. . . (4- . — ! !), 2002

12. | | 13.

12.1.
sing =c/2R, S=(absing)/2=abc/4R.
12.2.
ABD CBD AB/2sinADB BC/2sinBDC. , AB=BC sinADB=sinBDC.
12.3.
b=asinb/sina =asinb/sin(b+g), S=absing/2=a2sinbsing/2sin(b+g).
12.4.
(a+b)/c=(sina+sinb)/sing. , sina+sinb =2sin((a+b)/2)cos((ab)/2)=2cos(g/2)cos((ab)/2) sing =2sin(g/2)cos(g/2). .
12.5.
  A2BC2 A2B BC2 bcosg bcosa; BO A2BC2 90g 90a. BO A2C2 M. A2M=A2BsinA2BM/sinA2MB=bcosgcosa/sinC2MB=C2M.
12.6.
a =(a,c), b =(c,d) g =(d,b).
(AC/AS)/(BC/BS)=sina/sin(b+g), (BD/BS)/(AD/AS) = sing/sin(a+b).

(ACBD)/(BCAD) = sinasing/sin(a+b) sin(b+g).
12.7.
OA/PA=sinOPA/sinPOA OB/PB=sinOPB/sinPOB, (OA/OB)/(PA/PB)=sinPOB/sinPOA.
12.8.
D1A/D1B=sinB/sinA.
12.9.
O - , M N- , k - . MN kOMsinOMN kONsinONM. , OM/sinONM=ON/sinOMN.
12.10.
BX/sinBDX=BD/sinBXD=2R siny/sinBXD. , sinBDX=sinBDC=sinj; BXD : C D AB, BXD=pjy, , BXD = |jy|. , BX=2Rsinjsiny/sin|jy|.
12.11.
) A1- BC. AB2=AA12+A1B22AA1BA1cosBA1A AC2=AA12+A1C22AA1A1C cosCA1A , cosBA1A=cosCA1A, .
) ).

12.12.
a2(bc)2=2bc(1cosa)=4S(1cosa)/sina =4Stg(a/2).
12.13.
cosa =(b2+c2a2)/2bc. cos2(a/2)=(1+cosa)/2 sin2(a/2)=(1cosa)/2.
12.14.
a - . m2=a2+b2+2abcosa n2=a2+b22abcosa. m2n2=(a2+b2)2(2abcosa)2=a4+b4+2a2b2(12cos2a). , m2n2=a4+b4 , cos2a =1/2.
12.15.
M - AA1 BB1. AMB , AM2+BM2=AB2, .. 4(ma2+mb2)/9=c2. 12.11 ma2+mb2 = (4c2+a2+b2)/4.
12.16.
m=C1M j =C1MO. OC12=C1M2+OM22OMC1Mcosj BO2=CO2=OM2+MC2+2OMCMcosj =OM2+4C1M2+4OMC1Mcosj. BC12=BO2OC12 = 3C1M2+6OMC1Mcosj, .. c2=4BC12=12m2+24OMC1Mcosj. , 18 m2=2mc2=a2+b2c2/2 (. 12.21). a2+b2=2c2 , 18m2=3c2/2, .. c2=12m2. c2=12m2+24OMC1Mcosj, a2+b2=2c2 , C1MO=j =90, .. CC1^OM.
12.17.
BC K, - L. BC=BK+KC=rctg(b/2)+rctg(g/2) BC=BL+LC=ractgLBOa+ractgLCOa=ratg(b/2)+ratg(g/2). ,
cos(a/2)=sin
b

2
+ g

2


.
3.2 pb=BK=rctg(b/2) pb=CL=ratg(g/2).

AB AC P Q, p=AP=AQ=ractg(a/2).

12.18.
) 12.17 p=ractg(a/2) rctg(a/2)=pa; rctg(b/2)=pb ratg(b/2)=pc; rctg(b/2)=pa rbctg(b/2)=p. , .
) rp=ra(pa) rra=(pb)(pc), r2p=(pa)(pb)(pc). , S2=p(r2p).

) rra=(pb)(pc) rbrc=p(pa) .

