σχολικό βιβλίο α β λυκείου γεωμετρία
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Transcript of σχολικό βιβλίο α β λυκείου γεωμετρία
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2. , ++- !#!) # . /2!..2+ #20$0 # ..+2 /2!..2+ #20$0 #.21 *+0+!..2).!22#.) #!.2 ...2+ #20$0 #/0! #$!) ..2)2$ )* # 12 !0+.2../ # /2!.012 #0/RPRQRVRY)1$. ) .012-0.2! # 0)../ * #. !310/ 10 * # .2 3. + #0/0. +002!. $0 . /22) !). 0!+10 . #0 .2 12 1# 12 2 . 03!0 2.0!.12 )0.) 1*12.2 ##0/...2. !0121*$! 00./0#202. .#2)1*3 02..1.1#!.3 #0102 ./. )122 *2 0#0120)2. /10 2 # .2 2 # #0 # . ! 1 #2.#12!..2..2+11.0 2.)2.1#0 13!012..2./0.2 #2) #..2*11 2.2 !0.1)2. .2 .1% ! .#2)0.. 201.21# ! 10.. /.. .2+. /0$)0 2!)112 #2. .22.!0.0!12.1#12.2.2!..2 12 # ,. .0 . 0#$.!121 #0 0! 2 2 ..2 2.!0..2 0. #.!)130!010)2/!0.21#!.3 2 #! #2.22 # #20$0 ## 0) # .2 1.22 #.!.2!1012/.)!312 # #.2.20 2! !1 # 0 2 0*12 $0 .!.2!10 2 #1. 12 20!3.#2 *2 #! # 1#!.30 4. index-per.qxd22/01/2001 06:40 PMPage 349 1 . . . . . . . . . . . . . . .1 1.1 . . . . . . . . . . . . . . .3 1.2. . . .5 2 7 2.1. , 9 2.2. . . . . . . . . . . . . . . . .9 2.3. . . . . . . . . . . . . . . . .10 2.4. . . . . . . . . . . . . . .10 2.5. . . . . . . .10 2.6. . . . .11 2.7. . . . . . . . . . . . . . . . .11 2.8. . . . . . . . . . . . . . . . .12 2.9. . . . . . .13 2.10 . . . . . . . . . . . . . . . . . .13 2.11. . . . . . . . . . . . . . .14 2.12. . . . . . . . . . . . . . . . . .15 2.13. . . . . . . . . . .15 2.14. . . . . . . . . . . . . . . . .16 2.15. . . . . . .17 2.16. 18 2.17. 21 2.18. - . .23 2.19. . . . . .25 2.20. - . . . . . . . .28 3 . . . . . . . . . . . . . . . .33 3.1. . . .35 3.2. 1 36 3.3. 2 39 3.4. 3 39 3.5. . . . . . . . . . . . . . . . . .43 3.6. . . . . . . . . . . . . . . . .44 3.7. - . . . . . . . . . . . . . . .49 3.8. . . . . . . . .50 3.9. . . . . . . . . .51 3.10. . . . . . . . . . . . . . . . . . .533.11. 3.12. 3.13. 3.14. 3.15. 3.16. 3.17. 3.18. . . . . . . . . . . . . . . .54 . . . . . . . .54 . . . . . . . .58 . . . . . . . . . . . . . . .60 . . . . . . .62 . .63 67 .69 4 . . . . . . .73 4.1. . . . . . . . . . . . . . . . .75 4.2. . . . . . . . . . .75 4.3. . . . . . . . . . . . . . . . . .78 4.4. 