. . . (4- . — ! !), 2002

10. | | 11.

10.1.
AA1 BB1 M. BC > AC, A C AB, , CC1 M. , AM < BM, ..ma < mb.
10.2.
, , a > b. ma < mb (10.1). A1MB1C , 
a

2
+ mb

3
= b

2
+ ma

3

, ..(ab)/2=(mamb)/3. .
10.3.
, , BC > AC. MA < MB (. 10.1), BC+MB+MC > AC+MA+MC.
10.4.
) c a+b, c2 (a+b)2=a2+b2+2ab 2(a2+b2).
) M- ABC. ) MA2+MB2 AB2/2, ..
4ma2

9
+ 4mb2

9
c2/2

.

10.5.
) M- , O- ABC.
AO2+BO2+CO2 = (
AM

+
MO

)2+(
BM

+
MO

)2+(
CM

+
MO

)2=AM2+BM2+CM2+2(
AM

+
BM

+
CM

,
MO

)+3MO2

.

AM

+
BM

+
CM

=
0


, AO2+BO2+CO2=AM2+BM2+CM2+3MO2 AM2+BM2+CM2, ..3R2 4(ma2+mb2+mc2)/9.
) , (ma+mb+mc)2 3(ma2+mb2+mc2) (. .9).

10.6.
16S2=2a2b2+2a2c2+2b2c2a4b4c4. mc2=(2a2+2b2c2)/4 (12.11,)), mc2 ((a2+b2)/2c)2 mc2 > ((a2b2)/2c)2 16S2 4a2b2 16S2 > 0 .
10.7.
y=a2+b2+c2 y1=ma2+mb2+mc2. 3y=4y1 (12.11,)), y < 2x (9.7) 2x1+y1 < 2x+y, (ma+mb+mc)2 < (a+b+c)2 (. 9.2). 8x1+4y1 < 8x+4y 3y=4y1, 8x1 < y+8x < 10x, ..x1/x < 5/4.
M- ABC. AMB AMBN. AMN , (x/4)/(4x1/9) < 5/4, ..x/x1 < 20/9.

10.8.
, ha b, hb c, hc a, . ha+hb+hc < a+b+c.
10.9.
ha > 1 hb > 1. a hb > 1. S=aha/2 > 1/2.
10.10.
BH AC, , BH AC AM. AM BC BH. BH=AM=AC=BC. AC=AM, AC AM , ..C=90, AC=BC, ABC 45, 45, 90.
10.11.
,
1

ha
+ 1

hb
= a+b

2S
= a+b

(a+b+c)r

a+b+c < 2(a+b) < 2(a+b+c).
10.12.
aha=2S=r(a+b+c),
ha=r
1+ b

a
+ c

a


. ha,hb hc
x

y
+ y

x
2

, .
10.13.
hahb=2S(1/a1/b)=2S(ba)/ab 2S ab, hahb ba.
10.14.
12.21
2

ha
= 1

rb
+ 1

rc

. ,
1

rb
+ 1

rc
2/


rbrc

.
10.15.
2sinbsing =cos(bg)cos(b+g) 1+cosa,
ha

a
= sinbsing

sina
1+cosa

2sina
= 1

2
ctg a

2
.
10.16.
b/2R=sinb, 2p (a+b+c)(ha+hb+hc) 3sinb(a2+ac+c2). 6S, a(hb+hc)+b(ha+hc)+c(ha+hb) 3sinb(a2+c2). , , ahb=a2sing =a2c/2R, a(b2+c2)2b(a2+c2)+c(a2+b2) 0. f(x)=x2(a+c)2x(a2+c2)+ac(a+c). , f(a)=a(ac)2 0 蠠f(c)=c(ac)2 0. x a b c, f(b) 0.
10.17.
12.35,) la2=4bcp(pa)/(b+c)2. , 4bc (b+c)2.
10.18.
, ha/la=cos((bg)/2). 12.36,)
2r/R=8sin(a/2)sin(b/2)sin(g/2)=4sin(a/2)[cos((bg)/2)

cos((b+g)/2)]=4x(qx), 堠x=sin(a/2)蠠q=cos((bg)/2).
, 4x(qx) q2.

10.19.
) 10.17 la2 p(pa). , .
) la,lb lc (la+lb+lc)2 3(la2+lb2+lc2).

