. . . (4- . — ! !), 2002
<chap>. § <sec>
1. | | 2

1.1.
a) P Q AB CD, K L - PQ AC BD. PL=a/2 PK=b/2, KL=PLPK=(ab)/2.

) AD F , BF||CD. E MN BF.
MN=ME+EN= qAF

p+q
+b= q(ab)+(p+q)b

p+q
= qa+pb

p+q
.

1.2.
K, L, M N AB, BC, CD DA ABCD. KL=MN=AC/2 KL MN, . . KLMN . , KLMN - , AC BD ; , AC=BD; , AC BD .

1.3.
) AA1 BB1 O. B1BC A1A2||BB1. B1C/B1A2=1+p, AO:OA1=AB1:B1A2=B1C:qB1A2=(1+p):q.

) BA1:A1C=1:p AB1:B1C=1:q. AD:DA1=(1+p):q BD:DB1=(1+q):p.
a1= 1+p+q

1+p

,
b1= 1+p+q

1+q

c=(1+p+q)d.


1.4.
A2 A1B. CA1:A1A2=CP:PC1 A1A2:A1B=1:2, CA1:A1B=CP:2PC1. CB1:B1A=CP:2PC1=CA1:A1B.
1.5.
P QM AQD ( ). , 1.4 , P . , BC||AD.
1.6.
DAQP ~ DCQB, AQ:QC=AP:BC=1:n. AC=AQ+QC=(n+1)AQ.
1.7.
A1B1C1D1 B1D1 , AB CD. , BC AD. ⠗ ABCD.
1.8.
, AK:KM=BK:KD=LK:AK, .. AK2=LKKM.

.1.3

1.9.
AC , ABCD. AA1 CC1 BD (.1.3). , BA1=DC1. OP O BD. , P BD. AA1, OP, CC1 AO=OC, A1P=PC1. P BD, BA1=DC1.

1.10.
BO=PD, BO:OD=DP:PB=k. BC=1. AD=k ED=1/k. k=AD=AE+ED=1+(1/k), .. k2=1+k. , k2=AD2 1+k=BC2+BCAD.

1.11.
C, D, E, F AO, OB, BM, MA AOBM. AB=MO=R, R , 1.2. CDEF . CE^DF.

1.12.
) , , , . , O. OA:OB=OB1:OA1, OѠ:OA = OA1:OC1, OC:OB=OB1:OC1, , BC1||CB1 ( ).

) D E AB1 CA1, CB1 AC1. CA1:A1D=CB:BA=EC1:C1A. DCB1D ~ DEB1A, A1, B1 C1 .

1.13.
, , . a A1, AC AB, b B1 AB BC. , , A1M:B1M=p:q, p+q=1. M AC BC qa b . Ѡ , 1.1, ) M AB qa+pb.

.1.4

1.14.
, O BC; QN K (. 1.4). MO||PC, QM:MP=QO:OC, KO||BC, QO:OC=QK:KN. , QM:MP=QK:KN, .. KM||NP. MNP=KMO = QNM.

1.15.
M EF, CD ( E F BC AD). PL:PK=BL:KD OK:OL=KA:CL=KA:KF=BL:EL. KD=EL, PL:PK=OK:OL, , PL=OK.

1.16.
ABCD1. , D D1 ( ). AD1 CD1 S1 R1 , SS1||DD1 RR1||DD1. N PR1 QS1; N1 N2 , N DD1, PR QS .

N1N

= b
RR1

=ab
DD1





N2N

=a
SS1

=ab
DD1

.

N1=N2 PR QS. , PN1:PR=PN:PR1=b QN2:QS=a.

.   a = b . BP:BA=BQ:BC=a, PQ||AC PQ:AC=a. RS||AC RS:AC=1a. PR QS a:(1a).

1.17.
) A C AK CL BD. CBL=ABK CDL=KDA, DBLC ~ DBKA DCLD ~ DAKD. AD:DC=AK:CL=AB:BC.

) , BA1:A1C=BA:AC BA1+A1C=BC, BA1=ac/(b+c). BO ABA1, AO:OA1=AB:BA1=(b+c):a.

1.18.
O ABC, B1 AC, A1 BC. DBOA1 ~ DBCB1, BO:BA1=BC:BB1, ,
R=BO=a2/


4a2b2
.