79 4.5. . . . . . . . . . . . . . . . .80 4.6. . . .83 4.7. . . .84 4.8. - . . . . . . . . . . . . . . . . .85 5 - 95 5.1 . . . . . . . . . . . . . . . .97 5.2. . . . . . . . . .97 5.3. . . . . . . . . . . . . . .100 5.4. . . . . . . . . . . . . . . . .101 5.5. . . . . . . . . . . . . . .102 5.6. . . . .104 5.7. . . . . . .107 5.8. . . . .108 5.9. . . . . . . . . . . . . . . .109 5.10. . . . . . . . . . . . . . . .112 5.11. . . . . . . . .113 5.12. . . . . . . .116 6 . . . .121 6.1. - . . . . . .123 6.2. . . . . .123 6.3. . . . . . . . . . . . .124 6.4. . . . . . . . . . . . . . .126 6.5. 130 6.6 . .131 6.7. 5. index-per.qxd22/01/2001 06:40 PMPage 350 . . . . . . . . . . . . . . . . .135 7 . . . . . . . . . . . . . .143 7.1. . . . . . . . . . . . . . . .145 7.2. . . . . .145 7.3. - . . . . .145 7.4. . . . . . . . . . . . . . .147 7.5. 148 7.6. . . . . . . . . . . . .148 7.7. . . . . . . . .151 7.8. . . . . . . . . . . . . . . .158 7.9. . . . . . . .161 8 . . . . . . . . . . . . .169 8.1. .171 8.2. . . . . . . .172 9 . . . . . . 9.1. . . . . . . . . 9.2. . . 9.3. . . 9.4. . . . . . . . . . . . 9.5. . . . . 9.6. 9.7. . . . . . .. . . ..181 .183 .183 .186. . . ..189 .194 .196 .199 10 . . . . . . . . . . . . . .209 10.1. . . . . . . . .211 10.2. . . . . . . . . . . . . . . . .211 10.3. . . . . . . . . . . . . . . .212 10.4. . . . . . . . . . . . . . . .218 10.5. - . . . . .221 10.6. . . . . . . . . .225 11 . . . . . . . .231 11.1. . . . . . . . . . . . . . .233 11.2. . . . . .234 11.3. . . . . . . . . .238 11.4. 11.5. 11.6. 11.7. 11.8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..243 .244 .246 .246 .249 12 . . . . . . . . . . . . . . . .257 12.1. . . . . . . . . . . . . . . .259 12.2. . . . . . . . . . .260 12.3. . . . . . . . . . . . . . . .262 12.4. . . . . . . . .265 12.5. . . . . . . . .269 12.6. - . . . . . .274 12.7. - - . . . . . . . . . . . . . . . .277 12.8. - . . . . . . . . . . . .282 13 . . . . . . . .287 13.1. . . . . . . . . . .289 13.2. . . . . . . . . . . . .291 13.3. , . . . .292 13.4. . . . . . . .293 13.5. . . . . . . . . . . . . . .299 13.6. . . . . . . . . . . . . . . .301 13.7. . . . . . . .301 13.8. . . . . . . .304 13.9. . . . . . . . . . . . . . .304 13.10. . . . . .307 13.11. . . . . . . . . . . . . . .307 13.12. . . . . . . . .308 13.13. .310 13.14. . . . . . . .310 13.15. . . . . . . . . .312 13.16. 314 13.17. . . . . . . . . . .315 13.18. . . . . . . . . .316 13.19. . . . . . . .321 6. 1 1 7. 1Ellsworth Kelly (, 1923). 