10.20.
,



p(pa)
+


p(pb)
+mc


3p
.
, p=1; x=1a y=1b. mc2=(2a2+2b2c2)/4=1(x+y)+(xy)2/4=m(x,y).
f(x,y)=x+y+


m(x,y)

. , f(x,y) 3 x,y 0 x+y 1.
g(x)=f(x,x)=2x+


12x

.
g(x)= 1

x
1




12x

, x 0 1/3 g(x) 1 3, x 1/3 1/2 g(x) 3 2. d=xy q=x+y. , (xy)22q2(x+y)+q4=0, ..x+y=(d2+q4)/2q2.
f(x,y)=q+


1 q2

2
d2(2q2)

4q2
.
, q2=(x+y)2 2(x+y) 2, .. d2(2q2)/4q2 0. , q f(x,y) , d=0, ..x=y; x=y .
10.21.
,
1

a
+ 1

b
+ 1

c
=(ha+hb+hc)/2S

. , 9r ha+hb+hc (10.12) ha+hb+hc ma+mb+mc 9R/2 (10.5,)).
10.22.
, b+ca < 2bc/a. 2x=b+ca,2y=a+cb 2z=a+bc. , 2x < 2(x+y)(x+z)/(y+z), ..xy+xz < xy+xz+x2+yz. .
2bccosa =b2+c2a2=(b+ca)(b+c+a)2bc,
2bccosa

b+c
=b+ca+
(b+ca)a

b+c
2bc

b+c

.
, b+ca < 2bc/a.

10.23.
12.30 a2+b2+c2=(a+b+c)22(ab+bc+ac)=4p22r22p28rR=2p22r28rR abc=4prR. , 2p22r28rR < 2(14prR), p=1. .
10.24.
12.30 ab+bc+ca=r2+p2+4Rr. , 16Rr5r2 p2 4R2+4Rr+3r2 (10.34).
10.25.
r(ctga+ctgb)=c=rc(tga+tgb),
c2=rrc
2+ tga

tgb
+ tgb

tga

4rrc.
10.26.
12.36,) 10.45. , x(1x) 1/4, ..r/R 1/2.
10.27.
hc a hc b, 4S=2chc c(a+b). 6r(a+b+c)=12S 4ab+4S (a+b)2+c(a+b)=(a+b)(a+b+c).
10.28.

2

ha
= 1

ra
+ 1

rc

(12.21), 
ra

ha
=
ra

rb
+ ra

rc

/2

. rb/hb rc/hc . ,
x

y
+ y

x
2

, .
10.29.
Rr=RS/p=abc/4p (. 12.1), 27abc 8p3=(a+b+c)3.
(a+b+c)2 3(a2+b2+c2) a,b c, p2 3(a2+b2+c2)/4=ma2+mb2+mc2 (. 12.11,)). , ma2+mb2+mc2 27R2/4 (10.5,)).

10.30.
OA=r/sin(A/2), OB=r/sin(B/2) OC=r/sin(C/2), A/2, B/2 C/2 , A B C. , A 60 B 90, , sin(A/2) 1/2 sin(B/2) 1/2.
10.31.
C 120, a+b (12.21); , a+b 6r (12.27).
120, (a2+b2+c2)/2+23S (18.22,)). , (a2+b2+c2)/2 23S (10.53,)) 43S 36r2 (10.53,)).

10.32.
a =cos(A/2), b =cos(B/2) g =cos(C/2). 12.17,) a/ra=a/bg, b/rb=b/ga c/rc=g/ab. abg 3(a2+b2+g2) 4(b2g2+g2a2+a2b2). a2=(1+cosA)/2,b2=(1+cosB)/2 g2=(1+cosC)/2, cosA+cosB+cosC+2(cosAcosB+cosBcosC+cosCcosA) 3. 10.36 10.43.
10.33.
) 4R+r=ra+rb+rc (12.24) R2r 0 (10.26),
5Rr ra+rb+rc = pr((pa)1+(pb)1+(pc)1) =

= p(ab+bc+cap2)/S = p(2(ab+bc+ca)a2b2c2)/4S.
, 2(ab+bc+ac)a2b2c2 43S (10.54).
) , 4Rra=rb+rcr=pr/(pb)+pr/(pc)pr/p=(pa)(p2bc)/S. , 4(p2bc)=a2+b2+c2+2(abbc+ca)=2(ab+bc+ac)a2b2c2+2(a2+b2+c22bc) 43S+2(a2+(bc)2).

