. .

5- ,

2006

514.112 22.151.0 70 . . : . 5- ., . . .: : , 2006. 640 .: .ISBN 5-94057-214-6 711 . . 1900 150 . , . , . , , , .

70

22.151.0

ISBN 5-94057-214-6

. ., 2006 , 2006

1.

10 11

1. , (12). 2. (13). 3. (15). 4. (16). 5. , (17). 6. (18). (18). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.

30

1. , (31). 2. (32). 3. (33). 4. (34). 5. , (35). 6. (36). 7. (37). 8. (38). 9. (39). 10. (40). 11. (40). (41). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.

55

1. (56). 2. (57). 3. (58). 4. (59). 5. , (59). 6. (60). 7. (61). 8. , (61). 9. (62). 10. (63). 11. (65). (66). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

4. 1. (81). 2. (82). 3. , -

81

4

(83). 4. , (83). 5. (84). 6. , (85). 7. (86). 8. (87). 9. (88). (89). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

5.

101

1. (102). 2. (103). 3. (104). 4. 60 120 (105). 5. (106). 6. (106). 7. (109). 8. (111). 9. (113). 10. (115). 11. (116). 12. (117). 13. (119). (121). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

6.

151

1. (151). 2. (154). 3. (155). 4. (156). 5. (157). 6. (157). 7. (160). 8. (161). 9. (161). (162). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

7.

183

1. (183). 2. (184). 3. (185). 4. (186). 5. (186). 6. (186). 7. (187). 8. (187). 9. (188). (188). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

8.

197

1. (197). 2. (198). 3. (198). 4. (198). 5. (199). 6. (199). 7. (200). 8. (201). 9. (201). 10. (202). 11. (202). 12. (202). 13. (203). 14. (204). (205). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

9.

5 221

1. (222). 2. (222). 3. (223). 4. (223). 5. (224). 6. (224). 7. . (226). 8. (227). 9. (227). 10. (228). 11. (229). (230). . . . . . . . . . . . . . . . . . . . . 230 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232

10.

253

1. (253). 2. (253). 3. (254). 4. (254). 5. , (254). 6. (255). 7. (255). 8. (256). 9. (256). 10. (257). 11. (257). 12. (258). 13. (258). (259). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260

11.

273

1. (273). 2. (274). 3. (275). 4. (276). 5. (276). 6. (277). 7. (277). (278). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278

12.

289

1. (289). 2. (290). 3. , ; (291). 4. , , (291). 5. (292). 6. (292). 7. (293). 8. (294). 9. (295). 10. (295). (296). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297

13.

308

1. (309). 2. . (310). 3. (310). 4. (311). 5. (312). 6. -

6

(312). 7. (313). (314). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315

14.

325

1. (325). 2. (326). 3. (327). 4. (328). 5. (328). 6. (331). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332

15.

345

1. (345). 2. (346). (347). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347

16.

353

1. (354). 2. (354). 3. (355). (356). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356

17.

361

1. (361). 2. (362). 3. (363). 4. (363). 5. (364). 6. (364). (365). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365

18.

373

1. 90 (374). 2. 60 (374). 3. (376). 4. (377). (378). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379

19.

388

1. (389). 2. (389). 3. (390). 4. (391). 5. (391). 6. (393). 7. (394). 8. (394). (396). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396

20.

407

1. (407). 2. (408). 3. -

7

(408). 4. (409). 5. (409). 6. (410). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411

21.

419

1. , . . (419). 2. (420). 3. (421). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422

22.

430

1. (430). 2. (431). 3. (432). 4. (433). 5. (433). 6. (434). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435

23. , ,

453

1. (453). 2. (454). 3. (454). 4. (455). 5. (456). 6. (457). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 458

24.

469

1. (469). 2. (469). 3. (470). 4. (470). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471

25. , ,

479

1. (479). 2. , (480). 3. , (480). 4. (481). 5. , (481). 6. (482). 7. (483). 8. (483). 9. (484). 10. (485). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485

26. .

506

1. (506). 2. , (507). 3. (507). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 508

27. 1. (513). 2. (514). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

513514

8 28.

517

1. (518). 2. (518). 3. (519). 4. (520). 5. , , , (521). 6. (523). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524

29.

535

1. (535). 2. (537). 3. (538). 4. (542). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 542

30.

559

1. (559). 2. (561). 3. (564). 4. , (565). 5. (567). 6. (567). 7. (568). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 568

31. , ,

583

1. (583). 2. (584). 3. (586). 4. (587). 5. (589). 6. (590). 7. (591). 8. , (591). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593

611 625 632

. , . 57- . : . , . , . , . . . . , . , planimetry_bug@mccme.ru. , 5.137, . , 30.34 . 200 . 31, , . ( , .) : 2.11, 2.40, 2.84, 3.9, 3.49, 3.50, 3.57, 3.62, 3.65, 3.763.82, 4.33, 4.57, 5.13, 5.17, 5.24, 5.37, 5.53, 5.70, 5.71, 5.77, 5.96, 5.97, 5.126, 5.127, 5.1595.161, 6.41, 6.57, 7.17, 7.43, 8.45, 8.588.60, 9.10, 9.27, 9.48, 9.85, 9.95, 10.20, 10.58, 10.21, 12.17, 12.31, 12.77, 12.78, 12.83, 13.14, 13.15, 13.39, 14.38, 14.42, 14.4414.49, 14.53, 15.4, 17.23, 17.33, 17.40, 17.41, 17.42, 18.26, 18.31, 19.5019.52, 20.11, 20.28, 20.33, 22.3, 22.7, 22.14, 22.1522.23, 22.2422.31, 22.34, 23.16, 24.5, 24.6, 24.8, 24.10, 24.16, 24.17, 25.26, 25.37, 25.42, 28.8, 29.1429.19, 29.31, 29.32, 29.34, 29.40, 29.42, 30.34, 31.131.84. , : 2.5, 5.125, 10.46, 12.41, 14.60, 20.7, 23.15, 23.22, 24.7, 24.15, 24.18, 25.16, 25.63. Internet http://www.mccme.ru/prasolov/. .

, , . , , . . 30 , ( 2 14). . , . , . 15 . , , , . 70 , . . . , , , . 28 . . , 29 30 . . . . . . . 2, . 3. (.: , 1956). , . . , . . , . . , . . , . . . . . .

1

1. ABC A1 B1 C1 (: ABC A1 B1 C1 ) , : ) AB : BC : CA = A1 B1 : B1 C1 : C1 A1 ; ) AB : BC = A1 B1 : B1 C1 ABC = A1 B1 C1 ; ) ABC = A1 B1 C1 BAC = B1 A1 C1 . 2. A AB1 C1 AB2 C2 , AB1 : AB2 = AC1 : AC2 ( B1 B2 , C1 C2 ). 3. , . . , . . 4. , . . . , , SABC =1 AB AC sin A. 2

5. A1 A2 . . . An B1 B2 . . . Bn , A1 A2 : A2 A3 : . . . : An A1 = B1 B2 : B2 B3 : . . . : Bn B1 A1 , . . . , An B1 , . . . , Bn . ; .

1. ABC AA1 BB1 . , A1 C BC = B1 C AC. 2. ABC C CH. , AC2 = AB AH CH2 = AH BH. 3. , 2 : 1, . 4. BC ABC A1 , BA1 : A1 C = 2 : 1. CC1 AA1 ?

12

1.

5. ABC PQRS , P Q AB AC, R S BC. a ha .

1. , 1.1. AD BC ABCD a b (a > b). ) , . ) MN, AB CD AM : MB = DN : NC = p : q. 1.2. , . , , ? 1.3. ) A1 B1 BC AC ABC BA1 : A1 C = 1 : p AB1 : B1 C = 1 : q. AA1 BB1 ? ) BC AC ABC A1 B1 . AA1 BB1 D. a1 , b1 , c d A1 , B1 , C D AB. , 1 1 1 1 + = + . a1 b1 c d

1.4. P CC1 ABC AA1 BB1 ( A1 B1 BC CA). , A1 B1 AB. 1.5. , P ABCD Q AB CD, AD . , BC. 1.6. AD ABCD P , AP : AD = 1 : n; Q AC BP. , AQ : AC = 1 : (n + 1). 1.7. A1 B1 C1 D1 ABCD ( A1 AB, B1 BC . .). , . 1.8. BD ABCD K. AK BC CD L M. , AK2 = LK KM. 1.9. . , .

13

1.10. AD ABCD E , AE = BC. CA CE BD O P . , BO = PD, AD2 = BC2 + AD BC. 1.11. A B O 60 . M. , , MA OB, , MB OA. 1.12. ) A, B C , A1 , B1 C1 . , AB1 BA1 AC1 CA1 , BC1 CB1 . ) A, B C , A1 , B1 C1 , AB1 BA1 , AC1 CA1 BC1 CB1 . , A1 , B1 C1 . 1.13. ABC AA1 BB1 . , M A1 B1 AB M AC BC. 1.14. M N AD BC ABCD. DC D P; Q PM AC. , QNM = MNP. 1.15. AD BC ABCD A C K L. KL AB CD M N, AC BD O P. , KM = NL, KO = PL. 1.16*. AB, BC, CD DA ABCD P, Q, R S , BP : AB = CR : CD = a AS : AD = BQ : BC = b. , PR QS b : (1 b) a : (1 a).

2. 1.17. ) ABC BD . , AD : DC = AB : BC. ) , O ABC AA1 AO : OA1 = (b + c) : a, a, b, c . 1.18. a, b. . 1.19. , A ABCD, CD E BC F. , 1 1 1 + = . 2 2 2AE AF AB

1.20. BB1 CC1 ABC B2 C2 , AB2 C = AC2 B = 90 . , AB2 = AC2 .

14

1.

1.21. ABCD (BC AD) , AB CD K L , AD BC M N. ) Q BM AN. , KQ AD. ) , AK KB = CL LD. 1.22. BC CD ABCD ( ) AM AN. , MAN ABC. 1.23. l AB AD ABCD E F . G l AC. , AD AC AB + = . AE AF AG

1.24. AC ABCD. C AB AD CE CF . , AB AE + AD AF = AC2 . 1.25. ABC 3a + 2b = 180 . , a2 + bc = c2 . 1.26. AB CD , AB CD ; M AB CD. , AM BM . CM DM

1.27. P AC ABC AK CL , BC AB E F . , AK CL EF . 1.28. P. , P, a b. , 1 1 + . a b

1.29. BC ABC , K L, . , AK AL BC . 1.30. O ABC. AC BC M K , BK AB = BO2 AM AB = AO2 . , M, O K . 1.31*. , a1 = a2 b1 = b2 (. 1.1), x = y. 1.32*. MN AMN, NBM MNC (. 1.2). , ABC , M N.

15

. 1.1

. 1.2

1.33*. BE ABC , 3. ABC. . 5.52.

3. 1.34. AC ABC E. E DE BC EF AB (D E ). , SBDEF = 2 SADE SEFC . 1.35. AB CD ABCD M N , MN . MN, BC = a AD = b. 1.36. Q, ABC, , . , 1 , S2 S3 . , ABC S ( S1 + S2 + S3 )2 . 1.37. , , S, 3S/4. 1.38. ) , , ABCD, ABCD. ) , , , . 1.39. O, S, .

16

1.

.

4. 1.40. BC ABC C D E . , BC = 3AC, AEC, ADC ABC 90 . 1.41. K AB ABCD, L AC AL : LC = 3 : 1. , KLD . 1.42. A ABCD l1 l2 , . B D BB1 , BB2 , DD1 DD2 . , B1 B2 D1 D2 . 1.43. CA CB ABC D E , CD = CE. , D C AE, AB K L. , KL = LB. 1.44*. AB, BC, CD DA ABCD, a, b, c d, a c, b d, c a d b. , . 1.45*. ABCDEF R O, AB = CD = EF = R. , BOC, DOE FOA, O, R. * * * 1.46. BC CD ABCD BCK DCL. , AKL . 1.47. . , . 1.48*. ABC 2a, 2b 2g A , B C , a + b + g = 180 . , A B C a, b g. 1.49*. ABC AB1 C AC1 B BA1 C . , AB1 A1 C1 . 1.50*. ) AB AC ABC ABC1 AB1 C,

17

C1 = B1 = 90 , ABC1 = ACB1 = f; M BC. , MB1 = MC1 B1 MC1 = 2f. ) ABC . , , ABC. 1.51*. AB AC ABC AC1 B AB1 C f . ) M AA1 ( ), B1 C1 . , B1 MC1 = f. ) O BC, B1 C1 . , B1 OC1 = 180 f. 1.52*. ABCD , a A C. , , , , a. . 1.23, 3.1, 3.22, 5.15, 5.16, 7.247.26, 8.45.

5. , 1.53. AA1 BB1 ABC. , A1 B1 C ABC. ? 1.54. C ABC CH, H HM HN BC AC . , MNC ABC. 1.55. ABC BB1 CC1 . ) , A B1 C1 . ) , B1 C1 OA, O . 1.56. ABC A1 , B1 C1 , AA1 , BB1 CC1 H. , AH A1 H = BH B1 H = CH C1 H , H ABC. 1.57. ) , AA1 , BB1 CC1 ABC A1 B1 C1 . ) AB, BC CA ABC C1 , A1 B1 . , B1 A1 C = = BA1 C1 , A1 B1 C = AB1 C1 A1 C1 B = AC1 B1 , A1 , B1 C1 ABC. 1.58. ABC AA1 , BB1 CC1 . , , A1 AC, B1 C1 . 1.59. ABC AA1 , BB1 CC1 . , A1 B1 AB B1 C1 BC, A1 C1 AC.

18

1.

1.60*. p ABC, q , . , p : q = R : r, R r ABC.

6. 1.61. r. , , . r1 , r2 , r3 . , r1 + r2 + r3 = r. 1.62. ABC. x y , M AC MXM MYM , M x y AB BC , 1. 1.63. ABC H BC HE AC; O HE. , AO BE . 1.64. , , , . 1.65. AC B AB, BC, CA S1 , S2 , S3 AC. D S3 , BD AC. S1 S2 , F E . ) , EF S3 , D. ) , BFDE . 1.66*. M , ABCD, MQ MP MR MT . , PR QT , ABCD. 1.67*. , , . , , . , , . . 6.27.