2 2 , , 1974.2 8. 1.1 , , . , , , , . , . , , . . , . . , , . , . , . , . . , , , . , . , , . , , . , () . . , , , . : . , . , , . , . ; . , , . , . , , ( ), ( ). , . . 3 9. 1 , , , , , . : . . , . , , . , , . , (. ). , , . , , , . , . . . . , , . , . , , , . , , , . .4 10. 1.2 . , . , , , , . , , . . .. . , . , , , , . , , , . . , . , . , , 440 .., , , . , , , . (.. ), . , , 5 11. 1 , .. , 17 . ( ), . . 17 . , . H . 18 . , . 17 . . (1822) , ( ) . . . 19 . . (1829) . (1832). , , , . ( ), ( ) . 1854 , . 19 . , ( ) .6 12. kefal02.qxd22/01/2001 06:04 PMPage 72T : , , : , , , . , , , , .7 13. kefal02.qxd22/01/2001 06:04 PMPage 8 2Andrea Mantegna (, 1431 - 1506). Camera degli Sposi ( ), , .8 14. kefal02.qxd22/01/2001 06:04 PMPage 9 , , , . O : . . , .2.1 , . (.. , (.1)). , (.2). . , . () . . , , . 1 22.2 . , , . . , , , . . 9 15. kefal02.qxd22/01/2001 06:04 PMPage 10 22.3 3 4 , . (.3). (,,...) xx. . . , . 2.4 5 . xx (.4). x Ax . xx x (.5). x, y , (.6). 6 7: , , .102.5 ,. (.7) , . , . , , . .. (.7), , , , . , , . 16. kefal02.qxd22/01/2001 06:04 PMPage 11 2.6 . ( ) ( ) .2.7 , . : . x , = (.8). . x. , , , x (.8). . 8 . , , ( ) . , = (.9). . 911 17. kefal02.qxd22/01/2001 06:04 PMPage 12 2 . x. . x, =. , . (. 10). , =, (.10). , . (.10). 10 11 12 132.8 , . (i) = = (.11). , =+. . (. ). (ii) , = (.12). =. (iii) , , (.13). = = . =, , . .12 : i) + =+ () ii) (+) + = +( +) (). 18. kefal02.qxd22/01/2001 06:04 PMPage 13 2.9 - . . ( 7) ( ), = . , , . () , . , (.14). . 142.10 . , , . x = (.15). . . T . . 15A 1. : i) ii) iii) iv) . 2. : i) , ii) .3. ,, . , , , , . 4. Ox Ox ;5. ; ;13 19. kefal02.qxd22/01/2001 06:04 PMPage 14 2 1. :2. , , . : i) =2. , , : i) , ii) . 3. , , = . = . 4. , . , = 2. 1. , , . , , : i) = + , 2ii) A+=+. - , 2ii) = + . 23. i) ,,, +. 4. ii) , , , , ++. 1. ,, , , , = . 2 2. , . . ; ( 2). 2.11 1614 : 1 2, . 1, (.16) . , , , . (.16), 1 (,). 2 (,). 1 2 : , . , , (.16). 20. kefal02.qxd22/01/2001 06:04 PMPage 15 2.12 x y (.17), . , Ox, Oy . (Ox, B) (Oy, A) x y. ^ ^ ^ ^ ^ xOy yOx O AOB BOA (.17) . , ^ xOy, x Oy Ox Oy. , . . x y : x y , ^ x (.18) xOy ^ y , xO (.19) . A x, Oy (.20), xy . , , .2.13 ^ ^ AOB AOB , , (.21). : (i) , ^ ^ AOB = AOB. (ii) ^ ^ AOB, AOB ^ B AOBAOB. ^ ^ AO ^ (iii) AOB, ^ ^ AOB AOB ^ BAOB. ^ AO ^ ^ AOB ^ , 17 18 19 20 2115 21. kefal02.qxd22/01/2001 06:04 PMPage 16 2 ^ AOB. : , ^ ^ : AOB = (.21) ^ AOB, ^ ^ ^ ^ AOB:AOB (.21). ^ AOB, ^ ^ ^ ^ AOB: AOB (.21). 