12.19.
12.17 r=rctg(a/2)tg(b/2) p=rcctg(g/2).
12.20.
12.18,) ra=rp/(pa) rb=rp(pb). crarb=cr2p2/((pa)(pb)) ra+rb=rpc/((pa)(pb)), , crarb/(ra+rb)=rp=S.
12.21.
12.18,) 1/rb=(pb)/pr 1/rc=(pc)/pr.
1

rb
+ 1

rc
=a/pr=a/S=2/ha

.
12.22.
, 1/ha=a/2pr 1/ra=(pa)/pr. , .
12.23.
12.18,) 1/((pb)(pc))=1/rra. 12.22.
12.24.
12.1 4SR=abc. ,
abc=p(pb)(pc)+p(pc)(pa)+p(pa)(pb)

(pa)(pb)(pc)= S2

(pa)
+ S2

(pb)
+ S2

(pc)
S2

p
=S(ra+rb+rcr).
12.25.
12.18,) rarb=p(pc), rbrc=p(pa) rcra=p(pb). , .
12.26.
S=rp=ra(pa)=rb(pb)=rc(pc), (p3(pa)3(pb)3(pc)3)/S3=3abc/S3. , abc/S=4R (12.1).
12.27.
ABC 2a, 2b 2g. 12.36,) 12.37,) r=4Rsinasinbsing ra=4Rsinacosbcosg. (r+ra)(4R+rra)=16R2sina (sinbsing+cosbcosg)(1+sina(sinbsingcosbcosg))=16R2sinacos(bg)(1sinacos(b+g)) = 16R2sinacos(bg)cos2a. , 4Rsinacosa =a 4Rsin(b+g)cos(bg) = 2R(sin2b+sin2g)=b+c. .
12.28.
OA=r/sin(a/2) bc=2S/sina, OA2/bc=r2ctg(a/2)/S=r(pa)/S (. 12.17,). , r(pa+pb+pc)=rp=S.
12.29.
), ). ctg(j/2)=sinj/(1cosj), p2(1x)2=(1x2)(2R(1x)+r)2, x=cosj. x0=1 , ctg(j/2) ; , 1x, . 12.38, 12.41,), , (xcosa)(xcosb)(xcosg)=0, a,b g- . , j ; , 0 p.
12.30.
, 2pr=2S=absing=abc/2R, ..4prR=abc. : S2=p(pa)(pb)(pc), ..pr2=(pa)(pb)(pc)=p3p2(a+b+c)+p(ab+bc+ca)abc=p3+p(ab+bc+ca)4prR. p, .
12.31.
abc=4RS (12.1), (c+a+b)/4RS=2p/4Rpr=1/2Rr.
12.32.
, (pc)/p=r/rc (12.18,)), r=csin(a/2)sin(b/2)/cos(g/2) rc=ccos(a/2)cos(b/2)/cos(g/2) (12.17).
12.33.
12.1 S=abc/4R. Ѡ , S=aha/2. ha=bc/2R.
12.34.
aha=2S=2(pa)ra ra/a=cos(b/2)cos(g/2)/cos(a/2) (12.17,)), ha=2(pa)cos(b/2)cos(g/2)/cos(a/2). , (pa)ctg(b/2)=rc=(pb)ctg(a/2) (12.17,)), ha=2(pb)sin(b/2)cos(g/2)/sin(a/2).
12.35.
) AD ABC M. ADDM=BDDC , DABD ~ DAMC, ABAC=ADAM=AD(AD+DM)=AD2+BDDC. , BD=ac/(b+c) DC=ab/(b+c). , AD2=bcbca2/(b+c)2=4p(pa)bc/(b+c)2.
) . 4.47.

) AD- , AH- ABC. AH=csinb =2Rsinbsing. Ѡ , AH=ADsinADH = lasin(b+(a/2))=lasin((p+bg)/2)=lacos((bg)/2).

) , p=4Rcos(a/2)cos(b/2)cos(g/2) (12.36,) sinb+sing =2sin((b+g)/2)cos((bg)/2)=2cos(a/2)cos((bg)/2), ).