10.34.
a,b c- , F=(ab)(bc)(ca)=AB, A=ab2+bc2+ca2 B=a2b+b2c+c2a. , , F2 0. s1=a+b+c=2p,s2=ab+bc+ca=r2+p2+4rR s3=abc=4prR (. 12.30). , F2=s12s224s234s13s3+18s1s2s327s32.   , (s1s2)2F2 = (A+B+3abc)2-(AB)2=4AB+6(A+B)s3+9s32=4(a3b3+)+4(a4bc+)+6(A+B)s3+21s32. , 4s23=4(a3b3+)+12(A+B)s3+24s32,4s13s3=4(a4bc+)+12(A+B)s3+24s32 蠠18s1s2s3=18(A+B)s3+54s32.
s1,s2 s3 p,r R,
F2=4r2[(p22R210Rr+r2)24R(R2r)3] 0.
,
p2 2R2+10Rrr22(R2r)


R(R2r)
=

= [(R2r)


R(R2r)
]2 + 16Rr5r2 16Rr5r2

p2 2R2+10Rr+r2+2(R2r)


R(R2r)
=

= 4R2+4Rr+3r2[(R2r)


R(R2r)
]2 4R2+4Rr+3r2.

10.35.
ra+rb+rc=4R+r rarb+rbrc+rcra=p2 (10.24 10.25), ra2+rb2+rc2=(4R+r)22p2. 10.34 p2 4R2+4Rr+3r2, ra2+rb2+rc2 8R25r2. , r R/2 (10.26).
10.36.
) 12.38 cosa+cosb+cosg =(R+r)/R. , r R/2 (10.26).
) ) (. ).

10.37.
) , sina+sinb+sing =p/R. , p 33 R/2 (10.29).
) ) (. ).

10.38.
) 12.44,) ctga+ctgb+ctgg =(a2+b2+c2)/4S. , a2+b2+c2 43S (10.53,)).
) ) (. ).

10.39.
) 12.44,) ctg(a/2)+ctg(b/2)+ctg(g/2)=p/r. , p 33r (10.53,)).
) ) (. ). tga+tgb+tgg < 0; ., , 12.46.

10.40.
) 12.36,) sin(a/2)sin(b/2)sin(g/2)=r/4R. , r R/2 (10.26).
) ) (. ). cosacosbcosg < 0.

10.41.
) sinx=2sin(x/2)cos(x/2), , 12.36,) 12.36,), sinasinbsing =pr/2R2. , p 33R/2 (10.29) r R/2 (10.26).
) ) (. ).

10.42.
12.39,) cos2a+cos2b+cos2g =12cosacosbcosg. , cosacosbcosg 1/8 (10.40,)), cosacosbcosg < 0.
10.43.
, 2(cosacosb+cosbcosg+cosgcosa)=(cosa+cosb+cosg)2cos2acos2bcos2g. , cosa+cosb+cosg 3/2 (10.36,)) cos2a+cos2b+cos2g 3/4 (10.42).
10.44.
ABC a,b g A1,B1 C1. SABC=R2(sin2a+sin2b+sin2g)/2 SA1B1C1=R2(sin(a+b)+sin(b+g)+sin(g+a))/2. 12.72 10.26.
10.45.
, 2sin(b/2)sin(g/2)=cos((bg)/2)cos((b+g)/2) 1sin(a/2).
10.46.
A B AA1 BB1 ACB. AB AA1+BB1 = bsin(g/2)+asin(g/2).
10.47.
12.32 tg(a/2)tg(b/2)=(a+bc)/(a+b+c). a+b < 3c, a+bc < (a+b+c)/2.
10.48.
p2a > 0, p2b > 0, p2g > 0 (p2a)+(p2b)+(p2g)=p, p2a, p2b, p2g. , p2a, p2b, p2g, sin(p2a)=sin2a, sin2b, sin2g. p2a > p2b > p2g ó , sin2a > sin2b > sin2g.
10.49.
, cos2gcos(pab) = cos2acos2bsin2asin2b. cos2a+cos2bcos2g =cos2a+cos2bcos2acos2b+sin2asin2b.
acosj+bsinj


a2+b2

(. .9), 
(1cos2b)cos2a+sin2bsin2a+cos2b


(1cos2b)2+sin22b
+cos2b =2|sinb|+12sin2b

. , 2t+12t2 t=1/2 3/2. a =b =30, g =120.
10.50.
AB < CB, AX < CX SABX=SBCX, sinXAB > sinXCB. , XCB , .
10.51.
ABC a,b g, A1B1C1 (b+g)/2, (g+a)/2 (a+b)/2.
10.52.
M- AA1,BB1 CC1. AMB AMBN, BMC1=am AMC1=bm. , C1CB < g/2 B1BC < b/2. , am=C1CB+B1BC < (b+g)/2 < b. gm=A1AB+B1BA > (a+b)/2 > b.
, ABC . H AMC1. , AMB < AHB, ..pgm < pg, CMB > CHB, ..pam < pa. , a . CC1B , , am , ..am < a. M MX BC. gm > XMB > 180HAB > g.