22 23 21 21 21 24 25^ xOy , ^ ^ , xO = Oy (.22). . - ^ xOy (.23). ^ ^ xO Oy , . . 1, 2 1^2 (.24). , (.25).2.14 2616 xx (.26). A ^ xx, ^xx. 22. kefal02.qxd22/01/2001 06:04 PMPage 17 z (.26), ^ ^ , xx, xz = zx=1, ^ z xx. A, , , . : . (.27) . , (. .3.5), . xx xx (.27). 27 , , ^ xx ( ) xAx , ( ). . (.28). , . . , . (.29). . 28 292.15 , , ^ ^ , .. xO y yO z (.30) . 3017 23. kefal02.qxd22/01/2001 06:04 PMPage 18 2 ^^^ (. 31) , ^ ^ ^ ^ . , , . - ^ 31 32^(i) ^ (.31). . , ^ ^ ^ ^ , .. = ++ (.31). ^ ^ ^ (ii) (.32). , ^ ^ ^ (.32). ^ ^ ^ . ^ ^ ^ + = . . ^ (iii) ^ . $ $ $ $ = + 4444 , 1444 2 +...+ 4 3 ^^^^^.. = ++ = 3 ^ ^ = (.33). 3 33 .2.16 A . ^ ^ , .. (.34) . 3418 . (.35). 24. kefal02.qxd22/01/2001 06:04 PMPage 19 ( ) . . ^ ^ , (.35) ^ , . , . ^ ^ . , (.35) , ^ ^ ^ ^ ^ . , . , . ^ ^ ^ .. xOy xOy yOx ^ xOy (.36). 35 . ( ) , ..: : A , . : , . . I . ^ ^ xOy xOy (.36). ^ yOx. 36 II .19 25. kefal02.qxd22/01/2001 06:04 PMPage 20 2 ^ ^ xOy xOy ^ ^ ^ xOy. Ox = Oy. ^ ^ ^ ^ , O1 = Ox O2 = Oy ( ) . ^ ^ ^ O1 = O2, xOy. 37 III . 38 ^ ^ , (.38). ^ ^ ^ ^ + = 2 2+2 = 2 ^ ^ ^ + = 1 = 1. ^. A 1. : i) 1 , ii) 2 , iii) . . 2. , . 3. . ^ ^ 4. i) O AOB O ; ^ ^ ii) AO AOB ; .5. ; ^^1. xOy, yOz ^ ^ ^ zOt, xOz = yOt. ^ ^ xOy = zOt. 2. , , .3. . ; ( ) ; 1. .20 26. kefal02.qxd22/01/2001 06:04 PMPage 21 ^2. O, ^ O, . ^ ^ O+O ^ O = . 2 ^ 3. O, ^ O. ^ ^ O O ^ O = . 2 ^^^1. O, BO, O . x, y ^ ^ O, O , ^ ^ AO+O ^ xOy = . 2 ^ 2. O ^. , ^ ^ O BO , 1 ^ O = . 2 2.17 = (.39). . . (,) () (.40). , (,) (.40), . (,) =. , , . () . (,) , , = . 39 40 - , (.41). . 1, , 2.21 27. kefal02.qxd22/01/2001 06:04 PMPage 22 2 41 . , . , . , .41 , . (.41) , . . To (.41) . . . , (. 41) , . . , . 42 (,) x . (.42) , , . , . , (,) (.42) , , , . , (.43). , . 4322 28. kefal02.qxd22/01/2001 06:04 PMPage 23 2.18 E - . ^ , .44 xOy . . , . ^ , O . 44 . , = (.45). ( ) , (.45). . 45 , . I , . () (.46). , , . , ^ ^ . 4623 29. kefal02.qxd22/01/2001 06:04 PMPage 24 2 ^^. (). , ,,, ^ , ^ , . , = . I (i) . (ii) . (iii) , . B (.47) () , , , , (.47) , . 47 (.48) , , = . II 48 . , , (.49). ^ ^ A = , AOM MOB ^ AO. M, ^ , . 4924 30. kefal02.qxd22/01/2001 06:04 PMPage 25 ^ ^ xOy zOt - , ( ), (,R) (O,R) RR. = , = (.50). ^ ^ = , O = O, = . . 50A 1. (O,) . ; ; 2. ; 3. ; 4. ; 5. , ; ; 6. ; ; 7. ; , , ; 8. ^ ^ 1 = 2 . = ; E 1. , . ; ; 2. (,) (, R) R. (, R) (, ). 