12.36.
) O- , K- AB.
2Rsing=AB=AK+KB=r(ctg(a/2)+ctg(b/2)) =

= rsin((a+b)/2)(sin(a/2)sin(b/2)).
, sing =2sin(g/2)cos(g/2) sin((a+b)/2) = cos(g/2), .
) 3.2 pa=AK=rctg(a/2). pb=rctg(b/2) pc=rctg(g/2). , p(pa)(pb)× ×(pc)=S2=(pr)2, .

) ) ).

12.37.
)
rcos(a/2)sin(a/2)=pa, sin(a/2)sin(b/2)sin(g/2)=r/4R
(. 12.17,) 12.36,)), .
) 12.17,) ratg(g/2)=pb=rctg(b/2). r/4R=sin(a/2)sin(b/2)sin(g/2), .

12.38.
cosa+cosb =2cos((a+b)/2)cos((ab)/2) cosg =cos(a+b)=2cos2((a+b)/2)+1 , cos((ab)/2)cos((a+b)/2) = 2sin(a/2)sin(b/2), cosa+cosb+cosg =4sin(a/2)×
×sin(b/2)sin(g/2)+1= r

R
+1

(. 12.36,)).
12.39.
) cos2a+cos2b=2cos(a+b)cos(ab)=2cosgcos(ab) cos2g =2cos2g1=2cosgcos(ab)1 , cos(a+b)+cos(ab)=2cosacosb, .
) cos2a=2cos2a1 , ).

12.40.
sin2a+sin2b =2sin(a+b)cos(ab)=2sing×
×cos(ab) sin2g =2singcosg =2singcos(a+b) , cos(ab)cos(a+b) = 2sinasinb, .
12.41.
) , sin2a+sin2b+sin2g =(a2+b2+c2)/4R a2+b2+c2=(a+b+c)22(ab+bc+ca) = 4p22(r2+p2+4rR) (. 12.30).
) 12.39,) 2cosacosbcosg =sin2a+sin2b+sin2g2. ).

12.42.
abcosg =(a2+b2c2)/2. , .
12.43.
12.13 cos2(a/2)/a=p(pa)/abc. , p(pa)+p(pb)+p(pc)=p2 abc=4SR=4prR.
12.44.
) bccosa =2Sctga, a2=b2+c24Sctga. , .
) a2ctga =2R2sin2a =4SBOC, O- . . a SBOC .

12.45.
12.17 ctg(a/2)+ctg(b/2)=c/r tg(a/2)+tg(b/2)=c/rc. .
12.46.
, tgg =tg(a+b)=(tga+tgb)/(1tgatgb). 1tgatgb, .
12.47.

tg(g/2)=ctg
a

2
+ b

2

=[1tg(a/2)tg(b/2)] [tg(a/2)+tg(b/2)].

tg(a/2)+tg(b/2).
12.48.
) sinasinbsing. : cosg(sinacosb+sinbcosa)+sing(cosacosbsinasinb)=cosgsin(a+b)+singcos(a+b)=cosgsingsingcosg =0.
) sinasinbsing. : cosa(sinbsingcosbcosg)+sina(cosbsing+cosgsinb)=cos2a+sin2a =1.

12.49.
sin2a+sin2b+sin2g2 = 2cosacosbcosg (. 12.39,)) S=2R2sinasinbsing, , (tga+tgb+tgg)× ×cosacosbcosg =singsinbsina. 12.48,).
12.50.
A B- a b, P- , Q- , PA. AQB , PQB=2a. PQB+QPB=b+QBA, b =3a.
12.51.
4.47
1

b
+ 1

c
=2cos(a/2)/la

, cos(a/2)=1/2, ..a =120.
12.52.
M MD BC. MD=AH/2=BM/2.   BDM MD BM. MBC=MBD=30.
12.53.
ADBCsinADB BEACsinAEB , ABC. sinADB=sinAEB. .
1. ADB=AEB; A,E,D,B , EAD=EBD, ..A=B, .

2.ADB+AEB=180; ECD+EOD=180, O- . EOD=90+C/2 (5.3), C = 60.