a > am, a+(pam) > p, .. M AB1C1. , g =AB1C1 < AMC1=bm. a =CB1A1 > CMA1=bm, g+(pgm) < p.

10.53.
) , S2/p=(pa)(pb)(pc) ((pa+pb+pc)/3)3=p3/27. pr=S p2/33, ..r p/33. r, .
) (a+b+c)2 3(a2+b2+c2), S p2/33 = (a+b+c)2/123 (a2+b2+c2)/43.

10.54.
x=pa, y=pb, z=pc. (a2(bc)2)+(b2(ac)2)+(c2(ab)2)=4(pb)(pc)+4(pa)(pc)+4(pa)(pb) = 4(yz+zx+xy)
43S = 4


3p(pa)(pb)(pc)
=4


3(x+y+z)xyz
.
,
xy+yz+zx


3(x+y+z)xyz

.
x2y2+y2z2+z2x2 x2yz+y2xz+z2xy.
x2yz x2(y2+z2)/2,y2xz y2(x2+z2)/2 z2xy z2(x2+y2)/2, .
10.55.
) S=(absing)/2, S3=((abc)2singsinbsina)/8. 10.41.
) (hahbhc)2=(2S)6/(abc)2 (abc)2 (4/3)3S3, (hahbhc)2 (2S)6(3/4)3/S3=(3S)3.

(rarbrc)2=S4/r2 (12.18,) r2(3)3 S (10.53,)), (rarbrc)2 (3S)3.

10.56.
p=BA1/BC, q=CB1/CA r=AC1/AC. SA1B1C1/SABC=1p(1r)q(1p)r(1q) = 1(p+q+r)+(pq+qr+rp). (5.77) pqr=(1p)(1q)(1r), ..2pqr=1(p+q+r)+(pq+qr+rp); , (pqr)2=p(1p)q(1q)r(1r) (1/4)3. ,
SA1B1C1/SABC=2pqr 1

4

.
10.57.
, ABC 1. a+b+c=1u, u2 4abc. x=BA1/BC, y=CB1/CA z=AC1/AB. u=1(x+y+z)+xy+yz+zx abc=xyz(1x)(1y)(1z)=v(uv), v=xyz. u2 4v(uv), ..(u2v)2 0. .
10.58.
) x=BA1/BC, y=CB1/CA z=AC1/AB. , ABC 1. SAB1C1=z(1y), SA1BC1=x(1z) SA1B1C=y(1x). x(1x) 1/4, y(1y) 1/4 z(1z) 1/4, SAB1C1, SA1BC1 SA1B1C (1/4)3, , 1/4.
) x 1/2. y 1/2, C 2 A1 B1 BC AC, , SA1B1C SA1B1C1. , y1/2 z1/2. x=(1+a)/2, y=(1+b)/2 z=(1+g)/2. SAB1C1=(1+gbbg)/4,SA1BC1=(1+agag)/4 SA1B1C=(1+baab)/4, , SA1B1C1=(1+ab+bg+ ag)/4 SAB1C1+SA1BC1+SA1B1C 3/4.

10.59.
, AC < BC, ABC < BAC. AC < BC, BC A1 , A1C=AC. BAC > A1AC = AA1C > ABC.
10.60.
A1- BC. AA1 < BC/2=BA1=A1C, BAA1 > ABA1 CAA1 > ACA1, A=BAA1+CAA1 > B+C, ..A > 90. , AA1 > BC/2, A < 90.
10.61.
, , . A > A1 , BD > B1D1, ..C > C1. , B B1. AC A1C1, ..D > D1. 360=A+B+C+D > A1+B1+C1+D1=360. ; , B < B1 D < D1.
10.62.
B1 B M. , M BC, AH, .. BM, MBC=30. AH- , BC- . AB1=BC AB, ..ABB1 AB1B=MBC=30. , ABC=ABB1+MBC 30+30=60.
10.63.
, A > D. BE > EC EBA < ECD. EBC BE EC, EBC < ECB. B=ABE+EBC < ECD+ECB=C, . , A=B=C=D. B>E C < D. B=C=D=E.
10.64.
. MN M1 N1. , MN M1N1. M1 AB, N1- PQ. AM1N1+BM1N1=180, 90. AM1N1 90. AN1 M1N1, ó . , AN1 AP, AN1 AQ. , MN .
10.65.
, . , M N . .
1. M N . MN=2Rsin(MON/2) 2Rsin(AOB/2)=AB, MON/2 AOB/2 90.