1. (O,R) (,) R. ,,,. = = . 2. , .2.19 , .. B (.51) . T , 5125 31. kefal02.qxd22/01/2001 06:04 PMPage 26 2 52 53 , .. , (.51) . , , ^ ^ ^ , . (.52). + , + = . , 2 . , . = , - 1 1 , = . , , . 3.18 . (.53) , . = , 0. . .. 1 360 1. ( ), . . 5426 360 180 90 . 60 ( 60) 60 ( 60). ^ xOy (.54), (, ), 32. kefal02.qxd22/01/2001 06:04 PMPage 27 ^ . xOy ^ ^ . xOy (xOy) ^ y. xO , 90, 180 360. , , (.55), = 3 = 6 . : (i) , , ^ ^ ^ (ii) AO B, BO O. 55 (i) = 6 = 2 , + + = 360 9 = 360 = 40. = 240 = 80. ^ ^ (ii) AOB , AOB = = 240 ^ ^ ( ). BO = 80 O = 40.A 1. , : i) + , ii) + + , iii) . iii) 2 , iv) .3. :2. : i) 2 , ii) 2 + ,. . . . . . 4. ; ^ 5. = ( ), AB ;27 33. kefal02.qxd22/01/2001 06:04 PMPage 28 2 i) , ^ ^ ii) O OB ( ). 3. . , . 4. 6/5 , . ; . E 1. , , = . 1. i) 1 , = ( + ). 2 ii) , 1 = ( ). 22. = 80. : . . 2. 20. . 3. , , , ^ ^ ^ ^ O, O, O, O, 1, 2, 3, 4. . 2.20 - 56 5728 , .. , , , , , .56. , , . . , , , . , , . , . , (.56) , (.57) . , , 34. kefal02.qxd22/01/2001 06:04 PMPage 29 . (.56) , (.57) (.58) . , , , , .. , (.59). . , , , . 58 59 60 , (.60) (.60) , (.60) . (.60), (.60), (.60) , -. . . (.61). 61 . . . , ^ , .. x (. 60) ^ .29 35. kefal02.qxd22/01/2001 06:04 PMPage 30 2 1. , , , , , 2 2 , . = . 2 2. , . ,, , , , , . 3. ,, . . 2 4. (,R) , , , = 150, = 45 = 105. ^ O . 5. , . , . - 1. - ( 3), : i) , , : ) , ) . ii) ; iii) +1 . : .30 36. kefal02.qxd22/01/2001 06:04 PMPage 31 : , , : , . , . , , , . , . , . . , . ( ), , . , , . (,) , . , . . , ( , ). :, , - , , , , , - , , : , , 31 37. kefal02.qxd22/01/2001 06:04 PMPage 32 232 38. kefal03.qxd22/01/2001 06:06 PMPage 333T , . . , , . . . , , . .33 39. kefal03.qxd22/01/2001 06:06 PMPage 34 3 , 508 .. A. Tournaire34 40. kefal03.qxd22/01/2001 06:06 PMPage 353.1 (.1) , , , ^ ^ ^ B, , , . , , , , ^ ^ ^ ^ ^ ^ , , , . . + + , 2. , , : , ., : , (.2), , (.3). = , , (.4). , , , (.5), , (.6). , , (.7). . .8 . . , . .9 ^ . ^ 1 2 3 4 5 6^ . 735 41. kefal03.qxd22/01/2001 06:06 PMPage 36 3 8 , . , , . .10 . , . , . 9 10 , , . : . . . , . O .3.2 1 , , . 1136 I (1o ) , . AB = ^ ^ , = = (.11). AB, ^ ^ . = . , = = , . , . 42. kefal03.qxd22/01/2001 06:06 PMPage 37 I : . . = (.12). . ^ ^ = , 1 = 2 (), ^^, = . = , ^ ^ 1 = 2. ^ ^ ^ ^ 1 + 2 = 180 1 = 2 = 90, . 12I . I . (.13) . =, ^ ^ 1 = 2 = 90 (), , = . 