1.68. . ,

19

. ? 1.69. . , , . 1.70. 1. . . 1.71. A1 , B1 C1 ABC . , ABC = A1 B1 C1 . 1.72. , BAC = 2ABC, BC2 = (AC + AB) AC. 1.73. l A, B, C D. A B, C D . , ( ) l . 1.74. ABC AD A1 C1 . AD A1 C1 K. , 2A1 K = |b c|. 1.75. AD CD ABCD M N , MN AC. , SABM = SCBN . 1.76. AC ABCD P Q , AP = CQ. M , PM AD QM AB. , M BD. 1.77. AD BC O. EF, , AB CD. , AE : CF = = AO : CO. 1.78. , , , . , a, b c. 1.79. , , , x. x, a, b c. 1.80. P ABC, ABP = = ACP. AB AC C1 B1 , BC1 : CB1 = CP : BP. , , BP CP, ( ) B1 C1 , BC.

20

1.

1.1. a) P Q AB CD, K L PQ AC BD. PL = a/2 PK = b/2, KL = PL PK = (a b)/2. ) AD F , BF CD. E MN BF. MN = ME + EN = =qa + pb . p+q qAF q(a b) + (p + q)b +b= = p+q 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. B1 BC A1 A2 BB1 . B1 C/B1 A2 = 1 + p, AO : OA1 = AB1 : B1 A2 = B1 C : qB1 A2 = (1 + p) : q. ) BA1 : A1 C = 1 : p AB1 : B1 C = 1 : q. AD : DA1 = (1 + p) : q BD : DB1 = (1 + q) : p. a1 =1+p+q 1+p+q d, b1 = d c = (1 + p + q)d. 1+p 1+q

1.4. A2 A1 B. CA1 : A1 A2 = CP : PC1 A1 A2 : A1 B = 1 : 2, CA1 : A1 B = CP : 2PC1 . CB1 : B1 A = = CP : 2PC1 = CA1 : A1 B. 1.5. P QM AQD ( ). , 1.4 , P . , BC AD. 1.6. AQP CQB, AQ : QC = = AP : BC = 1 : n. AC = AQ + QC = (n + 1)AQ. 1.7. A1 B1 C1 D1 , B1 D1 , , AB CD. , BC AD. ABCD. 1.8. , AK : KM = BK : KD = LK : AK, . . AK2 = LK KM. 1.9. AC , ABCD. AA1 CC1 BD (. 1.3). , BA1 = DC1 . OP O . 1.3 BD. , P BD. AA1 , OP, CC1 AO = OC, A1 P = 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 + BC AD.

21

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 , OC : OA = OA1 : OC1 , OC : OB = OB1 : OC1 , , BC1 CB1 ( ). ) D E AB1 CA1 , CB1 AC1 . CA1 : A1 D = CB : BA = EC1 : C1 A. CB1 D EB1 A, A1 , B1 C1 . 1.13. , , . a A1 , AC AB, b B1 AB BC. , , A1 M : B1 M = p : q, p + q = 1. M AC BC qa pb . , 1.1 ) M AB qa + pb. 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 ( ). . 1.4 AD1 CD1 S1 R1 , SS1 DD1 RR1 DD1 . N PR1 QS1 ; N1 N2 , N DD1 , PR QS . # # # # # # N1 N = bRR1 = abDD1 N2 N = aSS1 = abDD1 . 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 = 1 a. PR QS a : (1 a). 1.17. ) A C AK CL BD. CBL = ABK CDL = KDA, BLC BKA CLD AKD. AD : DC = AK : CL = AB : BC. ) , BA1 : A1 C = BA : AC BA1 + A1 C = BC, BA1 = = ac/(b + c). BO ABA1 , AO : OA1 = = AB : BA1 = (b + c) : a.

22

1.

1.18. O ABC, B1 AC, A1 BC. BOA1 BCB1 , BO : BA1 = BC : BB1 , , R = BO = a2 / 4a2 b2 . 1.19. EAD = f, AE = AD/ cos f = AB/ cos f AF = AB/ sin f. 1 1 (cos2 f + sin2 f) 1 + = = . AE2 AF2 AB2 AB2

1.20. , AB2 = AB1 AC = AC1 AB = AC2 . 2 2 1.21. ) BQ : QM = BN : AM = BK : AK, KQ AM. ) O . CBA + BAD = 180 , ABO + BAO = 90 . AKO OKB, . . AK : KO = OK : KB. , AK KB = KO2 = R2 , R . CL LD = R2 . 1.22. ABC ( ), MAN ( ). , . ABC = MAN. ABM ADN ABM ADN, AM : AN = AB : AD = AB : CB, . . ABC MAN. 1.23. AC D B , BB l DD l. AB : AE = = AB : AG AD : AF = AD : AG. ABB CDD AB = CD, = AB = CD . AD CD + AD AC AB + = = . AG AG AG AG AB AD + = AE AF

1.24. B BG AC (. 1.5). ABG ACE AC AG = AE AB. AF CB , GCB CAF CBG ACF . AC CG = AF BC. , AC (AG + CG) = AE AB + AF BC. AG + CG = AC, . 1.25. a + b =90 (a/2), g =180 a b =90 +(a/2). AB D , ACD=90 (a/2), . . AC = AD. ABC CBD, , BC : BD = AB : CB, . . a2 = c(c b). 1.26. AB CD AMC , . AM/CM . BM/DM . 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. 1.28. C , A B . AC BC K L , PK BC PL AC. AKP PLB, AK : KP = PL : LB,

. 1.5

, (a p)(b p) = p2 , p = PK = PL. ,

1 1 1 + = . a b p

23

. 1.6

. 1.7

1.29. BC O, AK AL BC P Q. , BP < BQ. LCO LC AB. ABQ LCQ, . . BQ : QC = = AB : LC = 2 : 1. , BC = BQ + QC = 3QC. BC = 3BP. 1.30. BK:BO=BO:AB KBO=ABO, KOB OAB. 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 x + z1 a + z3 b y + z1 y a b + z2 = , = , = , = . b b a b + z2 a a a + z3 b

, x(y + z1 ) = y(x + z1 ), , x = y. 1.32. AMN = MNC BMN = MNA, AMB = ANC. , AM : AN = NB : NM = BM : CN. AMB ANC, , MAB = NAC. , BAC = MAN. . B1 C1 B C MN. AM : NB = MN : BM = MC : NC, MA MC1 = AM NC = NB MC = MB1 MC. M, A, C1 M, B1 , C . , A , BB1 CC1 . 1.33. AEB + BEC = 180 , ABE BEC, . . BE .

24

1.

: ABE = CBE ABE = BCE. , ABE = CBE. . 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 SADE SEFC . 1.35. MN = x; E AB CD. EBC, EMN EAD , SEBC : SEMN : SEAD = = a2 : x2 : b2 . SEMN SEBC = SMBCN = = SMADN = SEAD SEMN , x2 a2 = b2 x2 , . . x2 = (a2 + b2 )/2. . 1.8 1.36. Q 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 GQ IE FQ AI + IE + EC = + + = = 1, S AC AC AC AC

. . 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 = =SABCD . SEFGH 4 S S SABD + CBD = ABCD . SBEF + SDGH = 4 4 4 S S S = SABCD ABCD ABCD = ABCD . 4 4 2

) AC = BD, EFGH ( 1.2). ) SABCD = 2SEFGH = EG FH. 1.39. E, F, G H ABCD; E1 , F1 , G1 H1 O . EF E1 OF1 , SE1 OF1 = 4SEOF . SF1 OG1 = 4SFOG , SG1 OH1 = 4SGOH SH1 OE1 = 4SHOE . SE1 F1 G1 H1 = 4SEFGH . 1.38 ) SABCD = 2SEFGH . SE1 F1 G1 H1 = 2SABCD = 2S. 1.40. . BCMN MN P Q (. 1.9). ABC = = PDQ ACD = PMA. PAD ABC + ADC = PDQ + ADC = 45 . . DE = 1, EA = 2, EB = 2, AD = 5 BA = 10, DE : AE = EA : EB = AD : BA DEA AEB. , ABC = EAD. , AEC = CAE = 45 . ABC + ADC + + AEC = (EAD + CAE) + ADC = CAD + ADC = 90 .

25

. 1.9

. 1.10

1.41. L LM AB LN AD. KM = MB = ND KL = LB = DL, KML DNL . , DLK = NLM = 90 . 1.42. D1 A = B1 B, AD2 = BB2 D1 AD2 = B1 BB2 , D1 AD2 = = B1 BB2 . AD1 BB1 ( AD2 BB2 ) , B1 B2 D1 D2 . 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 ABC1 D1 A2 BCD2 ; P, Q, R S , AB, BC, CD DA. ABC + ADC = 180 , ADC = A2 BC1 . , , , RDS = PBQ RS = = PQ. QR = PS. , PQRS , RDS PBQ , ; QCR SAP. PQR + RSP = BQC + DSA = 180 , PQB = RSD RQC = PSA. , PQRS . 1.45. K, L, M FOA BOC, BOC DOE, DOE FOA; 2a, 2b 2g BOC, DOE FOA (. 1.11). K OB . 1.11

26

1.

BOC, OKB = 90 + a. OKA = 90 + g. a + b + g = 90 , AKB = 90 + b. AOB K, . L, COD, , OKB = CLO. , KOL = OKB. , COL = KBO, KOB + COL = 180 OKB = 90 a, , KOL = 2a + (90 a) = = 90 + a = OKB. , KL = OB = R. LM = MK = R. 1.46. A = a. , KCL ADL 240 a ( 120 + a). KC = BC = AD CL = DL, KCL = = ADL, , KL = AL. KL = AK. 1.47. P, Q R , DA, AB BC a A. , PAQ = 90 + a = RBQ, , PAQ = RBQ. AQ BQ , PQ QR. 1.48. , A, B C AB CA BC 360 , 360 . AC AC P, BC A (. 1.12). AB P = CB A , AB = CB , AP = CA PAB = 360 PAC C AB = 360 A BC C AB = A CB . , C B A = C B P, , 2A B C =PB A =AB C, PB A= = A B C, AB A . 1.49. BA:BC=BC1 :BA1 ABC=C1 BA1 , ABC C1 BA1 . ABC B1 A1 C. . 1.12 BA1 = A1 C, C1 BA1 = B1 A1 C. , AC1 = C1 B = B1 A1 AB1 = B1 C = C1 A1 . , AB1 A1 C1 . 1.50. ) P Q AB AC. MP = AC/2 = QB1 , MQ = AB/2 = PC1 C1 PM = C1 PB + BPM = B1 QC + CQM = B1 QM. , MQB1 = C1 PM, , MC1 = MB1 . , PMC1 + + QMB1 = QB1 M + QMB1 = 180 MQB1 , MQB1 = A + CQB1 = = A + (180 2f). , B1 MC1 = PMQ + 2f A = 2f. (, C1 PB + BPM > 180 , .) ) AB AC B C , AB : AB = = AC : AC = 2 : 3. M B C ABC. AB AC AB C1 AB1 C 60 B C . B1 C1 , AB AC; , ) MB1 = MC1 B1 MC1 = 120 . . ) ) , . 1.51. ) B AC AB1 , B1 ; C . AB : AC = AC1 : AB1 = AB : AC, B C BC. N

27

B C , , 1.50 ), NC1 = NB1 (. . N = M) B1 NC1 = 2AB B1 = 180 2CAB1 = f. ) BC BA1 C 360 2f A1 ( f < 90 , 2f). 360 , A1 B1 C1 180 f, f/2 f/2 (. 1.48). , , , A1 = O. 1.52. O1 , O2 , O3 O4 , AB, BC, CD DA; M AC. MO1 = MO2 O1 MO2 = a (. 1.50 ). MO3 = MO4 O3 MO4 = a. , a M O1 MO3 O2 MO4 . 1.53. A1 C = AC|cos C|, B1 C = BC|cos C| C ABC A1 B1 C , , |cos C|. 1.54. M N CH, CMN = CHN, AC HN, CHN = A. CNM = B. 1.55. ) l A . (l, AB) = (AC, CB) = (C1 B1 , AC1 ), , l B1 C1 . ) OA l l B1 C1 , OA B1 C1 . 1.56. AA1 , BB1 CC1 , A1 BH B1 AH, , AH A1 H = BH B1 H. BH B1 H = CH C1 H. ( , .) AH A1 H = BH B1 H = CH C1 H, A1 BH B1 AH, , BA1 H = AB1 H = f. CA1 H = CB1 H = 180 f. AC1 H = = CA1 H = 180 f AC1 H = AB1 H = f, f = 90 . . AA1 , BB1 CC1 . 1.57. ) 1.53 C1 A1 B = CA1 B1 = A. AA1 BC, C1 A1 A = B1 A1 A. , B1 B C1 C A1 B1 C1 A1 C1 B1 . ) AB, BC CA A1 B1 C1 , A1 A B1 A1 C1 , , AA1 BC. BB1 CC1 . 1.58. 1.57 ) , B1 A1 AC B1 C1 . 1.59. 1.53 B1 A1 C = BAC. A1 B1 AB, B1 A1 C = = ABC. BAC = ABC. , B1 C1 BC, ABC = BCA. ABC A1 C1 AC. 1.60. O ABC. OA B1 C1 (. 1.55 ), SAOC1 + SAOB1 = R B1 C1 /2. B C , SABC = qR. , SABC = pr. 1.61. , , BC, A AB AC, ABC : P1 + P2 + P3 = P. -

28

1.

, ri /r = Pi /P. , . 1.62. M = A. XA = A, AYA = 1. CXC = 1. , y = AYA x = CXC . BC D , AB MD (. 1.13). E CXC MD. XM M + YM M = XC E + YM M. ABC MDC, CE = YM M. XM M + YM M = XC E + CE = XC C = 1. 1.63. D BH. . 1.13 BHA HEA, AD : AO = AB : AH DAH = OAE. , DAO = BAH, , DAO BAH DOA = BHA = 90 . 1.64. AA1 , BB1 CC1 ABC. B1 B1 K B1 N AB BC B1 L B1 M AA1 CC1 . KB1 : CC1 = AB1 : AC = LB1 : A1 C, KLB1 C1 A1 C, , KL C1 A1 . MN C1 A1 . , KN C1 A1 (. 1.53 1.54). , K, L, M N . 1.65. ) O AC, O1 AB, O2 BC. , AB BC. O1 O1 K EF (K EO2 ). , DBO O1 KO2 . , O1 O2 = AC/2 = DO BO = (BC AB)/2 = KO2 . DBO O1 KO2 , BOD = = O1 O2 E, . . DO EO2 , D, EF. ) AC F, D, E , FAB = DAC = EBC FBA = = DCA = ECB, . . F AD, E DC. , AFB = BEC = = ADC = 90 , FDEB . 1.66. MQ MP . 1.14 , AD BC, MR MT , AB CD (. 1.14). M1 P1 RT QP . TM1 = RM = AQ TM1 AQ, AM1 TQ. AP1 RP. M1 AP1 = 90 , RP TQ. TQ RP, M1 A RP, P1 A TQ E, F, G . , E AC, , AFEG AM1 CP1 . ARF = AM1 R = M1 TG = M1 CT, a. AF = RA sin a = M1 A sin2 a, AG = M1 T sin a = M1 C sin2 a, AFEG AM1 CP1 .