2. M N AO BO. MN AOB.

3. M N , - AO BO. M AO, N- . MN ANO. , AO=NO=R AN AB.

10.66.
, A ( 1) , X AB . X DE (D E AB AC). AD AE AB DE < (DE+AD+AE)/2=AB, .. ADE AB. ADE ( DE), AB.

.10.1

10.67. , O A1A2A3A4A5. A1OA2, A2OA3,, A5OA1.   2p, , A1OA2, 2p/5. A1A2 OBC, BOC=2p/5 B C .  OBC BC, A1A2 BC.

O , A1OA2,, A5OA1 p, . 2p, .. 2p/5. .

O , p/4, , p/3. , p/4 < p/3 < 2p/5.

10.68.
BC, CA, AB A1 A2, B1 B2, C1 C2 , B1C2||BC, C1A2||CA,A1B2||AB (.10.1).  A1A2O, B1B2O, C1C2O A1A2, B1O, C2O . OP < A1A2, OQ < B1O, OR C2O, .. OP+OQ+OR < A1A2+B1O+C2O=A1A2+CA2+BA1=BC.
10.69.
c2=a2+b2, cn=(a2+b2)cn2 = a2cn2+b2cn2 > an+bn.
10.70.
2r. , 2r=a+bc (5.16).
10.71.
ch=2S=r(a+b+c)
c=


a2+b2

, 
r

h
=



a2+b2

a+b+


a2+b2
= 1

x+1

,
x= a+b




a2+b2
=


1+ 2ab

a2+b2

. 0 < 2ab/(a2+b2) 1, 1 < x 2. , 2/5 < 1/(1+2) r/h < 1/2.
10.72.
,
a+b 2


ab

c2=a2+b2 2ab.
c2

r2
= (a+b+c)2c2

a2b2
(2


ab
+


2ab
)22ab

a2b2
= 4(1+2)2.
10.73.
12.11, ) ma2+mb2=(4c2+a2+b2)/4=5c2/4. , 5c2/4 5(1+2)2r2 = (15+102)r2 > 29r2 (. 10.72).
10.74.
O- , A1,B1,C1- BC, CA, AB . ma=AA1 AO+OA1=R+OA1. mb R+OB1 mc R+OC1. ,
ma

ha
+ mb

hb
+ mc

hc
R
1

ha
+ 1

hb
+ 1

hc

+ OA1

ha
+ OB1

hb
+ OC1

hc
.
12.22 4.46.
10.75.
4.47
1

b
+ 1

c
= 2cos(a/2)

la
2

la

. , .
10.76.
M, O. ABC , O ( ); , AMB. AO+BO AM+BM, ..2R 2ma/3+2mb/3 ma+mb 3R. , COC1 (C1- AB) , CC1 CO, ..mc R.
.

10.77.
  hb lb mb (. 2.68), ha=lb hb mc=lb mb. , a b b c (. 10.1), ..c - , g- .
ha=mc , g 60 (. 10.62). g ABC 60, 60.

10.78.
1.60 A1B1C1 ABC r/R. , r R/2 (10.26).
. 12.72, , SA1B1C1/SABC=r1/2R1 1/4.

10.79.
AA1 BB1- OAH OBH. 2.1 A B, .. - AA1 BB1. AC > BC , AH > BH.
A1H/A1O=AH/AO > BH/BO=B1H/B1O,
.. OH : O, A1, B1, H. O ABH, AA1 BB1 BOH.
10.80.
90 a b g. CH- . I O, BC,CA,AB- K,L,M (.10.2).

.10.2

, O KCI. , CK KB BCO BCI. , CK=rctg(g/2) rctg(b/2)=KB 2BCO = 180BOC=1802a 180ab =g =2BCI. BCO=90a =ACH, CI CO CH. O- O , P- CH IL. CP CO=CO=R. , PH IM=r. , MIL=180a 90.