13I V , . (,) (.14). ^ ^ = . ^ ^ = (= ), = (= ) = . , = . 1437 43. kefal03.qxd22/01/2001 06:06 PMPage 38 3 , (.15). , , = . : ^ ^ (i) MAB = MBA, (ii) = . (i) 15 = , , ^ ^ B MA = MBA. ^ ^ (ii) = , = () M1 = M2 (), () , = . H .A 1. ^ ^ = = , = . = . 2. , , = = . . 3. . ^ 4. = = . ^ ^ = .38 1. . , , = , = , = , ^ ^ = . 2. . , , . , . 3. . , . ^ ^ = . 44. kefal03.qxd22/01/2001 06:06 PMPage 393.3 2 1 2 3 . , , . (2o ) , . (.16) = , ^ ^ ^ ^ = = . = . , .. . , = . = , = ^ ^ ^ ^ = , () , B = . ^ ^ ^ ^ = , B = , ^ ^ ^ B. , = . , , = , = ^ ^ = , () . : , (. 36). *3.4 3 16 , , . , , . (3o ) , . = , = , = (.17). ^ ^ = . . ^ ^ x, x = (.17) , = . ^ ^ , = , = = . ^ ^ = = . 1739 45. kefal03.qxd22/01/2001 06:06 PMPage 40 3 = = , , ^ ^ 1 = 1 (1). , = = , ^ ^ 2 = 2 (2). ^ ^ , (1) ^ ^ ^ ^ ^ ^ (2) = . = , = , . 3 : ^^i) B90 B 90. ^ ^ ii) B = 90 B = 90. . I I H , , . = (.18). =, ^ ^ =, () , 1 = 2, ^^ 1 = 2. . 18I II . 1940 (.19), , = . , , . 46. kefal03.qxd22/01/2001 06:06 PMPage 41 (3.2) . .I III A , , , . (,) , = . (.20) : = (= ), = (= ) = , ^ ^ () . , = , = . 20I IV , . IV , , . : : (), (), ().41 47. kefal03.qxd22/01/2001 06:06 PMPage 42 3 1 ^ xOy (,), (,R) R (.21). x, Oy , , , , , : (i) , (ii) , (iii) , ^ (iv) OM xOy. 21 ^ (i) = (= ), = (= R) O (), . ^ ^ ^ ^ (ii) 1 = 1 180 2 = 180 B 2 ^^^^^^ 2 = B 2 1 = 1. , = , 2 = B 2 ^^1 = 1 (), . (iii) (ii) =, = , = (), . ^ ^ (iv) , 1 = 2 , ^ xOy. 1 . 2 = , = = . . (.22). = , 22 = ( ) =, . , ^ ^ ^ (), = . , = , = = ^ , () .42 48. kefal03.qxd22/01/2001 06:06 PMPage 43 A 1. () () : i) . ii) . 2. . 3. : i) ................................................. ii) ........................................... iii) , ........................................... iv) , ...................... ................................................1. . 2. , (. ), . 1. = , = ^ ^ = . : i) = = ii) = I = I. ^ ^ 2. = , = = . : ^^i) = , ii) = = . 3. . .3. = ^ ^ ^ ^ = . = .1. . , . ^ ^ : i) = , ii) = , iii) , iv) = = . 2. . , . 3. ( = ). . = . : i) , ii) .3.5 2o . . . 43 49. kefal03.qxd22/01/2001 06:06 PMPage 44 3 xx, xx (.23). xx, . xx. xx ^ y, xMy ^ = AMx , = . , xx, xx ^ , . = M1 = ^ 23M2, , ^xx. xx . , , ^ ^ ^ ^ ^ = M1 = M2, 1 = 2. 1 ^^^= 90, 2 = 90, 1 + 2 = 180 ,, , , .3.6 24 25 , , 1 () 2 () : , , . (.24). , , (.25). . I , . 2644 ^ ^ = = 90, = ^ ^ = (.26). = . , .. . , = . 