29

1.67. O1 O2 . K, L; M, N (. 1.15). KM LN -

. 1.15 O1 O2 P1 P2 . , P1 = P2 . A, D1 , D2 KL MN, KM O1 A, LN O2 A . O1 AM + NAO2 = 90 , O1 MA ANO2 , AO2 KM AO1 LN. AD1 : D1 O1 = O2 P1 : P1 O1 D2 O2 : AD2 = O2 P2 : P2 O1 . AKO1 M O2 NAL AD1 :D1 O1 =D2 O2 :AD2 . , O2 P1 : P1 O1 = O2 P2 : P2 O1 , . . P1 = P2 . . . 3.70.

2

1. ABC, , , . O . 1 ABC = AOC,2

B O AC, ABC = 180 1 AOC, 2

B O AC. , , , , 180 . 2. AB , A, AB. 3. , , . 4. , , , , 180 . , . AB CD (: (AB, CD)) , AB , CD. , n 180 , . , CD AB AB CD ( 180 , , 0 ). : ) (AB, BC) = (BC, AB); ) (AB, CD) + (CD, EF) = (AB, EF); ) A, B, C, D, , , (AB, BC) = (AD, DC) ( : B D AC; B D AC).

1. ) A, , AB AC, . , -

31

BAC , . ) BAC . , BAC , BAC , A. 2. P, BAC, PC1 PB1 AB AC. , C1 AP = C1 B1 P. 3. , , n-, 180 /n. 4. ABC AB. ABC. 5. C ABC D. , AD = BD.

1. , 2.1. A ABC O . A AH. , BAH = OAC. 2.2. M K. M K AB CD , A C, B D. , AC BD. 2.3. M, A, MP MQ . A AK PQ. , PAK = MAQ. 2.4. ) B ABC M; O , Ob , AC. , A, C, O Ob M. ) O, ABC, , AO, BO CO BCO, ACO ABO. , O ABC. 2.5. ABC A , B C . , A . 2.6. AC ABCD ACK, B K

32

2.

AC. , BK = |AK CK|/ 2 DK = (AK + CK)/ 2. 2.7. ABC AA1 BB1 . , CAA1 = CBB1 , AC = BC. 2.8. ABC 120 . , , 120 . 2.8 .

2.9. n . , , M, , , . 2.10. A, B, M N. M MA1 MB1 , NB NA . , AA1 BB1 . 2.11. P Q. P A B, C D. , AQD = BQC. 2.12. ABCDEF , AB DE BC EF. , CD AF. 2.13*. A1 A2 . . . A2n . , , , . , n , n . 2.14*. ABC. , , ( ) A, B C. , . . 1.54.

2. . A, B, C, D . AC BD (AB + CD)/2, AB CD |AD CB|/2. ( , .)

2.15. A, B, C, D . M AB. MC MD AB E K. , KECD .

33

2.16. , . , , , 60 . 2.17. ABCD AB P. , O APB. 2.18. A, B, C, D ; A1 , B1 , C1 D1 AB, BC, CD DA . , A1 C1 B1 D1 . 2.19. ABC P , BPC = = A + 60 , APC = B + 60 APB = C + 60 . AP, BP CP ABC A , B C . , A B C . 2.20. A, C1 , B, A1 , C, B1 . ) , AA1 , BB1 CC1 ABC, A1 B1 C1 . ) , AA1 , BB1 CC1 ABC, A1 B1 C1 . 2.21*. T1 T2 , T2 , T1 . , , T1 T2 , , , T1 .

3. . AB , l , A, X (l, AB) = (XA, XB).

2.22. P Q. A AP AQ, B C. , A BC. 2.23. S1 S2 A P. A AB S1 , P CD, AB ( B C S2 , D S1 ). , ABCD . 2.24. S1 S2 A B. A AQ S1 ( Q S2 ), B BS S2 ( S

34

2.

S1 ). BQ AS S1 S2 R P. , PQRS . 2.25. A ABC BC E; AD ABC. , AE = ED. 2.26. S1 S2 A. A , S1 B, S2 C. C B , D. , BDC , A. 2.27. A B. A AM AN (M N ). , : ) ABN + MAN = 180 ; ) BM/BN = (AM/AN)2 . 2.28. M. AB , T. , MT AMB. 2.29. M, S, AB; M MP MQ , A B. , 1/PM + 1/QM , M. 2.30. S1 ABC A C. S2 AC C B; S1 M. , AM BC . 2.31. S S1 S2 A1 A2 ; B S, K1 K2 A1 B A2 B S1 S2 . , K1 K2 S1 , S2 . . 1.55 ), 6.49.

4. 2.32. ABCD A1 B1 C1 D1 . , AC = A1 C1 . 2.33. M, , MP MQ AB CD. , PQ M. 2.34. ABC B 60 , AD CE O. , OD = OE. 2.35. ABC B C 40 ; BD B. , BD + DA = BC.

35

2.36. AB S O C. AOC S D. , BC = CD. 2.37. ABCD M , MAC = = MCD = a. ABM. 2.38. A B ABC S, C . D S, BD = AB. CD S E. , EC S. 2.39*. , , . K ? 2.40*. ABC A . A , BC. X, AC AB B1 C1 . BC1 CB1 Y. , BY + CY = AX.

5. , 2.41. M BC ABC AB MN. , MAN = MCN. 2.42. ABCD AD BC O; B C B C BOC. , C AC = B DB. 2.43. AB CD ABCD P, BC AD Q. , AQB BPC . 2.44*. AB AC ABC M N. P MN B ( ). , : ) BPC = 90 ; ) SABP : SABC = 1 : 2. 2.45*. ABCD M , ABMD . , CBM = CDM, ACD = BCM. 2.46*. AP, BP CP ABC A1 , B1 C1 . A2 , B2 C2 BC, CA AB , (PA2 , BC) = (PB2 , CA) = (PC2 , AB). , A2 B2 C2 A1 B1 C1 . 2.47*. APQ ABCD, P Q BC CD

36

2.

; P Q AP AQ. , BQ C CP D . 2.48*. , ABCD CD = AD + BC, A B CD. 2.49*. AC CE ABCDEF M N , AM : AC = CN : CE = l. l, , B, M N . 2.50*. ABC A1 B1 C1 , AB A1 B1 . , , A1 BC AB1 C, C1 . 2.51*. ABC AA1 , BB1 CC1 . KL CC1 , K L BC B1 C1 . , A1 KL AC. 2.52*. O ABC MN CO, M N AC BC . AO BO ABC A B . , A N B M .

6. 2.53. A, B, C D. AB CD M. , AC AD/AM = BC BD/BM. 2.54. A, B C, B l, A, C. AC , B l, D. , AB2 = AC AD. 2.55. l AB C; M N A B l, D C AB. , CD2 = AM BN. 2.56. ABC AH, B C BB1 CC1 , A. , ABC HB1 C1 . 2.57. BC , ABC, P. AP BC Q. , 1/PQ = 1/PB + 1/PC. 2.58. BC CD ABCD E F , EAF = 45 . AE AF BD P Q. , SAEF /SAPQ = 2. 2.59. , C ABC, AB M,

37

N. , CM CN = AC2 CM/CN = = AM BM/(AN BN). 2.60. ABCD A. AB CB H K , CH = BC AK = AB. , : ) DH = DK; ) DKH ABK. 2.61. ) C A B. P, , PA1 , PB1 PC1 BC, CA AB. , PC2 = PA1 PB1 PA1 : PB1 = PB2 : PA2 . 1 ) O ABC OA , OB , OC ABC OA , OB , OC . , OA OB OC = = OA OB OC . 2.62. ABCDE . E AB, BC CD a, b c . E AD. 2.63. ABC AA1 , BB1 CC1 ; B2 C2 BB1 CC1 . , A1 B2 C2 ABC. 2.64. ABC A1 , B1 C1 , 2 : 1, . , A1 B1 C1 ABC. 2.65*. S1 AB S2 A C D. B , S2 M, S1 , S1 N. , MN2 = CN ND. 2.66*. C AB KL MN ( K M AB). KN ML AB Q P. , PC = QC ( ). 2.67*. ) , C, BC AC ABC A1 B1 , M. , AB1 M BA1 M. ) AC BC AA1 BB1 , ABC. M , CM A1 B1 . , CMO = 90 , O . . 2.27 ).

7. 2.68. ABC AC BC . , C

38

2.

, , , C = 90 . 2.69. , , , C, . . 2.70. , ABC AE AM AH. 2.71. ABC. AB P PM PN, AC BC ( M N BC AC); Q APN BPM. , PQ . 2.72. AD ABC E. D AB AC DP DQ. , SABC = SAPEQ .

8. ABCD , . : O ABCD, P .

2.73. , AOC ABCD . 2.74. R. ) AP2 + BP2 + CP2 + DP2 . ) ABCD. 2.75. , OP R. 2.76. A B CD, BD AC K L . , AKLB . 2.77. , ABCD (AB CD + BC AD)/2. 2.78. , O AB CD. 2.79. , , P BC, AD . 2.80. , ABCD P . 2.81. ) A, B, C D . , .

39

) KLMN ; A B KL LM. , AK BM = r2 , r . . 5.45.

9. 2.82. ABC ABC , AB C A BC, A , B C 180 . , . 2.83. ) BC, CA AB ABC ( ) A1 , B1 C1 , . , AB1 C1 , A1 BC1 A1 B1 C . ) A1 , B1 C1 BC, CA AB , A1 B1 C1 . , AB1 C1 , A1 BC1 A1 B1 C . ( , .) 2.84. A1 , B1 , C1 BC, CA, AB , A1 B1 C1 ( , ). , A1 B1 C1 , , A1 , B1 , C1 BC, CA, AB . 2.85. ABC X. AX, BX CX A1 , B1 C1 . , AB1 C1 , A1 BC1 A1 B1 C X, X ABC. 2.86*. BC, CA AB ABC A1 , B1 C1 . , A1 B1 C1 ABC , AB1 C1 , A1 BC1 A1 B1 C ABC. 2.87*. A , B C P BC, CA AB ABC. ) , AB C , A BC , A B C ABC . ) , A BC, AB C, ABC A B C Q.

40

2.

) I, J, K O A BC, AB C, ABC A B C . , QI : OI = QJ : OJ = = QK : OK. . 28.33, 28.34, 28.38.

10. 2.88. . ) , ( ). ) , , . 2.89*. AB, BC CA ( ) C1 , B1 A1 ; O, Oa , Ob Oc ABC, AB1 C1 , A1 BC1 A1 B1 C; H, Ha , Hb Hc . , : ) Oa Ob Oc ABC. ) OH, Oa Ha , Ob Hb Oc Hc . 2.90*. ABCD . , , , , . 2.91*. A, B, C D O. AB CD E, AEC BED E P. , : ) A, D, P O ; ) EPO = 90 . 2.92*. . , . . 19.46, 28.34, 28.36, 28.37.

11. 2.93. ABC AH; O . , OAH = |B C|. 2.94. H ABC, AA . , A H BC . 2.95. A B ABC , m n . , m n ABC. 2.96. ) A , S B C. , -

41

ABC , BC, S. ) , , B C ABC O , AB AC . 2.97*. AC BC ABC ACA1 A2 BCB1 B2 . , A1 B, A2 B2 AB1 . 2.98*. S1 S2 A B, S1 S2 . S1 C A B . , S2 . 2.99*. O OA l. l B C , AB = AC. B C , P Q, M N. PM QN l R S. , AR = AS ( ).

2.100. ABC AA1 BB1 ; M AB. , MA1 = MB1 . 2.101. ABCD A C . , AC = BD sin ABC. 2.102. AD, BE CF ABCDEF . , AB CD EF = = BC DE AF. 2.103. AB = BC = CD, M , K A D. , A, M, K D . 2.104. O1 O2 A B. O1 A O2 N. , O1 , O2 , B N . 2.105. S1 S2 A B. MN S1 M S2 N. A , MN. , O1 AO2 = 2MAN. 2.106. ABCD, , AB = BC. , SABCD = (DA + CD) hb , hb 2 ABD, B. 2.107. ABCD , AC DAB. , AC BD = AD DC + AB BC.1

42

2.

2.108. ABC C CM CH. HD HE AHC CHB. , C, D, H, E M . 2.109. . , , , . 2.110. BHC, H ABC, BHCD. , BAD = CAH. 2.111. ABC, BAC M , CMA = 30 BMA = a. ABM? 2.112. , , .

2.1. AD. CDA = CBA, , BAH = DAC, BHA = ACD = 90 . 2.2. (AC, CK) = (AM, MK) = = (BM, MK) = (BD, DK) = (BD, CK), . . AC BD. 2.3. P Q AM. QMA = = QPA , . PAK MAQ , , PAK = MAQ. 2.4. ) AOM = BAO + ABO = (A + B)/2 OAM = OAC + + CAM = A/2 + CBM = (A + B)/2, MA = MO. MC = MO. OAOb AOM = MAO = f, MAOb = MOb A = 90 f, , MA = MOb . MC = MOb . ) P ACO. COP = (180 CPO)/2 = 90 OAC. BOC = 90 + OAC. BOC = 90 + OAB, , OAB = OAC. , O B C. 2.5. O . O A BC, AOB = ACB = C. , A , , C. A O BC BA. , OA = BC sin f, f = OCA. f C 90 + C = 180 B, sin f 1, sin C sin B. , , A, ABC. 1. ABC, MON, 180 A.

