OB

+
OC

=
OB

+
OC




BC

=
BO

+
OC

=
OB

+
OC


,
(
AH

,
BC

)=OC2OB2=R2R2=0

, O - . , BH^AC CH^AB.

13.14. ai=(ai,a1). , a1, , a1,
a1=

i > 1
aicosai


0=

i > 1
aisinai

. ,
a12=

i > 1
ai2(cos2ai+sin2ai)+2

i > j > 1
aiaj(cosaicosaj+sinaisinaj) = a22++an2+2

i > j > 1
aiajcos(aiaj)

. , aiaj=(ai,a1)(aj,a1)=(ai,aj)=jij.

13.15.
a=
AD


,
b=
BD



c=
CD

.

BC2=|bc|2=BD2+CD22(b,c), U=2(b,c). , V=2(a,c) W=2(a,b). a =(a,b) b =(b,c). cos2a+cos2b+cos2(a+b)=2cosacosbcos(a+b)+1 (.12.39,)) 4uvw=4|a|2 |b|2 |c|2, .

13.16. O.
m=
OM


,
a=
OA

,,d=
OD


.
(
MA

,
MB

)(
MC

,
MD

)=(am,bm)(cm,dm)=(c+dab,m)+(a,b)(c,d)

. v=c+dab 0, M , (v,m) , , (c,d)(a,b). , v=0, ..

OC

+
OD

=
OA

+
OB


, ,

AC

=
DB


.

13.17. n1,,nk - , aM1,,Mk - . X, , i-
(
XMi

,ni)

. A B ,
k

i=1
(
AMi

,ni) = k

i=1
(
BMi

,ni)= k

i=1
(
BA

,ni) + k

i=1
(
AMi

,ni)

, ..


BA

, n

i=1
ni
=0.

, ,

ni=0

.

13.18. l - , n - , l. A B , l, n,
r(B,l)r(A,l)=(
AB

,n)

, r(X,l) - X l.

n1, n2, n3 n4 - , ABCD . X ABCD

(X)

.
0=
(B)
(A)=(
AB

,n1+n2+n3+n4).


(
BC

,n1+n2+n3+n4)=0.

A, B C , n1+n2+n3+n4=0. 13.5.

13.19.
a=
AB


,
b=
BC



c=
CD


.

AD

=a+b+c

,

AC

=a+b



BD

=b+c

. , |a|2+|b|2+|c|2+|a+b+c|2|a+b|2|b+c|2=|a|2+2(a,c)+|c|2=|a+c|2 0. , a=c, .. ABCD - .

13.20. a1,a2,a3,a4,a5 , . |a1+a2| > |a3+a4+a5|, |a1|2+2(a1,a2)+|a2|2 > |a3|2+|a4|2+|a5|2+2(a3,a4)+2(a4,a5)+2(a3,a5). , 4(|a1|2+)+2((a1,a2)+) > 6(|a1|2+)+6((a1,a2)), .. |a1+a2+a3+a4+a5|2 < 0. .

13.21. e1,,e10.

AB

=e1++e10

. , AB . ,
|
AB

ei|2=AB22(
AB

,ei)+|ei|2

, ..
(
AB

,ei)=(AB2+|ei|2|
AB

ei|2)/2

.
AB > |
AB

ei|

,
(
AB

,ei) > 0

, .. ei AB .

13.22.
ai=
OAi



x=
OX


. |ai|=R

XAi

=aix

.

XAi=
|aix|=
|aix||ai|/R
(aix,ai)/R =
(ai,ai)/R(x,
ai)/R

. , (ai,ai)=R2

ai=0

.

13.23. (a,b), (c,d), (e,f) (g,h). 360/4=90. 90, .

, .

13.24. . n=0 , , . , 2n+1 . 2n+3 (.. , ). , -

OP1




OP2n+3

.



OR

=
OP2

++
OP2n+2


1.

OR


P1OP2n+3,

OS

=
OP1

+
OP2n+3


. ,
|
OS

+
OR

| OR 1

.

13.25. , a, b c - , 1, ab, ac, bc , 1.   , a, b, c , 60, , 1 ( AB 1, BC 1 ABC 60, AC - AC 1).

a b. a b ab 90, |ab| 2, |a+b| 2.

13.26. , a , . , Oy a. - : e1,e2, (.13.2). k. , v1,,vk, i=1,,k vi+ei . .   , ( ). e1,v1,,ek,vk , n2k 1. n2k.

.13.2

e1,,ep . v1,,vp. , , Ox 0 90. , , , . Ox e1, e2,,ek, , ; ek v1. , Ox e2, v2 .. , Ox . ep+1,,ek, , ( ep+1 , , ).

13.27. X AB ,

AX

=l
AB


, ..

OX

=
OA

+
AX

= (1l)
OA

+l
OB


.

13.28. O

AiAj

=
OAj


OAi


.  

OAi


, .