50. kefal03.qxd22/01/2001 06:06 PMPage 45 = , = ^ ^ ^ ^ = , , = = 90, ^. A^ ^ ( ). . , , = , = ^ ^ = (). II , . ^ ^ (.27) = = 90, ^ ^ = = . = . , = = . . , , = = . = = . = , = = ( ^ ^ ), , = . (). 27I . I . (,), , (.28). ( = = ), 2845 51. kefal03.qxd22/01/2001 06:06 PMPage 46 3 , ^ ^ ^ ^ 1 = 2. 1 = 2 = . :. , : . . II . 29 (,) , (.29). ^ ^ , = = 90, = (= ) = ( = ). , = . . . ^ ^ = = 90, = = , , = = = . 2 2 IV K . 3046 ^ xOy (.30). MA^Ox MB^Oy. ^ ^ = = 90, ^ ^ O = O, = . . . MA^Ox MB^Oy = . ^ ^ , = = 90, ^ ^ = O = O, . 52. kefal03.qxd22/01/2001 06:06 PMPage 47 : . . . 1 . (.31) , = , =. , , : (i) , , (ii) = . 31 ^ ^ (i) ( = = 90) = ( ^ ^ ) 1 = 2 (). , . ^^^^ = = 90, = 1 = 2. (ii) =. = . = . 2 , (.32) , , . = . , = 32 , , ^ ^ . ( = = 90) = , ^ ^ M1 = M2 . , = . 47 53. kefal03.qxd22/01/2001 06:06 PMPage 48 3 A 1. . AB^ A^ (, ) : i) ii) iii) = A . 2. ( = ), : 1: . 2: . 3: . 1, 2 , 3, ; 3. . 4. . . . 5. : ..................... ........ ......... 6. , () , : 1 . = = , 2 . = , . = = , 1 1 . = = , 2 3 . = . . 7. ; 8. ; . 1. .482. : i) , ii) . 3. . 4. , . 1. ( = ) . : i) , ii) . 2. =, = = , . 3. =, = = , . ^ 4. ( = 1) . ^, . . 5. (, R), , . , : i) , ii) = = . ^ 1. . . . i) . ii) ^ , , , . iii) = = . 2. , , . 54. kefal03.qxd22/01/2001 06:06 PMPage 49 3.7 - - , , () . : (.33) , . (.34) , . (.35) , ( ) . . 33 34 35 , . , , (.36). =, . . . =, (,) . ,. . 3649 55. kefal03.qxd22/01/2001 06:06 PMPage 50 3 : , . . , .A . .i) ................. ii) .................. 1. , = A . 2. (,R). , = , , . 3.8 37 38 3950 2.10 , (.37). , (.38), . , . , , , . . A , (.39), 180 , . 56. kefal03.qxd22/01/2001 06:06 PMPage 51 , : (.40). (.40). (.40). 40 , . (.41), , , ^ ^ . = , OB = OB =, , = . . 41 . ^ ^ ^ ^ = , = , = ^ ^ 1 = 2. = , . . , .3.9 2.14 , () (.42). , (.43) , . . , , , . . , 42 4351 57. kefal03.qxd22/01/2001 06:06 PMPage 52 3 44 (.44) , , , . (.45). xx ^xx xx (.45). (.45). (=) (.45). (.45). 45 . , . 46 , . , = . ^ ^ , , 1 = 2. . ^ . ^ ^ ( 1 = 2), , = , . . , . 52 58. kefal03.qxd22/01/2001 06:06 PMPage 53 A 1. : , , , , , , , . 2. . , , , , , , . ^ ^ 3. xy xy, , ^ ^ ^ xy, xy = xy.4. . 5. . 6. , . , : i) = , ii) , , . . , .3.10 . . (.47) , , , ^ = . x ^ ^ ^ x = . E ^ ^ : = , = 1 = 2 , ^^ ^ ^ = . ^^^ ^ . . 47I (i) . (ii) 180. 53 59. kefal03.qxd22/01/2001 06:07 PMPage 54 33.11 . (.48). , =. ^ ^ 1 = 1 = . ^^^ , 1, ^ 1, 48^^^^ , 1. ^^^, . ^ ^ . . ^ ^ , = = ^ ^ , . I (ii) 3.2. : .