43

2. , A , A ( 31.63). 2.6. B, D K AC. KCA= f 45 . BK=AC sin(45 f)=AC(cos f sin f)/ 2 DK = AC sin(45 + f) = AC(cos f + sin f)/ 2. , AC cos f = CK AC sin f = AK. 2.7. B1 AA1 = A1 BB1 , A, B, A1 B1 . AB A1 B1 AB1 BA1 . AC = BC. 2.8. BC ABC A1 BC. P AA1 A1 BC. P ABC APC = 180 A1 PC = 180 A1 BC = 120 . APB = 120 . 2.9. , M , S OM (O ). S, O, n . , O, 180 /n, 360 /n. , O, 360 (n 1) 360 /n = = 360 /n. , S n . 2.10. , (AA1 , BB1 ) = (AA1 , AB1 ) + (AB1 , BB1 ) = (MA1 , MB1 ) + + (AN, BN). MA1 BN MB1 AN, (MA1 , MB1 ) = (BN, AN) = = (AN, BN). (AA1 , BB1 ) = 0 , . . AA1 BB1 . 2.11. , (AQ, QD) = (AQ, QP) + (PQ, QD) = (AB, BP) + (PC, CD), (CQ, QB) = (CQ, QP) + (PQ, QB) = (CD, DP) + (PA, AB). APB CPD , (AB, BP) = (PA, AB) (PC, CD) = (CD, DP). , (AQ, QD) = (CQ, QB). 2.12. AB DE, ACE = BFD, BC EF, CAE = = FDB. ACE BDF , . AC DF, . . CD AF. 2.13. n. , . , 2(n 1)-, 2n-. A1 . . . A2n 2n-, A1 A2 An+1 An+2 , . . . , An1 An A2n1 A2n . 2(n 1)- A1 A2 . . . An1 An+1 . . . A2n1 . n An1 An+1 = A2n1 A1 , n An1 An+1 A2n1 A1 . An1 An An+1 A2n1 A2n A1 . # # # # n . An1 An A2n1 A2n , An1 An+1 A2n1 A1 , An An1 An+1 = A1 A2n1 A2n An An+1 = A2n A1 , , . n . An1 An+1 = A2n1 A1 , . . A1 An1 An+1 A2n1 . An1 An An+1 A2n1 A2n A1 A1 An1 An+1 A2n1 , An1 An A2n1 A2n , An An+1 A2n A1 , .

44

2.

2.14. FG, GE EF A, B C, EFG , . . (GE, EF) = (EF, FG) = (FG, GE) = = 60 . (BE, EC) = (CF, FA) = (AG, GB) = 60 . , SE , SF SG , E, F G. E SE EFG. O EFG; P, R Q OE, OF OG SE , SF SG . , P, Q R , ABC ( , ), O PQR. , (CB, BP) = (CE, EP) = (EF, EO) = 30 , a (BP, CP) = (BE, EC) = (GE, EF) = 60 . (CB, CP) = (CB, BP) + + (BP, CP) = 30 . , P AB. Q R . PQR , ABC (. 1.50 ). , (PR, RQ) = 60 = (OE, OG) = (OP, OQ), . . O PQR. 2.15. , 2(KEC + KDC) = (MB + AC) + (MB + BC) = 360 , MB = AM. 2.16. , ABC , a. , , a . (a + a )/2 = BAC = 60 . a = a , , BC. a = a = 60 . 2.17. APB = (AB + CD)/2 = AOB, O APB. 2.18. O A1 C1 B1 D1 ; a, b, g d AB, BC, CD DA. 2.19. C A + CA = 2(180 APC) = 240 2B AB + BA = 240 2C, BA + + CA = 2A, C B = C A + AB = 480 2(A + B + C) = 120 . B A = C A = 120 . 2.20. ) , , AA1 C1 B1 . M . AMB1 = (AB1 + A1 B + BC1 )/2 = ABB1 + A1 AB + + BCC1 = (B + A + C)/2 = 90 . ) M1 M2 AA1 BC, BB1 AC. AM1 C BM2 C C, B1 BC = A1 AC, , B1 C = A1 C B1 C1 C = A1 C1 C, . . CC1 A1 C1 B1 . 2.21. T1 A, B C; BC, CA, AB A1 , B1 , C1 . T2 = A1 B1 C1 . AA1 , BB1 , CC1 T1 , O. AB C1 B1 K. , KO AC. AB1 O B1 C1 A1 OB1 = (A1 B + BB1 + C1 D + DD1 )/2 = (a + b + g + d)/4 = 90 .

45

( 2.20 ), . , AKO . KO AC , KOA = KAO = OAC. 2.22. l A . (l, AP) = (AQ, PQ) = (BC, PB), , l BC. 2.23. (AB, AD) = (AP, PD) = (AB, BC), BC AD. 2.24. , (AB, BS) = (AQ, QB) (BA, AQ) = (BS, SA). (BA, AS) = (AB, BQ), , PS QR. , (AP, PQ) = (AB, BQ) = = (AS, SR), PQ SR. 2.25. E BC. ABC = = EAC ADE = ABC + BAD = EAC + CAD = DAE. 2.26. P . (AB, DB) = = (PA, PB) (DC, AC) = (PC, PA). , (DC, DB) = (PC, PB) = (PC, CA) + (BA, PB); . 2.27. ) MAB = BNA, ABN MAN ABN. ) BAM = BNA BAN = BMA, AMB NAB, , AM : NA = MB : AB AM : NA = AB : NB. , . 2.28. A1 B1 MA MB . M , A1 B1 AB. A1 MT = A1 TA = B1 A1 T = B1 MT. 2.29. f AB , . AB = 2R sin f, R S. , PM = AM sin f QM = BM sin f. 1 1 AM + BM 2R + = AM BM = . PM QM sin f AM BM

AM BM AB. 2.30. AM S2 D. MDC = MCA = MAB, CD AB. , CAM = MCB = = MDB, AC BD. , ABCD , AD BC . 2.31. l1 , S1 A1 . K1 K2 S1 , (K1 K2 , K1 A1 ) = (K1 A1 , l1 ). , (K1 A1 , l1 ) = (A1 B, l1 ) = (A2 B, A1 A2 ). K1 K2 S2 , (K1 K2 , K2 A2 ) = (A1 B, A1 A2 ). , (K1 K2 , K1 A1 ) = (A2 B, A1 A2 ), (K1 K2 , K2 A2 ) = = (K1 K2 , A2 B) = (K1 K2 , A1 B) + (A1 B, A1 A2 ) + (A1 A2 , A2 B) = (A1 B, A1 A2 ). 2.32. AC A1 C1 ABC A1 B1 C1 , AC = A1 C1 . 2.33. O. P Q OM, . . O, P, Q M R/2. POQ = AOD, POQ = BOD = = 180 AOD, . . PQ . 2.34. AOC = 90 + B/2 (. 5.3), EBD + EOD = = 90 + 3B/2 = 180 , , BEOD . EO OD EBO OBD, EO = OD.

46

2.

2.35. BD D Q, ACQ = 40 . P AB QC. BPC = 60 D BCP. 2.34 AD = DQ. , BQC = BCQ = 80 . , BC = BD + DQ = BD + DA. 2.36. , ACD BCD B. , ACD = AOD = 2ABD. 2.37. M ABC, MAC < 45 < < MCD. , ABC ACD M , ACD. AMC = 180 MAC (45 MCD) = 135 . , M AB B. ABM = 2ACM = 90 2a. 2.38. O S. B ACD, CDA = ABC/2 = 30 , , EOA = 2EDA = 60 , . . EOA . , AEC = AED = AOB = 2AOC, E AOC. , EC = EO. 2.39. : , K ( K1 ), - () . O , O1 O2 , K2 K . A . , K1 A K2 A. , K2 O2 A = 2K1 OA. O , K2 OA = K2 O2 A/2 = K1 OA, . . K2 , K1 O . . 2.40. ABC BC , AC AB AB AC, . , Y ABC. BC1 CB1 B2 C2 . B1 AC1 , XAC = B1 AC = B1 CA = C2 CA. XAB = B2 BA. , BC B2 C2 , , BC2 B2 C . BY = YC2 , , BY + CY = CY + YC2 = CC2 = AX, CC2 AX AC. 2.41. N C AM. MAN MCN , . 2.42. BOC AC DB , , C AB = B DC . BO = B O, CO = C O AO : DO = CO : BO, AO B O = DO C O, . . AC B D C AB = B DC . 2.43. , . 2.1. , x = 90 . BMRN 180 f, a + f, b + f x, x = 90

47

(2a + f) + (2b + f) = 180 . , 2a + f = BAD 2b + f = BCD. 2.44. ) , P1 B ( ), BC 90 , P1 MN. P1 N CO, O , (P1 N, NC) = (P1 O, OC) = (180 A)/2 = = (MN, NC). ) BPC = 90 , BP = BC cos(B/2), SABP : SABC = (BP sin(B/2)) : (BC sin B) = = 1 : 2. 2.45. N , BN MC NC BM. NA CD, NCB = CBM = = CDM = NAB, . . A, B, N C . ACD = . 2.1 = NAC = NBC = BCM. 2.46. A2 , B2 , C P , (A2 B2 , B2 P) = (A2 C, CP) = (BC, CP). (B2 P, B2 C2 ) = (AP, BA). (A2 B2 , B2 C2 )=(BC, CP)+(AP, AB)=(B1 B, B1 C1 )+(A1 B1 , B1 B)= = (A1 B1 , B1 C1 ). , A1 B1 C1 A2 B2 C2 180 ; , (. 5.48). 2.47. Q C PQ, Q CQ= = Q PQ = 30 . , BCQ = 60 . CBQ = 60 , , BQ C . CP D . 2.48. BAD = 2a CBA = 2b; , a b. CD E , DE = DA. CE = CD AD = CB. C BCE 180 2a, CBE = a. DAE = b. B CD F. FBA = b = AED, ABFE , , FAE = FBE = a b. , FAD = b + (a b) = a, . . AF A. 2.49. ED=CB, EN=CM DEC= = BCA = 30 (. 2.2), EDN = CBM. MBC = NDE = a, BMC = END = b. , DNC = 180 b. BNC, BNC = 90 a. a + b =180 30 =150 , DNB= = DNC + CNB = (180 b) + (90 a) = = 270 (a + b) = 120 . B, O, . 2.2 N D (O ) . CO = CB = CD, . . C , , l = CN : CE = CB : CA = 1 : 3. 2.50. D A1 BC AB1 C. (AC, CD)=(AB1 , B1 D) (DC, CB)=(DA1 , A1 B).

48

2.

(A1 C1 , C1 B1 ) = (AC, CB) = (AC, CD) + (DC, CB) = (AB1 , B1 D) + + (DA1 , A1 B) = (A1 D, DB1 ), . . A1 , B1 , C1 D . , (A1 C1 , C1 D) = (A1 B1 , B1 D) = (AC, CD). , A1 C1 AC, . 2.51. M A1 AC. 1.58 M B1 C1 . (LM, MA1 ) = = (C1 B1 , B1 B) = (C1 C, CB) = (LK, KA1 ), . . M A1 KL. , AC A1 M. 2.52. PQ , AB, Q C AB; L QO ; M N LB LA AC BC. , M = M N = N. PA + AB + B Q = 180 , B Q = A, , B LQ = M AO. , AM OL M OA = M LA = B/2. CM O = (A + B)/2, . . M = M. N = N. 2.53. ADM CBM ACM DBM, AD : CB = DM : BM AC : DB = AM : DM. . 2.54. D1 BD , B. AB = AD1 , ACB = AD1 B = ABD1 . ACB ABD A , , ACB = ABD, ACB ABD. , AB : AC = AD : AB. 2.55. O . MAC = ACO = CAO, AMC = ADC. CDB = CNB. ACD CDB, CD2 = AD DB = AM NB. 2.56. B1 H AB, (AB, BC) = (AB, BH) = (AB1 , B1 H) = (B1 C1 , B1 H). (AC, BC) = (B1 C1 , C1 H). 2.57. BP P D , PD = CP. CDP CD QP. BP : PQ = = BD : DC = (BP + CP) : CP, . . 2.58. QE A B 45 , ABEQ . ABE = 90 , AQE = 90 . , AQE AE/AQ = 2. AF/AP = 2. 2.59. ANC = ABC = CAB, CAM CNA, , CA : CM = = CN : CA, . . CM CN = AC2 , AM : NA = CM : CA. BM : NB = = CM : CB. AM BM/(AN BN) = CM2 /CA2 = CM2 /(CM CN) = CM/CN. 2.60. AK = AB = CD, AD = BC = CH KAD = DCH, ADK = = CHD DK = DH. , A, K, H, C D . ADC . CK1 AD AH1 DC. K1 A = DC H1 C = AD. , K1 = K H1 = H, . . K H KAH KDH , . , , KDH . 2.61. a) PBA1 = PAC1 PBC1 = PAB1 , PBA1 PAC1 , PAB1 PBC1 , . . PA1 : PB = PC1 : PA,1 1 1 = + . PQ CP BP

49

PB1 : PA = PC1 : PB. , PA1 PB1 = PC2 , 1 , PA1 : PB1 = PB2 : PA2 . ) ) OA = OB OC , OB = OA OC , OC = OA OB . , . 2.62. K, L, M N , E AB, BC, CD DA. K N AE, (EK, KN) = (EA, AN). (EL, LM) = (EC, CM) = (EA, AN), , (EK, KN) = (EL, LM). (EN, NK) = (EA, AK) = (EC, CB) = (EM, ML). , EKN ELM, , EK : EN = EL : EM, . . EN = EK EM/EL = ac/b. 2.63. H , M BC. A1 , B2 C2 MH, (B2 A1 , A1 C2 ) = = (B2 M, MC2 ) = (AC, AB). , (A1 B2 , B2 C2 ) = (A1 H, HC2 ) = = (BC, AB) (A1 C2 , C2 B2 ) = (BC, AC). 2.64. M , H ABC. A1 , B1 C1 M , MH. , (A1 B1 , B1 C1 ) = (AH, HC) = (BC, AB). , . 2.65. BM DN S2 L C1 . , DC1 CN AN. BN NA, , CNB = BND. CB BD . C1 M CL AN, , , MDC1 = CML. , CNM = MND. , MCN DMN, . . CN : MN = MN : DN. 2.66. Q QK1 QN1 KL NM, P PM1 PL1 NM KL. , . .QC2 QK1 QN1 = . KNC = MLC NKC = LMC, QN1 : PL1 = PC2 PL1 PM1 QC2 QK QN AQ QB (AC QC) (AC + QC) AC2 QC2 = = = = . PC2 PL PM PB AP (AC PC) (AC + PC) AC2 PC2 QC QK1 QN1 = = , PC PL1 PM1

= QN : PL QK1 : PM1 = QK : PM.