13.29. e1, e2 e3 - ,

OA


,

OB




OC


; a =BOC, b =COA g =AOB. ,
e1sina+e2sinb+e3sing =
0


. A1B1C1, OC, OA OB.

0

=
A1B1

+
B1C1

+
C1A1

=2R(e1sina+e2sinb+e3sing)

, R - ABC.

13.30. a b - , OA OB, l =OA m = OB. AB X,

OX

=t
OA

+(1t)
OB

=tla+(1t)mb

. x0 y0, x0/l =t=1(y0/m) l m. x0=p/c y0=q/c.   , p/OA+q/OB=c, AB X,

OX

=(pa+qb)/c

.

13.31.
a=
MA


,
b=
MB



c=
MC


.
e=
MC1

=pa+(1p)b



MA1

= qc+(1q)b=qa+(12q)b

. Ѡ ,

MA1

=ae

.
be=
MB1

=rb+(12r)a

. , 1+(1/a)+(1/b)=0. apa+a(1p)b=ae=qa+(12q)b, ap=q a(1p)=12q, , 1/a =13p. bp=12r b(1p)=r, , 1/b =3p2.

13.32.

AA2

+
BB2

+
CC2

=
0




A2B2

+
B2C2

+
C2A2

=
0


,

AB2

+
BC2

+
CA2

=
0


. ,

AB2

=l
C2B2


,

BC2

=l
A2C2




CA2

=l
B2A2


. E - BC, A2E||AA1.

BA1

=l
EA1




EC

=l
EA1


,

A1C

=
EC


EA1

= (l1)
EA1


. ,

A1C
/

BA1
=(l1)/l

.

AB1
/

B1C
=

BC1
/

C1A
=(l1)/l

.

13.33. O - ,
a=
OA


,
b=
OB


,
c=
OC



d=
OD


. Ha - BCD,

OHa

=b+c+d

(. 13.13).

OMa

=(a+b+c+d)/2=
OMb

=
OMc

=
OMd


.

13.34. O - ,
a=
OA


,
b=
OB


,
c=
OC



d=
OD


. Hd - ABC,

OHd

=a+b+c

(13.13).

) K ,

OK

=a+b+c+d

.
KHd=|
OK


OHd

|=|d|=R

, .. K Sd. , K Sa, Sb Sc.

) Od - ABC, .. OHd.

OOd

=
OHd

/2 = (a+b+c)/2

. X ,

OX

= (a+b+c+d)/2

. XOd=|d|/2=R/2, .. X ABC. , X BCD, CDA DAB.

13.35. X1 - X l.
a
AA1

+b
BB1

+g
CC1



a
AX1

+b
BX1

+g
CX1


, l. ,
a
AX1

+b
BX1

+g
CX1

=a
AX

+b
BX

+g
CX

+(a+b+g)
XX1



a
AX

+b
BX

+g
CX

=
0


(13.29), .

13.36.
a=
A1A2

+
A3A4

++
A2n1A2n


, a 0. , Ox a.

A1A2

,
A3A4

,,
A2n1A2n


Oy , a Ox ; , a , . , , , .. 2.

13.37. ( ) BC, CA AB. , , BC; BC. P - A BC, N - BC.

PN

=
PC

+
CN

=(b2+a2c2)/2a(a/2)=(b2c2)/2a

(PC AB2BP2=AC2CP2). NM:NA = 1:3,

MO


BC

PN

/3=(b2c2)/6a

. , a3na+b3nb+c3nc BC
b3singc3sinb= b3 cc3 b

2R
= abc

2R
b2c2

a
=2S b2c2

a
.

13.38. AB, BC CA U, V W. ,

OZ

= 3R

r

ZK


, ..

OZ

= R

r
(
ZU

+
ZV

+
ZW

)

. , , ( ) BC ; BC. N - O BC.

OZ


BC

NV

=
NC

+
CV

=(a/2)(a+bc)/2=(cb)/2

.

ZU

+
ZV

+
ZW




ZU

+
ZW


, ..
rsinVZU+rsinVZW=rsinB+rsinC=r(cb)/2R.

13.39. Oxy. lj - , O j (0 < j < p) Ox, .. A lj A , AOX=j; l0=lp=Ox.

a a Ox ( Ox a), a lj |a||cos(ja)|. 0p |a||cos(ja)|dj =2|a| a.

a1,,an,b1,,bm Ox a1,,an,b1,,bm. |a1| |cos(ja1)|++|an| |cos(jan)| |b1| |cos(jb1)|++|bm| |cos(jbm)| j. j 0 p, |a1|++|an| |b1|++|bm|.

.
1

ba
ab f(x)dx

f [a,b].

p

0
|a||cos(ja)|dj =2|a|
, a 2|a|/p, , f(j), a lj, [0,p] 2|a|/p.

13.40. . , . , 13.39 , .. , , .

13.41. L, 13.39 2L/p.

f [a,b] c,
c= 1

ba

b

a
f(x)dx < (ba)c

ba
=c.
l, 2L/p.

l . , L/p. , , , , L/p.