1.19.
EAD=j, AE=AD/cosj = AB/cosj AF=AB/sinj.
1

AE2
+ 1

AF2
= (cos2j+sin2j)

AB2
= 1

AB2
.

1.20.
, AB22=AB1AC=AC1AB=AC22.
1.21.
) BQ:QM=BN:AM=BK:AK, KQ||AM.

) O . CBA+BAD=180, ABO+BAO=90. DAKO ~ DOKB, .. AK:KO=OK:KB. , AKKB=KO2=R2, R . CLLD=R2.

1.22.
ABC ( ), MAN ( ). , . ABC=MAN. ABM ADN ABM ADN, AM:AN=AB:AD=AB:CB, .. DABC ~ DMAN.
1.23.
AC D B, BB||l DD||l. AB:AE=AB:AG AD:AF=AD:AG. ABB CDD AB=CD, AB=CD.
AB

AE
+ AD

AF
= AB

AG
+ AD

AG
= CD+AD

AG
= AC

AG
.

1.24.
BG AC (.1.5). ABG ACE ACAG=AEAB. AF CB , GCB CAF CBG ACF . ACCG=AFBC. , AC(AG+CG)=AEAB+AFBC. AG+CG=AC, .
1.25.
a+b = 90(a/2), g = 180 ab = 90 +(a/2). AB D , ACD=90(a/2), .. AC=AD. DABC ~ DCBD, , BC:BD=AB:CB, .. a2=c(cb).
1.26.
AB CD AMC , . AM/CM . BM/DM .

.1.5

.1.6

1.27.
O, AK FP FE Q M, CL EP FE R N (.1.6). , FM:FE=FQ:FP=LO:LC=1:3, .. FM=FE/3. EN=FE/3.

.1.7

1.28.
A B . AC BC K L , PK||BC PL||AC. DAKP ~ DPLB, AK:KP=PL:LB, , (ap)(bp)=p2, p=PK=PL. ,
1

a
+1

b
=1

p
.

1.29.
BC O, AK AL BC P Q. , BP < BQ. LCO LC||AB. DABQ ~ DLCQ, .. BQ:QC=AB:LC=2:1. , BC=BQ+QC=3QC. BC=3BP.

1.30.
BK:BO=BO:AB KBO=ABO, DKOB ~ DOAB. KOB=OAB. AOM=ABO. , KOM=KOB+BOA+AOM=OAB+BOA+ABO=180, .. K,O M .

1.31.
, , a1=a2=a b1=b2=b (.1.7).
x

b
= x+z1

b

,
a+z3

a
= b

b+z2

,
y+z1

a
= y

a

,
a

a+z3
= b+z2

b

. , x(y+z1)=y(x+z1), , x=y.

1.32.
AMN=MNC BMN=MNA, AMB=ANC. , AM:AN=NB:NM=BM:CN. DAMB ~ DANC, , MAB=NAC. , BAC=MAN. .

B1 C1 B C MN. AM:NB=MN:BM=MC:NC, MAMC1=AMNC=NBMC=MB1MC. , A , BB1CC1.

1.33.
AEB+BEC=180, ABE BEC, .. BE .

: ABE=CBE ABE=BCE. , DABE=DCBE. .   ABC=ABE+CBE=ABE+BAE=90.   ABC , 1:3, 90,60,30.

1.34.

SBDEF/2SADE=SBDE/SADE=DB/AD=EF/AD=


SEFC/SADE

.
SBDEF=2


SADESEFC

.

.1.8

1.35.
MN=x; E AB CD. EBC,EMN EAD , SEBC:SEMN:SEAD=a2:x2:b2. SEMNSEBC=SMBCN=SMADN=SEADSEMN, x2a2=b2x2, ..x2=(a2+b2)/2.

1.36.
Q, ABC, DE,FG HI BC,CA AB , F H BC, E I AC, D G AB (.1.8). : S=SABC, S1=SGDQ,S2=SIEQ, S3=SHFQ.



S1

S
+


S2

S
+


S3

S
= GQ

AC
+ IE

AC
+ FQ

AC
= AI+IE+EC

AC
=1,
..
S=(


S1
+


S2
+


S3
. )2

1.37.
M ABC; A1 M BC. CMA1 ABC, 2:3. 9SCMA1/4. , SCMA1=S/3 (. 4.1).

1.38.
E,F,G H AB,BC,CD DA.

) ,
SAEH+SCFG= SABD

4
+ SCBD

4
= SABCD

4

.
SBEF+SDGH= SABCD

4

.
SEFGH=SABCD SABCD

4
SABCD

4
= SABCD

2
.