QC = PC. 2.67. ) CAM = CBM CB1 M = CA1 M, B1 AM = A1 BM AB1 M = BA1 M. ) M1 S CO, CM1 A1 B1 ; M2 S ABC; A2 B2 BC AC. , M1 = M2 . ) AB2 M2 BA2 M2 , B2 M2 : A2 M2 = AB2 : BA2 . CA1 = p b = = BA2 CB1 = AB2 , B2 M 1 sin B2 CM1 sin CA1 B1 CB1 AB2 = = = = . A2 M 1 sin A2 CM1 sin CB1 A1 CA1 BA2

A2 CB2 S X, B2 X : A2 X = k (. 7.14), M1 = M2 .

50

2.

2.68. O , M AB, H CH, D , A B, C. OD CH, DCH = MDC. , MCD = DCH = MDC = ODC = OCD, . . M = O AB . 2.69. a = A < B. C = 90 . CM ABC . ACM = A = a, MCB = 3a, , a + 3a = 90 , . . a = 22,5 . A = 22,5 , B = 67,5 , C = 90 . 2.70. D , AE . D BC. MD AH, A D MH. , E MH. 2.71. , (AQ, QP) = (AN, NP) = (PM, MB) = (QP, QB). Q , AB 2(AC, CB), QP AB . 2.72. P Q AD; BC F (F D, AB = AC). , (FC, CE) = (BA, AE) = (DA, AQ) = (DF, FQ), . . EC FQ. BE FP. , , , . 2.73. AOB = a COD = b. a/2 + b/2 = ADP + PAD = 90 . 2SAOB = R2 sin a 2SCOD = R2 sin b, R , SAOB = SCOD . SBOC = SAOD . 2.74. AOB = 2a COD = 2b. a + b = ADP + PAD = 90 . (AP2 + BP2 ) + (CP2 + DP2 ) = AB2 + CD2 = 4R2 (sin2 a + cos2 a) = 4R2 . BC2 + AD2 = 4R2 . 2.75. M AC, N BD. AM2 = AO2 OM2 , BN2 = =BO2 ON2 , AC2 +BD2 =4(R2 OM2 )+4(R2 ON2 )=8R2 4(OM2 +ON2 )= = 8R2 4OP2 , OM2 + ON2 = OP2 . 2.76. BLP BDC , . , BLP = BDC = BAP. , AK BL AL BK. AKLB . 2.77. D , DD AC. DD BD, BD , , D AB = D CB = 90 . SABCD = SABCD = (AD AB + BC CD )/2 = = (AB CD + BC AD)/2. 2.78. AE. BEA = BCP ABE = BPC = 90 , EAB = CBP. , EB CD, , EB = CD. EBA = 90 , O AB EB/2. 2.79. , P BC, BC H AD M (. 2.3). BDA = BCA = BPH = MPD. MDP MPD , . 2.3

51

MP APD. , APM = 90 MPD = 90 MDP = PAM, . . AM = PM = MD. 2.80. ABCD (. 1.2), . K L AB CD, M KP CD. 2.79 PM CD, , M P CD M KL. . 2.81. ) , A, B, C D , 180 , . f , A B, 180 AOB, y , C D, 180 COD. AOB + COD = 180 , f + y = 180 . . , f + y = 180 , AOB + + COD = 180 , . . AC BD. ) O . AKO + BMO = 90 , AKO = BOM AKO BOM. , AK BM = BO AO = r2 . 2.82. , A BC AB C P , C. (PA, PB) = (PA, PC) + (PC, PB) = (B A, B C) + (A C, A B) = (C A, C B), . . P ABC . , A BC AB C , . . P = C, : PC . 2.83. ) 2.82 AB1 C1 , A1 BC1 A1 B1 C, A1 B1 C1 , . ) P . , (AP, PC) . (AP, PC) = (AP, AB) + + (AB, BC) + (BC, PC), (AB, BC) , , (AP, AB) + (BC, PC) . , (AP, AB) + + (BC, CP) = (AP, AC1 ) + (CA1 , CP) = (B1 P, B1 C1 ) + (B1 A1 , B1 P) = = (B1 A1 , B1 C1 ), . , (AP, PB) (BP, PC) . , P . 2.84. 2.83 ) AB1 C1 , A1 BC1 A1 B1 C P. A1 B1 C1 PA1 . , BC. PB1 PC1 . 2.85. AB1 C1 X, BXC = 180 A. , X , ABC BC. , , , . 2.86. 2.83 ), A1 B1 C1 , ABC. H A1 B1 C1 , . .

52

2.

ABC. A1 B1 CH, H A1 B1 C. , A1 BC1 AB1 C1 . 2.87. ) X ABC AB C . (XB , XC) = (XB , XA) + (XA, XC) = (C B , C A) + + (BA, BC). AC = AP = AB , C AB , C AB = 2A, (C B , C A) = A 90 . , (XB , XC) = A 90 + B = 90 C = (A B , A C), . . X A B C. A BC . ) X A B C A BC. , ABC . , (XB, XC ) = (XB, XA ) + (XA , XC ) = (CB, CA ) + + (B A , B C ). A1 , B1 C1 PA , PB PC . (CB, CA ) = (CP, CA1 ) = (B1 P, B1 A1 ), (B A , B C ) = (B1 A1 , B1 C1 ) (AB, AC ) = (AP, AC1 ) = (B1 P, B1 C1 ). , (XB, XC ) = = (AB, AC ). , X AB C. ) QA O I, QA OI. QB OJ QC IJ. OJI B QC, OIJ A QC , , sin OJI = sin B QC sin OIJ = sin A QC. , OI : OJ = sin OJI : sin OIJ = sin B QC : A QC. , sin QJI sin(QJC/2) sin QB C QI = = = , QJ sin QIJ sin(QIC/2) sin QA C

C Q IJ, Q, B C J. , sin B QC : sin QB C = = B C : QC sin A QC : sin QA C = A C : QC, OI QI BC AC : = : = 1. OJ QJ QC QC

2.88. ) , . AB, AC, BC D, E, F (. 2.4). P ABC CEF, C. , P BDF. , (BP, PF)= =(BD, DF). , (BP, PF)=(BP, PC)+ + (PC, PF) = (BA, AC) + (EC, EF) = = (BD, AC) + (AC, DF) = (BD, DF). , P ADE. . 19.46. . 2.4

53

) . 2.4. ) ABC, ADE BDF P, ABP, ADP BDP. , , P (. 5.106). , , P. , , P. 2.89. ) P AB, BC, CA A1 B1 . PA, PB, PC P Sa , Sb , Sc (AB1 , B1 P) = (AC1 , C1 P), (BC1 , C1 P), (CA1 , A1 P). (AC1 , C1 P) = (BC1 , C1 P) = (CA1 , A1 P) = f, f P PA, PB PC Sa , Sb Sc , , f 90 POa , POb POc . , POa /PA = POb /PB = POc /PC = = 1/(2 sin f). , f 90 P 1/(2 sin f) ABC Oa Ob Oc . ) ) O ABC O Oa Ob Oc , H ABC H Oa Ob Oc . OO H OO H M. OH/OM = OH/O H = 2 sin f HOM = (HO, O H ) = f 90 , MH = MO, . . M OH. , OOa Ob Oc M : O Oa , Ob , Oc , M (. 13.35). 2.90. , AB DC E, BC AD F. P BCE CDF. CPE = ABC CPF = ADC. CPE + CPF = 180 , . . P EF. 2.91. ) (AP, PD) = (AP, PE) + (PE, PD) = (AC, CD) + + (AB, BD) = (AO, OD), A, P, D O . 6) , (EP, PO) = (EP, PA) + (PA, PO) = (DC, CA) + (DA, DO) = = 90 , , , . 2.92. . 2.4. P CA CB CE CF. , P ABC CEF (. 5.105 ). 2.93. A A BC. OAH = AOA /2 = ABA = |B C|. 2.94. AA , A C AC, BH A C. CH A B. , BA CH . 2.95. l , ; D m n. (AD, DB) = (m, AB) + (AB, n) = = (AC, l) + (l, CB) = (AC, CB), , D ABC. 2.96. ) O S, ABC. CBO = BCO,

54

2.

BCO = ABO. BO ABC, . . O ABC. , S, ABC, . ) , S BAC. D ABC. DB = DO = DC (. 2.4 ), . . D S. 2.97. C , : C A1 B, A2 B2 , AB1 . C = 90 , ACA1 A2 BCB1 B2 C C1 . (AC1 , A2 C1 ) = (A2 C1 , A1 C1 ) = (A1 C1 , C1 C) = = (C1 C, C1 B1 ) = (C1 B1 , C1 B2 ) = (C1 B2 , C1 B) = 45 ( 45 ; , ). (AC1 , C1 B1 ) = = 4 45 = 180 , . . AB1 C1 . A2 B2 A1 B C1 . 2.98. P O S1 S2 ; a = APC, b = BPC; AC BC S2 K L. OAP = OBP = 90 , AOB = 180 a b. , LOB = 180 2LBO = 2CBP = 180 b. KOA = 180 a. LOK = LOB + KOA AOB = 180 , . . KL . 2.99. M , P , Q R , M, P, Q R OA. C B OA, P Q C. : (CS, NS) = (Q Q, NQ) = (Q P , NP ) = (CP , NP ) , P , M R , (CR , P R ) = (MM , P M ) = = (MN, P N) = (CN, P N). , C, N, P , S R . S, R C , S = R .

3

1. , , . A, , ; B C , O . : ) AB = AC; ) BAO = CAO; ) OB AB. ( AB, AB. , ) : , , .) 2. l1 l2 , A, B1 , C1 B2 , C2 . AB1 AC1 = AB2 AC2 . , AB1 C2 AB2 C1 ( , : A A ). l2 , . . B2 = C2 , AB1 AC1 = AB2 . 2 , , . 3. , , . 4. , . , . 0 . 5. 6 , : . 5.51 7.42. 6. V . . . ( , ) . : , , . 1882 . , p , . . . , , , .

56

3.

-, 3.39 ( ): , , . , , .

1. , A, , , (. . A ) . 2. A B. X AB, AB. , , X , . 3. R r (. . ). . 4. a b , c . , : ) (a + b c)/2; ) , , (a + b + c)/2.

1. 3.1. PA PB O (A B ). , PA PB X Y. , XOY . 3.2. ABC BC K, L. , CK = BL = = (a + b c)/2, a, b, c . 3.3. AB ABC E, ACE ECB , CE M N. MN, AE BE. 3.4. ABCD , , BAD BC CD. , AB + BC = AD + DC. 3.5. R r A B C. , AC CB = Rr. 3.6*. AB CD. , ABCD , .

57

3.7*. ABCD. ABD AD AB M N. , MN BC CD BCD. 3.8*. ABCD , , . 3.1. , , ABCD . 3.9*. ; . 3.1 , , , . , , , , , . , , , . . 1.21 ), 1.61, 1.65, 1.67.

2. 3.10. P, AB , KM LN . , KLMN . 3.11. A B; MN . , AB MN . 3.12. OA A, BC OA. OB OC K L. , KL OA . 3.13. ABCD AC BD; M AC, BCDM . , BD ABM ADM. 3.14. S A B . l, A S M N, BMN. , , B. 3.15. S, A B C AB. S , AB C

58

3.

S P Q, M AB PQ. , M S . . 1.32, 2.29.

3. 3.16. A. () , C D. , CAD = 90 . 3.17. S1 S2 O1 O2 A. A , S1 A1 S2 A2 . , O1 A1 O2 A2 . 3.18. S1 , S2 S3 . , , S1 S2 , S3 , . 3.19. O1 O2 R O. OO1 O2 . 3.20. S1 S2 S A B, S1 S2 AB. , S1 S2 S. 3.21. S1 S2 , A, R r (R > r). , S2 B S1 , , AB = a. ( .) 3.22. AB C. , C, AC BC K L, AB M N. , KM = LN. . 3.2 3.23. S1 , S2 , S3 S4 , Si Si+1 i = 1, 2, 3, 4 (S5 = S1 ). , . 3.24*. ) A, B, C, l, , . 3.2. b c A, B, C. , a, 1/ c = 1/ a + 1/ b.

59

) ( ). a, b, c, d , a = 1/a, b = 1/b, g = 1/c d = 1/d. , 2(a2 + b2 + g2 + d2 ) = (a + b + g + d)2 .

4. 3.25. R H; A, B C , H. , : ) H ABC; ) ABC R.

. 3.3 3.26*. , . 3.3, . , AB1 + BC1 CA1 = 180 , . 3.27*. P; A, B Q . Q C D. ABQ CDP , ABCD (. 3.4). , ABCD . 3.4 .

5. , 3.28. A AB AC O. , M AO 90 , OB OC . 3.29. A AB AC O. X BC KL,

60

3.

XO ( K L AB AC). , KX = XL. 3.30. KL O A, AP AQ; M PQ. , MKO = MLO. 3.31*. A AB AC , D E; M BC. , BM2 = DM ME DME DBE DCE; , BEM = DEC. 3.32*. ABCD , B D K, AC. ) , AB CD = BC AD. ) , KB, BA, BD BC P, Q R. , PQ = QR. * * * 3.33. S l, . P, l, PA PB S. , AB . P S, PA PB , AB P S.

3.34*. S1 S2 A B, O S1 S2 . , O, AB P, S2 C. , P C S1 .

6. 3.35. C D AB. AC BD, AD BC P Q. , AB PQ. 3.36*. PC PD AB (C D ). , , P AC BD, AB. 3.37*. AB C, AB. ( ) C AB, : ) C ; ) C . 3.38*. Oa , Ob Oc PBC, PCA PAB. , Oa Ob PA PB, Oc PC.

61

7. 3.39. , , . 3.5. , . 3.40*. , . . , , . , . 3.5 , (. 3.6). 3.41*. OA, OB OC ( B AOC) . , , O, () ABC.

. 3.6

. 3.7

3.42*. ABC . (. 3.7). , , ABC.

8. , 3.43. AB S . S1 AB M N.

62

3.

, : ) MN P ; ) PQ S1 PA. 3.44. D S DC AB. S1 CA E, CD S. , DE ADC. 3.45*. , AB , M N. , MN C AB. 3.46*. AB S K , S L. SA SB S, LK AB, , SA AK A1 , SB BK B1 . , A1 LB1 = 45 . 3.47*. , AC BC ABC M N, ( ). , MN ABC. 3.48*. ABC1 ABC2 S, AC2 BC1 . S1 AC2 M2 , BC1 N1 S. , ABC1 ABC2 M2 N1 . 3.49*. BC ABC D. S1 BD DA , S2 CD DA . I, I1 , I2 r, r1 , r2 S1 , S2 ; f = ADB. , I I1 I2 , I1 I : I2 I = tg2 r = r1 cos2

f + r sin2 f (). 2 2 2

f . , 2

3.50*. ABCD . ra , rb , rc , rd BCD, ACD, ABD, ABC. , ra + rc = rb + rd . . 5.102, 6.104, 19.15, 28.23, 28.26.