13.42. l AB. , A B A1 B1 . A1B1 AB, .. A1B1, aA1B1 < d . l 2AB, 2d.

2P/p, P - (.13.39). , , 2P/p < 2d, .. P < pd.

13.43. 13.39 |a|+|b|+|c|+|d| |a+d|+|b+d|+|c+d| , .. , a, b, c d - , , .. , a+b+c+d=0. , d 0, .

, a b c. : 1)a,b,c 0; 2)a 0 b,c 0 3)a,b 0, c 0. . |d| |b|, |b| |d| |a| |a| |d| ( , |d|=|a|+|b||c| |a|+|b|).

13.44. 13.39 .

OA1

,,
OAn


l ( ) a1,,an. a1,,an : x1 x2 xk 0 y1 y2 ynk 0. yi=yi. x1++xk=y1++ynk=a, , x1 a/k y1 a/(nk). 2(x1+y1).

OAi


x1++xk+y1++ynk=2a.
2(x1+y1)

x1++ynk
2((a/k)+(a/(nk)))

2a
= n

k(nk)
,
, k(nk) k=n/2 n k=(n1)/2 n.

13.45. - . P l, dl. 1e < dl < 1 l. , e . , l 2dl.

2dl 2P/p (.13.39). 22e < 2P/p < 2, .. ppe < P < p. e . , p.

13.46. , . , .

13.47. ) l < 0, (la)b= l|a||b|sin(a,b)= l|a||b| sin(a,b)=l(a b). l > 0 .

)
a=
OA


,
b=
OB



c=
OC


. , Oy OA. A=(0,y1), B=(x2,y2) C=(x3,y3). a b=x2y1, a c=x3y1 a (b+c)=(x2+x3)y1=a b+a c.

13.48. e1 e2 - , Ox Oy. e1 e2=e2 e1=1 e1 e1=e2 e2=0. a b=(a1e1+a2e2) (b1e1+b2e2)=a1b2a2b1.

13.49. ) ,

AB


AC

=
AB

(
AB

+
BC

)=
BA


BC

=
BC


BA


.

)

AB


AC

=(
AD

+
DB

) (
AD

+
DC

) =
AD


DC

+
DB


AD

+
DB


DC

=
DC


DA

+
DA


DB

+
DB


DC


.

13.50. , .. t=0,

AB

=v



AC

=w

. t

AB

=v+t(ba)



AC

=w+t(ca)

, a, b c - A, B C. a, b c , (ba)(ca) = 0, ,
|S(A,B,C)|=|
AB


AC

|/2=|x+yt|

, x y - . |x|=2, |x+5y|=3, , ABC t |2+(t/5)| |2t|. t=10 4 8.

13.51. v(t) w(t) - , t. , v(t) = ta+b w(t)=tc+d. , v(t)||w(t), .. v(t) w(t)=0. f(t)=v(t) w(t)=t2 a c+t (a d+b c)+b d , f(0) 0. , , .

13.52.

OC

=a

,

OB

=la

,

OD

=b



OA

=mb

.
2SOPQ=
OP


OQ

= ((a+mb)/2) ((la+b)/2)=(1lm)(ab)/4

2SABCD=2(SCODSAOB) = (abla mb)=(1lm)ab.

13.53.
aj=
P1Aj


. P
(x+a1)(x+a2)+(x+a3) (x+a4)++(x+a2n1) (x+a2n),

x=
PP1


; P1 x(a1a2+a3a4++a2n1a2n)=xa. x a=0
x=
P2P1



x=
P3P1


, . , a=0, .. xa=0 x.

13.54.
a=
AP


,
b=
BQ



c=
CR


.

QC

=aa

,

RA

=bb



PB

= gc

, (1+a)a+(1+b)b+(1+g)c=0. ,

AB


CA

=
PQ


RP


. (a+gc)(c+bb)(gc+b)(a+bb) = ac+bab+ab+gac=a[(1+g)c+(1+b)b] = a(1+a)a=0.

13.55.
ai=
A4Ai



wi=
A4Hi


. 13.49,) , a1a2+a2a3+a3a1 = w1w2+w2w3+w3w1. a1w2 a2w1 a3, , .. (a1w2)(a2w1)=0. (a2w3)(a3w2)=0 (a3w1)(a1w3)=0, .

13.56. x=x1e1+x2e2. e1x = x2(e1e2) xe2=x1(e1e2), ..
x=((xe2)e1+(e1x)e2)/(e1e2).
(e1e2)y,
(xe2)(e1y)+(e1x)(e2 y)+(e2e1)(xy)=0.(1)

e1=
AB


,
e2=
AC


,
x=
AD



y=
AE


. S=a+xe2+d = c+ye2 +a=d+xe1+b, .. xe2=Sad, ye2=Sca xe1=Sdb. (1), .




File translated from TEX by TTH, version 3.00.
On 22 Nov 2002, 13:46.

13. | | 14.

Copyright ©2002 ! !
.
: biblio@mccme.ru.
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