10.81. B2C2- B1C1 BC. B1C1 B2C2=BCBC1cosbCB1cosg. A1C1 ACAC1cosaCA1cosg A1B1 ABAB1cosaBA1cosb. cosa, cosb cosg . B1C1cosa+C1A1cosb+A1B1cosg acosa+bcosb+ccosg(acosbcosg+bcosacosg+ccosacosb). c=acosb+bcosa, ccosg=acosbcosg+bcosacosg. , acosbcosg+bcosacosg+ccosacosb =(acosa+bcosb+ccosg)/2.

10.82.
cos2a+cos2b+cos2g+2cosacosbcosg =1 (12.39,)), ABC , cos2a+cos2b+cos2g < 1, .. sin2a+sin2b+sin2g > 2. 4R2, .
10.83.
, p2(2R+r)2 = 4R2cosacosbcosg (. 12.41,)).
10.84.
A B C. ABC , CC1 < AC < AA1 A1 C1 BC AB. , A1,B1 C1, . , x, :
ha x < max
(b,c)=c

,
hb x < max
(a,c)=c


hc x < max
(a,b)=b

. ,
max
(ha,hb,hc)=ha

,
min
(b,c)=b

ha < b.
10.85.
A B C. , ABC . l, AB, l c hb, hb a, a hc, hc b, b ha ,, ha c. hb < a, x, hb < x < a. , x .
, ABC . l, AB, l c hb, hb hc; hc ha, ha c. .

10.86.
M N AB AC . C , AB. N1- MN. N1O:MO=2, NO N1O, NO:MO 2.
10.87.
S, ABC, ABC. , ABC, ABC, ABC, S . rABC=rABC < rABC.
10.88.
lc ABC , alcsin(g/2) blcsin(g/2). aha=2S=lc(a+b)sin(g/2). , a/(a+b) 1/2 sin(g/2).
10.89.
, ctgA+ctgB=c/hc c/mc. M - , N- AB. AMB , MN=AB/2. , c=2MN=2mc/3.
10.90.
BNBA=BM2 BM < BA, BN < BM, , AN > CN.
10.91.
B AB. F- AC (.10.3). , AD, BM CH , AB=CF.   , L- BM CH. AD L , BA:AM=BL:LM, BL:LM=FC:CM = FC:AM.

.10.3

AF ABF (ABF=90) CF=AB, BAC ABC . , ACB . B BP AF. ACB , FP > FC=AB, ..BFsinA > BFctgA. , 1cos2A=sin2A > cosA, ..cosA < (51)/2.   ,
90 > A > arccos((


51
)/2) 5150.

10.92.
, (ab)(ab) 0, (bc)(bg) 0 (ac)(ag) 0. ,
2(aa+bb+cg) a(b+g)+b(a+g)+c(a+b)=(a+b+c)paabb-cg,
.. p/3 (aa+bb+cg)/(a+b+c).
,
a(b+ca)+b(a+cb)+g(a+bc) > 0,
..a(b+ga)+b(a+gb)+c(a+bg) > 0. a+b+g =p, a(p2a)+b(p2b)+c(p2g) > 0, ..(aa+bb+cg)/(a+b+c) < p/2.

10.93.
OB OC C1 B1, OC1=OC OB1=OB. B2 C2- B1 C1 , AO. BOsinAOC+COsinAOB=B2C2 BC. , . , B1C1^AO,C1A1^BO A1B1^CO , O- .
10.94.
CBD=C/2 B A, ABD=B+CBD (A+B+C)/2=90.
10.95.
BM:MA=BC:CA BK:KC=BA:AC. BM:MA=BK:KC, ..
AB

AM
=1+ BM

MA
< 1+ BK

KC
= CB

CK
.
, M AC, K, ..AKM > KAC=KAM KMC < MCA = MCK. AM > MK MK > KC (. 10.59).
10.96.
, 2. SABO+SAOC < 2SXBO+2SXOC=2SOBC,SABO+SOBC < 2SAOC SAOC+SOBC < 2SABO. , . , 2.
10.97.
S,S1 S2 r,r1 r2. AB1C1 A2BC2 ABC, r1/r r2/r . S1 S2 AB1C1 A2BC2. , , S1 S2 . AB1+A2B > AB, ..r1+r2 > r.



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On 22 Nov 2002, 13:45.

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