9. 3.51. R1 R2 , d. , , d2 = R2 + R2 . 1 2

63

3.52. A, B C. , ABC . 3.53. O1 O2 A B. A , M1 , M2 . , BO1 M1 = = BO2 M2 .

10. 3.54. S P. , P, A B. , PA PB . , P P , P S.

3.55. , P, S, S , . 3.56. , P S d2 R2 , R S, d P S. 3.57. f(x, y) = 0, f(x, y) = x2 + + y2 + ax + by + c. , (x0 , y0 ) f(x0 , y0 ). 3.58*. S1 S2 . , , S1 S2 , . S1 S2 .

3.59*. , . 3.60*. , . . , . .

3.61*. . . , . 3.62*. S1 S2 .

64

3.

3.63*. S1 S2 . , , , , ( ) . 3.64*. ) , . ) . , . 3.65*. S AB C, C CH AB. , S S1 C CH CH . 3.66*. BC AC ABC A1 B1 ; l , AA1 BB1 . , : ) l H ABC; ) l C, AB1 : AC = BA1 : BC. 3.67*. AB CD ABCD F, BC AD E. , AC, BD EF , ABE, CDE, ADF BCF. 3.68*. A1 A2 , B1 B2 , C1 C2 . , A1 B2 B1 C2 C1 A2 = A2 B1 B2 C1 C2 A1 . 3.69*. BC ABC A . A B AB M, A C AC N. , , A MN, ABC. 3.70*. 1.67, . 3.71*. . , , . 3.72*. ) ABC AA1 , BB1 CC1 . AB A1 B1 , BC B1 C1 , CA C1 A1 C , A B . , A , B C . ) ABC A , B C . , A , B C , -

65

, ABC. 3.73*. , AD, BE CF ABCDEF (). 3.74*. S1 , S2 , S3 S4 , Si Si+1 i = 1, 2, 3, 4 (S5 = S1 ). , S1 S3 S2 S4 . 3.75*. ) S1 S2 A B. P S1 S2 p, P AB h, d. , |p| = 2dh. ) A B BCD ACD pa pb . , |pa |SBCD = = |pb |SACD . . 3.763.82, 8.90, 14.56 ), 28.6.

11. , : . , , , , .

3.76*. ) , . ) , . 3.77*. f(x, y) = x2 + y2 + a1 x + b1 y + c1 g(x, y) = x2 + y2 + + a2 x + b2 y + c2 . , l = 1 f lg = 0 , f = 0 g = 0. 3.78*. , ( ), ( ), ( ). (. . ).

3.79*. , , , . 3.80*. , , .

66

3.

3.81*. , , , . .

3.82*. , , .

3.83. , R, . ? 3.84. A, R, AB AC, B C . 2 BC = a. , 4R2 = r2 + ra + a2 /2, r ra ABC. 3.85. . , , A D, B C. , AB : BC : CD = 2 : 4 : 3. 3.86. R, 1 < R < 2, 2. , ? 3.87. AB C AB, AC BC ( AB). , , , . 3.88. BC ABC A1 A2 , AC B1 B2 , AB C1 C2 . , AC1 BA1 CB1 = C1 B A1 C B1 A AC2 BA2 CB2 C2 B A2 C B2 A1

.

3.89. A AB AC; PQ ; l Q. PA, PB PC l A1 , B1 C1 . , A1 B1 = A1 C1 .

3.1. XY Z. XOA XOZ , XOA = = XOZ. ZOY = BOY. , XOY = XOZ + ZOY = = (AOZ + ZOB)/2 = AOB/2.

67

3.2. M N AB BC. BK + AN = BM + AM = AB, CK + CN = a + b c. P Q AB BC. AP = AB + BP = AB + BL AQ = AC + CQ = AC + CL. AP + AQ = a + b + c. , BL = BP = AP AB = (a + b c)/2. 3.3. 3.2 CM = (AC + CE AE)/2 CN = (BC + CE BE)/2. , AC = BC, MN = |CM CN| = |AE BE|/2. 3.4. AB, BC, CD DA P, Q, R S. CQ = CR = x, BP = BC + CQ = BC + x DS = DC + CR = = DC + x. , AP = AB + BP = AB + BC + x AS = AD + DS = = AD + DC + x. , AP = AS, . 3.5. AB O1 O2 C D. O1 AO2 = 90 , AO1 C O2 AD . O1 C : AC = AD : DO2 . , AD = CB (. 3.2). , AC CB = Rr.

. 3.8

. 3.9

3.6. AB CD O. , A D , B C , OB < OA (. 3.8). M A D ABCD , AOD, N B C , BOC (. 2.96 ). ABCD , M N . 3.7. R BD, P Q MN BC CD (. 3.9). DMQ = BPN, DQM = BNP DMQ = BNP, MDQ, PBN PCQ . CP = CQ, DQ = DM = DR BP = BN = BR. , P, Q R BCD (. 5.1).

68

3.

3.8. , . 3.10. .

. 3.10 (ST ). . ; (. . ST QR). AB + CD = BC + AD AP = AQ. 3.9. ABCD . . . YZ , tA , tB , . . . , tZ , . A B tA tB . (tA tB ) + (tB tC ) + . . . + (tY tZ ) + (tZ tA ) = 0. 3.10. O ABCD. ABCD , AOB DOC, . . OA OC = OB OD. ALBN AMBK , PL PN = PA PB = PM PK. KLMN . 3.11. O AB MN. OM2 = = OA OB = ON2 , . . OM = ON. 3.12. OA BC ; M KL OA. LOM = LCB = OKM, , KOM OLM. , OM : KM = LM : OM, . . OM2 = = KM LM. , MA2 = MK ML. MA = OM.

69

3.13. O AC BD. MO OC = = BO OD. OC = OA BO = OD, MO OA = BO2 MO OA = DO2 . , OB ABM OD ADM. 3.14. C AB BMN, B; AP S. AB AC = AM AN = AP2 , , AC = AP2 /AB, . . C l. . , , A S, AB. 3.15. , MC2 = MP MQ = MA MB, M AB, AC > BC, BA, AC < BC. M AB. (MB + BC)2 = (MB + BA) MB. , MB = BC2 /(AB 2BC), , M S . 3.16. M CD A. MC = MA = MD. A CD. 3.17. O1 , A O2 , A2 AO2 = A1 AO1 . AO2 A2 AO1 A1 , A2 AO2 = AA2 O2 A1 AO1 = AA1 O1 . , AA2 O2 = AA1 O1 , . . O1 A1 O2 A2 . 3.18. O1 , O2 O3 S1 , S2 S3 ; A, B, C S2 S3 , S3 S1 , S1 S2 ; A1 B1 CA CB S3 . B1 O3 CO1 A1 O3 CO2 . O1 , C O2 , A1 , O3 B1 , . . A1 B1 S3 . 3.19. A1 , A2 B O O1 , O O2 , O1 O2 ,. O1 O2 = O1 B + BO2 = O1 A1 + O2 A2 . OO1 + OO2 + + O1 O2 = (OO1 + O1 A1 ) + (OO2 + O2 A2 ) = OA1 + OA2 = 2R. 3.20. O, O1 O2 S, S1 S2 ; C S1 S2 , AB. AOB, AO1 C CO2 B , OO1 CO2 OO1 = O2 C = O2 B, , AO = AO1 + O1 O = AO1 + O2 B. 3.21. O1 O2 S1 S2 ; X AB S2 . BA BX. 3.17 BO1 XO2 , AB : BX = O1 A : O1 O2 AB BX = AB2 O1 O2 /R = a2 (R r)/R, . 3.22. O, O1 O2 AB, AC BC. , KO = OL. , O1 KO = O2 OL. , O1 K = AC/2 = O2 O, O1 O = BC/2 = O2 L KO1 O = OO2 L = 180 2a, a KL AB. 3.23. Oi Si , Ai Si Si+1 . O1 O2 O3 O4 ; a1 , a2 , a3 a4 . , Ai1 Ai Ai+1 = (ai + ai+1 )/2, A1 + A3 = (a1 + a2 + a3 + a4 )/2 = A2 + A4 . 3.24. ) A1 , B1 C1 A, B C l; C2 C AA1 . CC2 = AC2 AC2 , 2 2

70

3. . . A1 C2 = (a + c)2 (a c)2 = 4ac. 1 B1 C2 = 4bc A1 B2 = 4ab. A11 + C 1 1 + C11 = A1 B1 , ac + bc = ab, . . 1/ b + B + 1/ a = 1/ c. ) A, B, C , D (. 3.11). BDC b + c + d, cos2 sin2d(b + c + d) BDC = , 2 (b + d)(c + d) BDC bc = 2 (b + d)(c + d)

(. 12.13). a + b + g = 180 , sin a + sin b sin g + 2 sin b sin g cos a = 0 ( ). a = BDC/2, b = ADC/2 g = ADB/2, 2 2 2

. 3.11

a bcd(b + c + d) bc ac ab +2 = 0, (b + d)(c + d) (a + d)(c + d) (a + d)(b + d) (a + d)(b + d)(c + d)

. .a+d b+d c+d +2 a b c d(b + c + d) = 0. bc

d,

a b g d + 2 bg + gd + db = 0.

(a + b + g + d)2 = (a b g d)2 + 4(ab + ag + ad) =

= 4(bg + gd + db) + 4(ab + ag + ad) = 2(a + b + g + d)2 2(a2 + b2 + g2 + d2 ),

. . 2(a2 + b2 + g2 + d2 ) = (a + b + g + d)2 . 3.25. A1 , B1 C1 (. 3.12). A1 BC1 H , , BA1 HC1 . B1 A HC1 , B1 A BA1 B1 ABA1 . ) A1 B1 CH A1 B1 AB, AB CH. , BC AH CA BH. ) , , B1 A BA1 , , B1 C BC1 A1 C AC1 ; , R. BA1 C BA1 CO. AB1 CO . AO = BO = CO = R, . . O ABC, R. 3.26. , AB1 B1 A1 = . 3.12 = AC1 + C1 A1 , BC1 + C1 B1 = BA1 B1 A1 C1 A1 CA1 = C1 B1 B1 C, . , AB1 + + BC1 CA1 = AC1 + BA1 CB1 . ,

71

ABC BA1 CA1 , AB1 CB1 BC1 + AC1 , 360 . 3.27. AP + BP + PQ = 180 (. 3.26), AB = 180 PQ. CD = 180 PQ, . . AB = CD, , AB = CD. , PQ AB PQ CD (. 3.25), AB CD. 3.28. M, B C AO. , OB OC . 3.29. B X KO, XKO = XBO. XLO = XCO. XBO = XCO, KOL , OX . 3.30. , AK AL = AM AO. , K, L, M O , MKO = MLO. AOP APM, AM AO = AP2 ; , AK AL = AP2 . 3.31. , DBE . O ; D E D E AO. 28.7 ED E D M. BDM = EBM BEM = DBM, , BDM EBM. , BM : DM = EM : BM. , DME = DE = 2DBE. BEM = DBM , BEM = DBC = DEC. 3.32. ) KAB KBC, AB : BC = KB : KC. AD : DC = = KD : KC. , KB = KD, . ) , PQ sin PBQ sin ABD sin ABD AD = = = = , BQ sin BPQ sin KBA sin ADB AB QR CD = . BQ CB

3.33. O S OM l. , X, AB OM, . A, B M PO. AMO = ABO = BAO, , AMO XAO, O . , AO : MO = XO : AO, . . OX = OA2 /MO . 3.34. OBP = OAB = OCB, OBP OCB, , OB2 = = OP OC. C CD S1 . OD2 = OB2 = OP OC. , ODC OPD OPD = ODC = 90 . 3.35. BC AD APB, PQ, Q , AB. 3.36. AC BD, BC AD K K1 . KK1 AB, , C D KK1 . , C KK1 . M C KK1 . ABC CKK1 , . CAB = CK1 K. , KCM = ABC, CMK . CMK1 KM = CM = K1 M, . . M KK1 . , D KK1 .

72

3.

3.37. ) AC A A1 , BC B B1 . A = A1 ( B = B1 ), AC ( BC) . , AB1 BA1 ABC , C AB1 BA1 . ) C1 , , AB. D E. P DC AB, F PE . AB C F. CF . 3.38. PA Ob Oc , PA Oa , POa Oa Ob Oc . B C. , P Oa Ob Oc , , POc Oa Ob . 3.39. 2a 2b , 2c . pa2 + pb2 + SABC pc2 . , p(a2 + b2 c2 ) = 0. 3.40. , (. 3.13). . 3.13 , , 90 ; S s . , S = 4s. , S 2s = 2s, 2s. 3.41. , OB OC, OA OC, OA OB, A1 , B1 , C1 (. 3.14). OA1 B = OA1 C = 90 , B, A1 C , , BA1 = A1 C.

. 3.14

. 3.15

73

A1 , B1 , C1 ABC, BA1 = C1 B1 BC1 = A1 B1 . , BA1 C1 B1 , C1 B B1 A1 . A1 B1 C1 A1 B1 C1 B, . . ABC. 3.42. , , . x, y, z u ; a, b, c, d, e f , ; p, q r , (. 3.15). x + (a + b) = u + p + q + (c + f), y + (c + d) = u + q + r + (e + b) z + + (e + f) = u + r + p + (a + d). , x + y + z = = 2(p + q + r + u) + u. 3.43. ) O O1 S S1 . MO1 N PON , MO1 N = PON. , P, M N . ) , PQ2 = PM PN = PM (PM + MN). K AB. PM2 = PK2 + MK2 PM MN = AM MB = AK2 MK2 . PQ2 = PK2 + AK2 = PA2 . 3.44. 3.43 ) BE = BD. DAE + ADE = DEB = = BDE = BDC + CDE. DAB = BDC, ADE = CDE. 3.45. O1 O2 , CP CQ . CO2 = CP2 + PO2 = CP2 + O1 M2 , CQ = CA = CP 1 1 ( 3.43 ), CO2 = CQ2 + QO2 = CP2 + O2 M2 . , CO2 CO2 = 2 2 1 2 2 2 = MO1 MO2 , , CM O1 O2 (. 7.6). MN C. . , , ; MN . 3.46. LAB = a LBA = b (a + b = 90 ). 3.43 ) AB1 = AL, AB1 L = 90 a/2. BA1 L = 90 b/2. , A1 LB1 = (a + b)/2 = 45 . 3.47. A1 B1 BC AC; O . A1 B1 CO (. 2.20 ) MN CO, , MN A1 B1 . M N CA CB , M N A1 B1 . M N L, B1 M A1 N , ABC. , MN O, L (. 2.52). 3.48. . , O1 ABC1 M2 N1 . A1 A2 BC1 BC2 , B1 B2 AC1 AC2 ; PQ S, AB, Q C1 AB. O1 AA1 BB1 , L S S1 3.43 )

74

3.