12.54.
B- AC AB. AB=CB ABC=180275=30. AB=CB=2CH=AB, ..B=B B=30.
12.55.
, AKDM- L- . AD^BC AM^BA, DAM=ABC. , KAD=ACB. A AP KM. B < C. P KL. AKP MKA AK: AP=MK:MA. AKAM=APMK=APAD = 2APAL. AL2=AKAM, , AL=2AP, ..ALP=30. , KMA=ALP/2=15. ABC 15 75.
12.56.
CD- . BD=ac/(a+b). Ѡ , DBDC ~ DBCA, BD: BC=BC:BA, ..BD=a2/c. , c2=a(a+b)=3a2. ABC a,2a 3a, 30,90 60.
12.57.
ABC=2x, A ABE ABE+AEB=x+a. , KAEBAK=(180xa)(1802x2a)=x+a. , AE- AABK. BE B , E- , AK. AKE=AKC/2=90a.
12.58.
A1 A18- .   ABC A14A1A9. 6.62, ) A1A12,A2A14 A9A18 , AMC=(A18A2+A9A14)/2=70.
12.59.
A1 A18- , O- .   ABC A1OA18. A2A14 A18A6 A1A10, A2A14 A1A12 A9A18 (. ), ADE=(A1A2+ A12A14)/2=30.

.12.6

12.60. BDE=50 CED=30, BOC=EOD=1805030=100. , BB CC , BOC=100, A BC. D- BB AC, E- CC AB (.12.6). A B C OE , OD , OED , ODE . A, CED=OED=30 BDE=ODE=50.

, ABC A=50,B=70, C=60 . A1 A18- .   ABC A2A14A9. A1A12 E (. 12.58). F- A1A12 A5A14; A9A16 A1A12 A5A14, F.  CDF CE C, FE- F. DE- ADB, ..ODE = (A2A14+A5A9)/4=50.

12.61.
D,E F- BP,PQ QC; BOD=90B=90C=COF=a, DOP=POE=b EOQ=QOF=g. 180=BOC=2a+2b+2g, ..a+b+g =90. BPO=DPE/2= (180DOE)/2=90b QOC=g+a =90b, BPO=COQ. , PBO=OCQ. DBPO ~ DCOQ, ..PBCQ=BOCO=BC2/4.
12.62.
P Q- BC CD . P Q BC CD. , PF+GQ=FG.   , FG- , FG , PF+GQ=FG, F=F G=G.
, 2. GD=x. BF:EB=AD:GD, BF=2/x.
CG=2x, GQ=x1,CF=2 2

x
, FP= 2

x
1

, ..PF+GQ=x+2/x2
FG2=CG2+CF2 = (2x)2+
2 2

x

2


=44x+x2+4 8

x
+ 4

x2
=
x+ 2

x
2
2


=(PF+GQ)2.

.12.7

12.63. , .12.7. O- , H- , K- AA1. ꠠ=tgAHB=2=tg A1HD1, H AA1. a =AHB = A1HD1. ABA1D1=(AHA1H)sina =2KHsina =2OHsin2a. tga =2 1+ctg2a =1/sin2a, sin2a =4/5. 8h/5.

12.64. BM ABC K L, BK=KL=LM=x. AC MC. , BM B M , , MC BC. , BC=MC=AC/2, ..b=2a.

BM2=(2a2+2c2b2)/4 (. 12.11,)), 9x2=(2a2+2c24a2)/4=(c2a2)/2. P- BC. BP=(a+cb)/2 = (ca)/2. Ѡ , BP2=BKBL, ..BP2=2x2. 2x2=((ca)/2)2. 9x2=(c2a2)/2 ((ca)/2)2=2x2, (c+a)/(ca)=9/4, ..c:a=13:5.   , a:b:c=5:10:13.

12.65.
2a 2b- . , ,



(a+2b)2+b2

. Ѡ , 2a. , (a+2b)2+b2=2a2, ..
a=2b


4b2+5b2
=(23)b

. a=5b.
12.66.
P Q- AC AB, R- S1; a=AC/2, b=BC/2, x- S1. , PR=a+x, QR=a+bx PQ=b. PQR RH. R CD x, PH=ax, , QH=|ba+x|. , (a+x)2(ax)2=RH2=(a+bx)2(ba+x)2, ..ax=b(ax).   x=ab/(a+b). S2 .
12.67.
x- S, S1 S2 AB, y- S, S2 S3 BA. , S1 AB (S3 BA) , , x=y.
S AD, (x+1)2(x1)2=(3+x)2(5x)2, ..x=4/3.