.1.9

) AC=BD, EFGH (1.2). a)SABCD=2SEFGH=EGFH.

1.39.
E,F,G H ABCD; E1,F1,G1 H1 O . EF E1OF1, SE1OF1=4SEOF. SF1OG1=4SFOG, SG1OH1=4SGOH SH1OE1=4SHOE. SE1F1G1H1=4SEFGH. 1.38,) SABCD=2SEFGH. SE1F1G1H1=2SABCD=2S.

1.40.
. BCMN MN P Q (.1.9). DABC=DPDQ DACD=DPMA. PAD ABC+ADC=PDQ+ADC=45.

. DE=1, EA=2, EB=2,AD=5
BA=


10

, DE:AE=EA:EB=AD: BA DDEA ~ DAEB. , ABC=EAD. , AEC=CAE=45. ABC+ADC+AEC=(AED+CAE)+ADC=CAD+ADC=90.

1.41.
L LM AB LN AD. KM=MB=ND KL=LB=DL, KML DNL . , DLK=NLM=90.
1.42.
D1A=B1B, AD2=BB2 D1AD2=B1BB2, DD1AD2=DB1BB2. AD1 BB1 (AD2 BB2) , B1B2^D1D2.
1.43.
AC C M , CM=CE (.1.10). ACE C 90 BCM. MB AE, , CL. MC=CE=DC DK,CL MB , KL=LB.
1.44.
AB BC ABC1D1 A2BCD2; P,Q,R S , AB,BC,CD DA. ABC+ADC=180, DADC=DA2BC1, , DRDS=DPBQ RS=PQ. QR=PS. , PQRS , RDS PBQ , ; QCR SAP. PQR+RSP=BQC+DSA=180, PQB=RSD RQC=PSA. , PQRS .

.1.10

.1.11

1.45.
K,L,M FOA BOC, BOC DOE, DOE FOA; 2a,2b 2g BOC,DOE FOA (.1.11). K OB BOC, OKB=90+a. OKA=90+g. a+b+g =90, AKB=90+b. AOB K, . L, COD, , DOKB=DCLO. , DKOL=DOKB.   , COL=KBO, KOB+COL=180OKB=90a, , KOL=2a+(90a) = 90+a =OKB. , KL=OB=R. LM=MK=R.
1.46.
A=a. , KCL ADL 240a (120+a). KC=BC=AD CL=DL, DKCL=DADL, , KL=AL. KL=AK.
1.47.
P,Q R , DA,AB BC a A. , PAQ=90+a =RBQ, , DPAQ=DRBQ. AQ BQ , PQ^QR.

.1.12

1.48.
, A,B C ABCABC 360, 360. AC ACP, BCA (.1.12). DABP=DCBA, AB=CB, AP=CA PAB=360PACCAB = 360ABC-CAB=ACB. , DCBA=DCBP, , 2ABC=PBA=ABC, PBA=ABC.

1.49.
BA:BC=BC1:BA1 ABC=C1BA1, DABC ~ DC1BA1. DABC ~ DB1A1C. BA1=A1C, DC1BA1=DB1A1C. , AC1=C1B=B1A1 AB1=B1C=C1A1. , AB1A1C1 .

1.50.
) P Q AB AC. MP=AC/2=QB1, MQ=AB/2=PC1 C1PM=C1PB+BPM=B1QC+CQM=B1QM. , DMQB1=DC1PM, , MC1=MB1. , PMC1+QMB1=QB1M+QMB1=180MQB1, MQB1=A+CQB1=A+(1802j). , B1MC1=PMQ+2jA=2j. (, C1PB+BPM > 180, .)

) AB AC B C, AB:AB=AC:AC=2:3. M BC ABC. AB AC ABC1 AB1C j =60. B1 C1 , AB AC; , ) MB1=MC1 B1MC1=120.

. ) ) , .

1.51.
) B AC AB1, B1; C . AB:AC=AC1: AB1=AB:AC, BC||BC. N BC, , 1.50, NC1=NB1 (..N=M) B1NC1=2ABB1=1802CAB1=j.