A1 N1 B2 M2 (. 3.16). C1 AB = 2a, C1 BA = 2b, C1 AC2 = = 2f. A1 A2 = 2f = B1 B2 , . . A1 B2 B1 A2 . A1 B2 BC1 (B2 C1 + A1 B)/2 = b f + a. , BC1 AC2 (C1 C2 + AB)/2 = 2f + 180 2a 2b, M2 N1 BC1 AC2 a + b f, , M2 N1 A1 B2 . , M2 N1 AC2 BC1 , M2 N1 A1 B2 . A1 N1 B2 M2 L . L S M2 N1 . AB L1 , . 3.16 M2 , N1 AC2 L1 B2 , BC1 L1 A1 , M2 N1 A1 B2 O1 M2 N1 . Q1 S, 2(PQ, PQ1 ) = (PC2 , PC1 ), L1 Q1 O1 S. , L1 . B1 Q = 2a, B2 Q1 = 2(a 2f) = C2 A1 . AM2 O1 L1 , , M2 O1 A = M2 L1 A = B2 A1 A, . . M2 O1 B2 A1 . N1 O1 B2 A1 . 3.49. E1 E2 , I1 I2 BC. 3.48 I , E1 AD S1 , , E2 AD S2 . F1 E1 I1 E2 I, F2 E2 I2 E1 I. , DI1 E1 I, DI2 E2 I DI1 DI2 . I1 D F1 E2 I2 D F2 E1 . , E1 I1 : I1 F1 = E1 D : DE2 = F2 I2 : I2 E2 . , I I1 I2 , I1 I : I2 I = E1 F1 : E2 F2 = E1 E2 tg

f2

: E1 E2 ctg

f2

= tg2

f2

.

E I BC. r = IE. 1.1 )I1 E1 ctg

f2

IE =tg

f2

+ I2 E2 tg

f

+ ctg

f2

2 = r cos2 1

f2

+ r2 sin2

f2

.

3.50. f = AOB, O AC BD. , , rab , rbc , rcd , rad , ABCD CO DO, DO AO, AO BO,

BO CO. ( 3.49) 2ra = rad sin2 rc = rbc sin2

75

f2

+ rab cos2 + rcd cos2

f2

, ,

rb = rab cos2 rd = rcd cos2

f f2 2

+ rbc sin2 + rad sin2

f2

, .

f2

f2

f2

ra + rc = (rad + rbc ) sin2 + (rab + rcd ) cos2 = rb + rd . 2 2 3.51. O1 O2 A. O1 A O2 A A, , O1 AO2 = 90 , . . O1 O2 = O1 A2 + O2 A2 . 2 3.52. A1 , B1 C1 , A, B C B1 C1 , C1 A1 A1 B1 . A1 B = A1 C, B1 A = B1 C C1 A = C1 B, A, B C A1 B1 C1 (. 5.1). , A1 B, B1 C C1 A ABC. # # 3.53. , Oi B Oi Mi ( ) 2(AB, AMi ). , (AB, AM1 ) = = (AB, AM2 ). 3.54. P , A1 B1 . PAA1 PB1 B, PA : PA1 = PB1 : PB. 3.55. P PC. PAC PCB, PA : PC = PC : PB. 3.56. , P , A B. PA = d + R PB = |d R|. PA PB = |d2 R2 |. , d2 R2 P S . 3.57. a = a/2, b = b/2 R = a2 + b2 c. f(x, y) = (x a)2 + + (y b2 ) R2 , . . (a, b) S, R . , (x0 , y0 ) S (x a)2 + (y b2 ). 3.56 (x0 , y0 ) S f(x0 , y0 ). 3.58. R1 R2 . , (a, 0) (a, 0). 3.56 (x, y) (x + a)2 + y2 R2 (x a)2 + y2 R2 . 1 2 , x = (R2 R2 )/4a. 1 2 , , . 3.59. , . , . 3.60. , . , . 3.61. 3.59 , , . 3.60

f

f

76

3.

, . . 3.62. S, . S1 S S2 S. S1 , S2 S. - . . 3.63. O1 O2 , r1 r2 . S r O Si , r2 = OO2 ri2 , . . S i O Si . , . 3.64. ) . ) ABCD AB. AD BC , O AC . AC A1 C1 , OA1 OA = OC1 OC, , OA1 = OC1 AA1 = CC1 . 3.65. M CH. , M S S1 , . . . S S1 2R 2r. M S1 CM2 4r2 = 3r2 , S OM2 4R2 , O AB. , OH2 = 4R2 4r2 , OM2 = OH2 + HM2 = 4R2 4r2 + r2 = = 4R2 3r2 . , OM2 4R2 = 3r2 . 3.66. ) SA SB AA1 BB1 ; S AB. S SA , S SB AHa BHb , ( ) H. 3.61 SA SB AHa BHb . ) SA SB A1 Ha B1 Hb (. . C) , CB1 CHb = CA1 CHa ( ). CHb = (a2 + b2 c2 )/2b CHa = (a2 + b2 c2 )/2a, CB1 /b = CA1 /a. 3.67. CDE CC1 DD1 ; H . AC BD C1 D1 , H CD ( C1 D1 ). , H AC, BD EF , . . H. . . (. 4.56), .

77

3.68. A1 A2 , B1 B2 C1 C2 O (. 3.61). A1 OB2 B1 OA2 , A1 B2 : A2 B1 = OA1 : OB1 . B1 C2 : B2 C1 = OB1 : OC1 C1 A2 : C2 A1 = = OC1 : OA1 . , . 3.69. B C A M A N , A BC (. 3.17). A BM A NC , ABC = A B C . AM BM = = A M B M, M S S , ABC A B C , . . 3.17 N, MN S S . S S , . A , S , MN , S. 3.70. AC BD ; E K AC BD, AB CD; O1 O2 (. 3.18).

. 3.18 AB O1 E, O1 E O2 E O2 E CD, AB CD, , K S1 S2 AC BD. K S1 S2 ; , O1 O2 . O1 A O1 B S1 S2 , O1 . O2 . 3.71. S1 , . . . , Sn . Si Mi , X, Si S1 , . . . , Sn . Mi . , Mij X,

78

3.

Si Sj . Mij , , Si Si Sj . Mi Mij , . , Mij Si , Mi Si . - S1 , . . . , Sn , Mi . Mi , , . 3.72. ) B1 C1 BC, A A1 B1 C1 ABC A . , A ABC. B C . ) A1 B1 C1 , ABC ( A1 B1 C1 ). ) A , B C ABC A1 B1 C1 . , , . . A1 B1 C1 . , A1 B1 C1 ABC (. 1.57 ). 3.73. ABCDEF R, Q, T, S, P, U ( R AB, Q BC . .). a > 0 BC EF Q P , # # QQ = PP = a, QQ PP # # CB EF. R , S , T , U (. 3.19; RR = SS = TT = = UU = a). S1 , BC EF Q P . S2 S3 . . 3.19 , B E S1 S2 . BQ = QQ BQ = RR BR = BR ( QQ < BQ, BQ = BQ QQ = BR RR = BR ) EP = EP + PP = ES + SS = ES . , FC AD S1 S3 , S2 S3 . , AD, BE CF . 3.74. Ai Si Si+1 , X A1 A4 A2 A3 . X S2 S4 (. 5.73). A1 A2 A3 A4 ( 3.23), XA1 XA4 = XA2 XA3 , , X S1 S3 . 3.75. ) O , , Ox . P (x, y); R r S1 S2 ;

79

a = d/2. (x + a)2 + y2 = R2 p = (x a)2 + y2 r2 = ((x + a)2 + y2 R2 ) 4ax r2 + R2 = R2 r2 4ax. A (x0 , y0 ). (x0 + a)2 + y2 R2 = 0 = (x0 a)2 + y2 r2 , . . x0 = (R2 r2 )/4a. 2dh = 4a|x0 x| = 0 = |R2 r2 4ax| = |p|. ) d ACD BCD; ha hb A B CD. ) |pa | = 2dha |pb | = 2dhb . , SBCD = hb CD/2 SACD = ha CD/2, . 3.76. , ) ). S O R l. S1 O1 R1 S l , O1 , l O, , A l AO2 R2 = AO1 R2 , . . R2 = AO2 AO2 + R2 1 1 1 ( , A l). , , , , . , AO2 R2 = AO1 R2 1 AO2 R2 = AO2 R2 , AO2 R2 = AO2 R2 , l 2 1 1 2 S1 S2 . 3.77. l = 1, 1 a la2 b l b2 c lc2 (f lg) = x2 + y2 + 1 x+ 1 y+ 1 . 1l 1l 1l 1l 1 1 (f lg) = (f 1l 1m

3.57 f lg =0 f mg = = 0

mg).

l = m,

f = g. , f = 0 g = 0. 3.78. 3.76 , , . 3.79. ; . ; . . A m, . , , k A . k > 0. O m , AO2 = k. . 3.80. S O R, S1 O1 R1 . , OO2 = R2 + R2 . O S1 1 1 OO2 R2 , S S1 , 1 1 O S1 R2 .

80

3.

, S S1 S2 . O S1 S2 R2 . O . O , S1 S2 , R2 . 3.81. S O R . , 3.80, O , S, R2 . , , . 3.82. O , 0, . . O . , , , .

4

1. S ABC : ) S = aha /2, a = BC, ha , BC; ) S = ) S = pr, p , r . , O , S = SABO + SAOC + SOBC =1 (c + b + a)r = pr. 2 1 bc sin A, b, c , A ; 2

2. , . 3. , , .

1. , . 2. E F BC AD ABCD. , AE, ED, BF FC, , ABCD S. 3. r. , pr, p . 4. X ABCD. , SABX + SCDX = SBCX + SADX . 5. A1 , B1 , C1 D1 CD, DA, AB, BC ABCD, S. , AA1 , BB1 , CC1 DD1 .1 d d sin f, d1 d2 , 2 1 2

f

1. 4.1. , . 4.2. ABC. P, ABP, BCP ACP .

82

4.

4.3. ABC O, BOL, COM AON ( L, M N AB, BC CA, OL BC, OM AC ON AB; . 4.1). 4.4. ABC # # 1 , #B1 # C1 , A # # AB1 = 2AB, BC1 = 2BC CA1 = 2AC. A1 B1 C1 , , ABC S. . 4.1 4.5. DA, AB, BC, CD ABCD # # # # # # # A1 B1 , C1 , D1 , DA1 = 2DA, AB1 = 2AB, BC1 = 2BC # , CD1 = 2CD. A1 B1 C1 D1 , , ABCD S. 4.6*. ABCDEF . AD, BE CF . , ABCDEF ACE. 4.7*. ABCD O, OAB, OBC, OCD ODA . , .

2. 4.8. , , 4. , , 5. 4.9. ABCDE . ABCDE. 4.10. ABCD , K AB. , ABCD. 4.11*. ABC E . 4.2 BC, D AC, AC = 1, BAC = 60 , ABC = = 100 , ACB = 20 DEC = 80 (. 4.2). ABC CDE?

4.12*. Ta = A1 A2 A3 Tb = B1 B2 B3 , Tb Tc = C1 C2 C3 , Ta Tc . Tb Ta Tc . 4.13*. ABC A1 , B1 C1 , BA1 : A1 C = p, CB1 : B1 A = q AC1 : C1 B = r. AA1 , BB1 CC1 , . 4.3. PQR ABC.

83

. 4.3

3. , 4.14. ABCD O. , SAOB = SCOD , BC AD. 4.15. ) ABCD P. ABP, BCP, CDP. ADP. ) , . , . 4.16*. ABCD P, S2 + S2 = S2 + S2 . , P ABP CDP BCP ADP . 4.17*. ABCD P1 , P2 , P3 , , ABPi CDPi BCPi ADPi i = 1, 2, 3. , ABCD .

4. , 4.18. K, L, M N AB, BC, CD DA ABCD; KM LN O. , SAKON + SCLOM = SBKOL + SDNOM . 4.19. K, L, M N AB, BC, CD DA ABCD, KM LN . O. , KBLO MDNO , O AC.

84

4.

4.20. AB CD ABCD M N , AM : MB = CN : ND. AN DM K, BN CM L. , SKMLN = SADK + SBCL . 4.21. AB ABCD A1 B1 , CD C1 D1 , AA1 = BB1 = pAB CC1 = = DD1 = pCD, p < 0, 5. , SA1 B1 C1 D1 /SABCD = 1 2p. 4.22*. (. 4.4). , () 25 . 4.23*. . . , . 4.24*. K M AB CD ABCD, . 4.4 L N BC AD , KLMN . , ABCD KLMN. 4.25*. , . , , .

5. 4.26. ABCD M. , SACM = |SABM SADM |. 4.27. AB BC ABC ; P , AB BC. AC , BP. , . 4.28*. O, , . . , .

85

4.29*. AD BC ABCD O; M N AB CD, P Q AC BD. , : ) SPMQN = |SABD SACD |/2; ) SOPQ = SABCD /4. 4.30*. AB CD ABCD E F. K, L, M N DE, BF, CE AF. , KLMN E F. 4.31*. AC, BD, CE, . . . ABCDEF . , . 4.32*. PQ RS A. C , B , BC PQ BC = RA. A B AK BL CQ. , SACK = SBCL . 4.33*. ABCD O; P Q . , S S SAOP = ACP ABD . SBOQ SBDQ SABC

* * * 4.34*. O, ABC, , . AA1 , BB1 CC1 ABC . 4.5 (. 4.5). , , A, B C, . 4.35*. A ABC A1 , AA1 = p a = (b + c a)/2, A1 la , . lb lc , ABC , . , . . 3.393.42, 13.5513.59, 16.5, 24.7.