S2 S3, , AB2=(3+4)212=48. Ѡ , S AD BC (y+3)2(5y)2 = 16(y1) (4+y)2(4y)2=16y . ,
4


y1
+4y=


48

, ..y=4/3.

.12.8

12.68. , ag, a, a+g, g 0. 180, a =60. 2Rsin(ag), 2Rsina,2Rsin(a+g). ó , sin(ag) sina sin(a+g).

) sin(ag) sina sin(a+g) , sina =(sin(a+g)+sin(ag))/2=sinacosg, ..cosg =1, g =0. , 60.

) sin(ag) sina sin(a+g) , sin2a =sin(ag)sin(a+g)=sin2acos2gsin2gcos2a sin2acos2g. cosg =1, .. 60.

12.69.
ABC ABCE (.12.8). BC=x AD=y. (ba)h=2SAED=xysin45 (ba)2=x2+y22xycos45=x2+y22xysin45. a2+b2=(AO2+BO2)+(CO2+DO2)=(BO2+CO2)+(DO2+AO2)=x2+y2. , (ba)2=x2+y22xysin45=a2+b22(ba)h, ..h=ab/(ba).
12.70.
BK=(a+cb)/2 KC=(a+bc)/2 (. 3.2), BKKC=(a2(bc)2)/4=Stg(a/2) (. 12.12).
12.71.
(b+c)/a=cos((bg)/2)/sin(a/2) (12.4), cos((bg)/2)=cos(a/2), ..bg=a. b=g+a, b=90, b+a=g, g=90.
12.72.
, SABC=2R2sinasinbsing. SA1B1C1=2R2sin((b+g)/2)sin((a+g)/2)sin((a+b)/2)=2R2cos(a/2)cos(b/2)cos(g/2). SABC/SA1B1C1=8sin(a/2)sin(b/2)sin(g/2)=2r/R (. 12.36,).
12.73.
(a2+b2+c2)/4S (12.44,)). , ma2+mb2+mc2=3(a2+b2+c2)/4 (12.11,)) , ABC, 3/4 ABC (1.37).
12.74.
Ai , , , A1A2A3 (). l1, l2 l3 ; l1=(b2+c2a2)/2,l2=(a2+c2b2)/2 l3=(a2+b2c2)/2, a=A2A3,b=A1A3 c=A1A2. 5.47,) A1A42=4R2a2, R- A1A2A3. l4=A1A42l1=4R2(a2+b2+c2)/2 = A2A42l2=A3A42l3.
,

1/li=0

. (b2+c2a2)/2 = bccosa =2Sctga, 1/l1=tga/2S. , 2/(a2+b2+c28R2)=(tga+tgb+tgg)/2S (12.49).

12.75.
(a1,b1), (a2,b2) (a3,b3)- .
(xa1)2+(yb1)2=(xa2)2+(yb2)2,

(xa1)2+(yb1)2=(xa3)2+(yb3)2.
, , , .
12.76.
AB CD K L, . , AK CL S. O S Ox Oy, OB OD. S 1. AK CL y=(x+1)/3 y=2x1. x0=4/5 y0=3/5. , x02+y02=1.
12.77.
d- . , R=d+2r. (0,0), (2a,0) (0,2b)- . (a,b) - , (r,r) - , r=a+bR. , d2=(2ra)2+(2rb)2=a2+b24r(a+br)+4r2=(R2r)2, a2+b2=R2.
12.78.
Ox, l. : A(x,y), B(y,x), C(x,y) D(y,x); l y=a. Q ((x+y)/2, (yx)/2), P (y,a). , (t,t+a/2), t=(xy)/4. , xy



2(x2+y2)
=AB

AB.



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On 22 Nov 2002, 13:45.

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