) BC BA1C 3602j A1 (j < 90, 2j). 360, A1B1C1 180j,j/2 j/2 (. 1.48).  , , , A1=O.

1.52.
O1,O2,O3 O4 , AB,BC,CD DA; M AC. MO1=MO2 O1MO2=a (. 1.50). MO3=MO4 O3MO4=a. , a M O1MO3 O2MO4.
1.53.
A1C=AC|cosC|, B1C=BC|cosC| C ABC A1B1C , , |cosC|.
1.54.
M N CH, CMN=CHN, AC^HN, CHN=A. CNM=B.
1.55.
) l A . (l,AB)=(AC,CB)=(C1B1,AC1), , l||B1C1.

) OA^l l||B1C1, OA^B1C1.

1.56.
AA1,BB1 CC1 , AA1C BB1C C, . , DA1BH ~ DB1AH, , AHA1H=BHB1H. BHB1H=CHC1H.

AHA1H=BHB1H=CHC1H, DA1BH ~ DB1AH, , BA1H=AB1H=j. CA1H=CB1H=180j. AC1H=CA1H=180j AC1H=AB1H=j, j =90 ..AA1,BB1 CC1 .

1.57.
) 1.53 C1A1B=CA1B1=A. AA1^BC, C1A1A=B1A1A. , B1B C1C A1B1C1 A1C1B1.

) AB,BC CA A1B1C1, A1A B1A1C1, , AA1^BC. BB1 CC1 .

1.58.
1.57,) , B1A1 AC B1C1.
1.59.
1.53 B1A1C=ABC. A1B1||AB, B1A1C=BAC. BAC=ABC. , B1C1||BC, ABC=BCA. ABC A1C1||AC.
1.60.
O ABC. OA^B1C1 (.1.55, ), SAOC1+SAOB1=RB1C1/2. B C , SABC=qR. Ѡ , SABC=pr.
1.61.
, , BC, A AB AC, ABC:P1+P2+P3=P. , ri/r=Pi/P. , .

.1.13

1.62.
M=A. XA=A, AYA=1. CXC=1. , y=AYA x=CXC . BC D , AB||MD (.1.13). E CXC MD. XMM+YMM=XCE+YMM. DABC ~ DMDC, CE=YMM. XMM+YMM=XCE+CE=XCC=1.

1.63.
D BH. DBHA ~ DHEA, AD:AO=AB:AH DAH=OAE. , DAO=BAH, , DDAO ~ DBAH DOA=BAH=90.

1.64.
AA1,BB1 CC1 ABC. B1 B1K B1N AB BC B1L B1M AA1 CC1. KB1:CC1=AB1: AC=LB1:A1C, DKLB1 ~ DC1A1C, , KL||C1A1. MN||C1A1. , KN||C1A1 (.1.54). , K,L,M N .
1.65.
) O AC, O1 AB, O2 BC. , AB BC. O1 O1K EF (K EO2). , DBO O1KO2 .   , O1O2=DO=AC/2 BO=KO2=(BCAB)/2. DBO O1KO2 , BOD=O1O2E, .. DO EO2 , D, EF.

) AC F,D,E , FAB=DAC=EBC FBA=DCA=ECB, ..F AD, E DC. , AFB=BEC=ADC=90, FDEB .

.1.14

1.66.
MQ MP , AD BC, MR MT , AB CD (.1.14). M1 P1 RT QP .

TM1=RM=AQ TM1||AQ, AM1||TQ. AP1||RP. M1AP1=90, RP^TQ.

TQ RP, M1A RP, P1A TQ E,F,G . , E AC, , AFEG AM1CP1 . ARF=AM1R=M1TG=M1CT, a. AF=RAsina =M1Asin2a, AG=M1Tsina M1Csin2a, AFEG AM1CP1 .

1.67.
O1 O2. K, L; M, N (.1.15). KM LN O1O2 P1 P2 . , P1=P2. A,D1,D2

.1.15

KL MN, KM O1A, LN O2A . O1AM+NAO2=90, O1MA ANO2 , AO2||KM AO1||LN. AD1:D1O1=O2P1:P1O1 D2O2:AD2=O2P2:P2O1. AKO1M O2NAL AD1: D1O1=D2O2:AD2. , O2P1:P1O1=O2P2:P2O1, ..P1=P2.

1. | | 2

Copyright ©2002 ! !
.
: biblio@mccme.ru.
Rambler's Top100