6. , 4.36. MN, CD ABCD, ( M N BC AD). , A B

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4.

CD BC AD, a b. , MN2 = (ab + c2 )/2, c = CD. 4.37. . , , , , , 1/4 . 4.38*. l . , , l, , 1 + 2. 4.39*. , . 4.40*. ) , , , . ) . 4.41*. A B S1 S2 , , S1 , . , S2 , A B, S1 . 4.42*. . , A B , AB O . . 2.73, 6.55, 6.56, 16.8, 18.33.

7. 4.43. ABCD P. A, B P CD a, b p. , ABCD ab CD/2p. 4.44. ABCD R, f . , S ABCD 2R2 sin A sin B sin f. 4.45*. , , , tg f |a2 + c2 b2 d2 |/4, a, b, c d , f . 4.46*. ) , ABCD S2 = (p a)(p b)(p c)(p d) abcd cos2B+D , 2

p , a, b, c, d . ) , ABCD , S2 = = (p a)(p b)(p c)(p d).

87

) , ABCD , S2 = = abcd sin2 ((B + D)/2). . 11.34.

8. 4.47. , , , ( ). 4.48. , AD ABC 4.49. ABC O; AO, BO CO A1 , B1 C1 . , : ) 4.50. (2n 1)- A1 . . . A2n1 O. Ak O An+k1 An+k Bk . , An+k1 Bk /An+k Bk (k = 1, . . . , n) 1. 4.51. A1 A2 . . . An . A1 A2 B1 D2 , A2 A3 B2 D3 . . , A1 B1 C1 D1 , . . . , An Bn Cn Dn , A1 C1 , . . . , An Cn O. , A1 B1 A2 B2 . . . An Bn = A1 D1 A2 D2 . . . An Dn . 4.52. . , . 4.53. X BC ABC AB AC db dc . , db /dc = BX AC/(CX AB). 4.54*. , r, ( ). , r. 4.55*. M, ABCD, PR QS, BC AB ( P, Q, R S AB, BC, CD DA ). , BS, PD MC . 4.56*. , , , , , ( ). 4.57*. BC DC ABCD D1 B1 , BD1 = DB1 . BB1 DD1 Q. , AQ BAD. 4.58*. ABC BB1 CC1 AB AC K L , OA1 OB1 OC1 AC1 BA1 CB1 + + = 1; ) = 1. AA1 BB1 CC1 C1 B A1 C B1 A 2bc a cos . b+c 2

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4.

AK = BC1 AL = CB1 . , AO, O ABC, KL . 4.59*. AA1 CC1 ABC M. , A1 BC1 M , AB = BC. 4.60*. ABC O. O BC, CA, AB da , db , dc , O A, B, C Ra , Rb , Rc . , : ) aRa cdc + bdb ; ) da Ra + db Rb + dc Rc 2(da db + db dc + dc da ); ) Ra + Rb + Rc 2(da + db + dc ) (); ) Ra Rb Rc (R/2r)(da + db )(db + dc )(dc + da ). . 1.60, 5.5, 5.34, 6.5, 6.31, 6.38, 6.40, 6.83, 9.26, 10.6, 10.52, 10.99, 11.21, 12.35, 22.49.

9. 4.61. , . 4.62. . , .

. 4.6 4.63*. AB CD ABCD 1 n , AD BC m . ) , . 4.6, . ) , . 4.6, . ? 4.64*. ) (. 4.7). , 12 .

) , , 1, 3. . 2.77, 3.41, 4.35, 9.44.

89

4.65. ABCD AB BC = = AD DC. , ABC ADC . . 4.7 4.66. , , , ( ) ? 4.67. , , , . 4.68. , , , , . 4.69. S, P. , , ( ) h. , . 4.70. AB ABC D E , ACD = DCE = ECB = f. CD : CE, AC BC f. 4.71. AA1 , BB1 CC1 ABC. , SA1 B1 C1 /SABC = 2abc/((a + b) (b + c) (c + a)). 4.72. M N AB CD ABCD. , AN NB + CM MD, AB = CD = BC + AD. 4.73. R, , . : d1 , d2 d3 . , d1 d2 d3 /4R. 4.74. AB, BC CA ABC C1 , A1 B1 ; C2 , A2 B2 . , SA1 B1 C1 = SA2 B2 C2 . 4.75. ABC P. , P ,

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A1 , B1 C1 . , , AA1 , BB1 CC1 , A1 B1 C1 .

4.1. M ABC. BM ABC AMC , SABM = SBCM . SBCM = SCAM . 4.2. ABP BCP , A C BP . BP AC, . . 4.8. 4.3. LO AC L1 . SLOB = SMOC MOC = L1 OC, SLOB = SL1 OC . . 4.8 LOB L1 OC , LO = L1 O, . . O , A. , O , B C, . . O . , . 4.4. SA1 BB1 = SA1 AB = SABC , SAA1 B1 = 2S. SBB1 C1 = = SCC1 A1 = 2S. SABC = 7S. 4.5. AB = BB1 , SBB1 C = SBAC . BC = CC1 , SB1 C1 C = SBB1 C = SBAC SBB1 C1 = 2SBAC . SDD1 A1 = 2SACD , SBB1 C1 + SDD1 A1 = 2SABC + 2SACD = 2SABCD . SAA1 B1 + SCC1 D1 = 2SABCD , SA1 B1 C1 D1 = SABCD + SAA1 B1 + SBB1 C1 + SCC1 D1 + SDD1 A1 = 5SABCD . 4.6. O . AD, BE CF , SABO = SDEO = SAEO , SBCO = SEFO = SCEO , SCDO = SAFO = SACO . , SABCDEF = 2(SABO + SBCO + SCDO ) SACE = SAEO + SCEO + SACO . , SABCDEF = 2SACE . 4.7. E F AC BD. SAOB = SAOD , O AF. O CF. , . AF CF F, O = F. , O = E. . 4.8. AC ABCD AD 5. ACB ACBE. ABCD DBE. BH DBE. EH2 = BE2 BH2 = 52 42 = 32 ED = BE2 /EH = 25/3. SDBE = ED BH/2 = 50/3. 4.9. SABE = SABC , EC AB. . P BD EC.

91

SBPC = x, SABCDE = SABE + SEPB + SEDC + SBPC = 3 + x (SEPB = SABE = 1, ABPE ). SBPC : SDPC = BP : DP = SEPB : SEPD , x : (1 x) = 1 : x, , x = ( 5 1)/2 SABCDE = ( 5 + 5)/2. 4.10. (. 1.7), KL. M N , BC. M N KL. O . O1 O BC. BO1 = CO1 MO1 = NO1 , , BM = NC. , SKLM + SKLN = SKBCL , , (SKBM + SLCM ) + (SKBN + SLCN ) = SKBCL = BC(KB + CL)/2 = BC AB/2. , KB BM + KB BN = KB BC, LC CM + LC CN = LC BC KB BC + LC BC = AB BC. 4.11. C l AB. A , B E A, B E l. AA C , ACB = BCB = B CA = 20 . EE C DEC 20 , EC . , SABC + 2SEDC = SABC + 2SEE C . E BC, 2SEE C = SBE C = SBB C /2. SABC + 2SEDC = = SAA C /2 = 3/8. 4.12. Ta , Tb Tc a, b c. Ta Tc , , , O. k a/c. , SA1 B3 O : SC1 B3 O = A1 O : C1 O = k. , a : b = k, , b = ac. 4.13. 1.3 ), , BQ p + pq = , BB1 1 + p + pq B1 R qr = , BB1 1 + q + qr CR q + qr = , CC1 1 + q + qr SRB1 C SABC CP pr = . CC1 1 + r + pr

, SPQR QR PR = SRB1 C RB1 RC

=

B1 C B1 R . AC BB1

SPQR QR PR B1 C QR PR CC1 B1 C = = . SABC BB1 RC AC BB1 CC1 CR AC

, QR p + pq qr 1 rq (1 + q)(1 pqr) =1 = = BB1 1 + p + pq 1 + q + rq 1 + p + pq 1 + q + rq (1 + p + pq)(1 + q + qr)

PR (1 + r)(1 pqr) = , CC1 (1 + q + qr)(1 + r + pr) SPQR (1 pqr)2 = . SABC (1 + p + pq)(1 + q + qr)(1 + r + pr)

4.14. SAOB = SCOD , AO BO = CO DO. AOD COB AD BC. . 4.15. ) SADP : SABP = DP : BP = SCDP : SBCP , SADP = SABP SCDP /SBCP .

92

4. ) ) SADP SCBP = SABP SCDP .

4.16. sin2 f/4, f , (AP BP)2 + (CP DP)2 = = (BP CP)2 + (AP DP)2 , . . (AP2 CP2 )(BP2 DP2 ) = 0. 4.17. , ABCD ; , AB CD . 7.2 P, ABCD, SABP + SCDP = SBCP + SADP = SABCD /2, . , P1 , P2 P3 . . 4.18. , SAKON = SAKO + SANO = (SAOB + SAOD )/2. SCLOM = = (SBCO + SCOD )/2. SAKON + SCLOM = SABCD /2. 4.19. KBLO MDNO , OK OL = = OM ON. , ON = KA OM = LC, KO : KA = LC : LO. , KOA LCO, , O AC. . 4.20. h1 , h h2 A, M B CD. 1.1 ) h = ph2 + (1 p)h1 , p = AM/AB. SDMC = h DC/2 = (h2 p DC + h1 (1 p) DC)/2 = SBCN + SADN . SDKN + SCLN , . 4.21. 4.20 SABD1 + SCDB1 = SABCD . SA1 B1 C1 D1 = = SA1 B1 D1 + SC1 D1 B1 = (1 2p)SABD1 + (1 2p)SCDB1 = (1 2p)SABCD . 4.22. 4.21 , , AB CD, . , , (. 1.16), , 4.21, . 4.23. AB, BC, CD AD K, L, M N . , KM AD. K, M, N L BC. KLM . , LN AB, SAKN + SBKL + SCLM + SDMN = SABCD /2, . . SKLMN = SABCD /2. 4.24. L1 N1 BC AD . KL1 MN1 ABCD (. 1.38 ). , KLMN KL1 MN1 . , , , , KM , LL1 NN1 BC AD. KM ABCD BC AD, KLM KL1 M, KM, , . . KLMN KL1 MN1 . 4.25. l1 l2 , S1 , S2 , S3 S4 , , , S1 + S2 S3 + S4

SABP SCBP SCDP SADP = (SADP SCBP )2 .

93

S2 + S3 S1 + S4 . S1 = S2 = S3 , S1 + S2 = S2 + S3 . , l1 +90 90 l2 , . , l1 ( , l2 ) . , . l1 l2 90 , , , . 4.9 (

. 4.9 ). l1 l2 , a, a + b, a + 2b + c a + b, a, b c . , . . 4.26. AM. hb , hc hd B, C D AM. # # # AC = AB + AD, hc = |hb hd |. 4.27. , P , AB BC, . . ABPQ CBPR. , SACRQ = SABPQ + SCBPR . 4.28. a. 3a, a O , a2 3 3/4 (. 4.47). . 4.29. ) PMQN BC AD sin a/4, a AD BC. ABD ACD, B C, OB sin a OC sin a, |SABD SACD | = = |OB OC| AD sin a/2 = BC AD sin a/2. ) AD BC. PN AO QN CO, N OPQ. SOPQ = SPQN + + SPON + SQON =SPMQN S S S SACD + SACD + SBCD S + ACD + BCD = ABD = ABCD . 2 4 4 4 4

4.30. KM LN CED AFB, EF. , KM = CD/2, LN = AB/2 KM LN a AB CD. KLMN AB CD sin a/8.

94

4.

4.31. ABCDEF , . 4.10. , A1 B1 C1 D1 ABCD. , . A1 C1 B1 D1 BDF ACE, . , D1 E1 F1 A1 DEFA. 4.32. a = PQC. PC AK, CK = AP cos a. 2SACK = CK AK = . 4.10 = (AP cos a) (AQ sin a) = AR2 sin a cos a = 2 = BC sin a cos a = BL CL = 2SBCL . . 1, 4.33. , CBD = SABD AO SABCD AC S = = ACP . SABD AO SAOP S CO

(1)

, S SABCD BD = = BDQ . SABC BO SBOQ

(2)

(2) (1), . 4.34. Sa , Sb Sc , A, B C; S . , SACC1 + SBAA1 + SCBB1 = SABC S + Sa + Sb + Sc . , SABC = SAOC + SAOB + SBOC = SACC1 + SBAA1 + SCBB1 . 4.35. O ABC, B1 AC. ABC AOB1 OAB1 . OB1 la . . , ABC . , , . 4.36. AD BC O. SCDO : SMNO = c2 : x2 , x = MN, SABO : SMNO = ab : x2 , OA : ON = a : x OB : OM = b : x. , x2 c2 = ab x2 , . . 2x2 = ab + c2 . 4.37. , , , . 4.11. S. S3 + (S2 + S7 ) = S/2 = S1 + S6 + (S2 + S7 ), S3 = S1 + S6 . S/2 = S1 + S2 + S3 + S4 , S/2 = 2S1 + S2 + S4 + S6 2S1 , . . S1 S/4.

95

. 4.11

. 4.12

4.38. l B, A C. C1 , C; l K L, K1 L1 C1 K C1 L, A (. 4.12). , l , K1 KLL1 , C1 KL. SC1 KL , . . AB (KL + K1 L1 ) BC KL. K1 L1 = SK1 KLL1 = KL (AB + BC)/BC, AB + AB/BC) BC. (2 , BC/AB 1 + 2. AB/BC 1 + 2 ( , A C). 4.39. S. l . , l Ox. S(a) , y = a. a + S(a) 0 S, S(a) = S/2 a, . . y = a . , l . , S1 , S2 , S3 S4 (. 4.13). S1 + S2 = S3 + S4 S1 + S4 = S2 + S3 , S1 = S3 = A S2 = S4 = B. l 90 A B, B A. A B l , . 4.13 A = B, . . . 4.40. ) , ABC, AC BC P Q . ABC O,

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4.

r. SABQOP = r(AP + AB + BQ)/2 SOQCP = r(QC + CP)/2. PQ , AP + AB + BQ = QC + CP, SABQOP = SOQCP . , SABQP = SQCP . SOQP = 0, . . QP O. ) . 4.41. S2 S1 , , AA1 S1 S2 K, A, AK > A1 K. KA1 K S1 A1 , BK > A1 K, . .