(i) , . (ii) , . (iii) , .3.12 . . . 54 60. kefal03.qxd22/01/2001 06:07 PMPage 55 . + (.49). , , = . ^ , ^ ^ ^ ^ = 1 1. ^^ , + . + + . , , , , . E: 49 + , I . 1 , : ^ ^ (i) (ii) + +. 50 (i) (.50) ^ ^ . 1. ^^^^ ^ 1 , 1. . (ii) + + + . : + + + ( + ) + + + . 2 . : (i) , (ii) , (iii) , . (.51). - 5155 61. kefal03.qxd22/01/2001 06:07 PMPage 56 3 . ( = , ^ ^ ^ ^ ^ ^ = , 1 = 2 ). = (1) 1 = . 1 = ^^ ^ = (2), , 1 = 2 = . (1) (2) =. , = . 3 , . ^ ^ = , = (.55). ^ ^ B. , ^ Ax , ^ ^ 52 x = . x , = . (). , = . ^ , , = . , + + . . = , = ^ ^ ^ ^ . = , = , , ^ ^ , . , . 4 , , (.53). (i) , + = + . + ; (ii) , , (.53) . , . , ; (i) , , = 53 + = + (1). , + (2), 56 62. kefal03.qxd22/01/2001 06:07 PMPage 57 + = (3). (1), (2) (3) + = + + , = 0, 0 . (ii) (i).A 2. = ^ ^ 1. () () A = , = . : ^ i) ^ . ^ ii) ^ . iii) 180. iv) ( ), ^ ^ = . v) = ( ), ^ ^ = . 2. : . = 7 . =1 . 17 . 7 . 01 . 33 53. = = ; . ^ ^ 1. 11 . ^ 190. ; ^ ^ 3. =. i) ^ ; ii) , . 4. ^ x . ^ , , , , . 5. , . ^ 6. ( =90), ^ . . 7. . . = = . 8. ( = ) , . ^ ^ , . 9. ( = ) ^ ^ , . : i) , ^ ii) . 10. 1, 2, 3 ( ), 7, 6 10 km . 1 1231 1. 57 63. kefal03.qxd22/01/2001 06:07 PMPage 58 3 48 km. ; . 6. (,R) , . = 2 2. 7. . 1. . i) N . 21. , ^ ^ ^ + . = ; 2 2 2. ^ ^ . . 3. . : ^ ^ i) M M , + ii) , 2 2 iii) + + 2.4. (,R) . . ( ). 5. . , : i) A = 2AB, ii) . + + + + + + . 2 ii) + + + ; 2. () = = . . : i) , ^ ii) . 3. . : i) , ii) A++ ++, iii) . ^ 4. xOy x, Oy , . 2.3.13 (.54) . o . . , , . . 5458 64. kefal03.qxd22/01/2001 06:07 PMPage 59 A , , . (.55). To , , = . . = . , ( 3.12) = . : (i) . (ii) , . ^ (i) (.56), . , , . (ii) . , , , (.57). , , . (.57). ^ . , , ^ ^ , = 1, ^ . ^ , , . . . = , = , . , , . . 55 (i) , : . 56 5759 65. kefal03.qxd22/01/2001 06:07 PMPage 60 3 A , , : 1. i) = .................. ii) ................. 2. () () . i) ii) = iii) 2. . , .3. , . 1. , i) , . : i) , ii) . . ii) , ; 3.14 5860 (,R) xx = xx (.58). R : R, = R R. . R (.58). , , OMOAR. , xx . = R (.58). , xx (,R), = R. , xx . . , xx (,R) . : 66. kefal03.qxd22/01/2001 06:07 PMPage 61 . . R (.58). . x , = R. T , = R. x, , , , , , . x , . , xx . xx, . . : R, . =R, . R, . . . I . (,) , , , (. 59). = (= ) = (= ), , , , . , , . 59 . 4.5 , .61 67. kefal03.qxd22/01/2001 06:07 PMPage 62 33.15 (, ) . 6.7 . , (.60), . : II , . ^ ^ (.60) = = 90, = (= ), , = . 60I , : (i) , (ii) . A 1. , -. () () : ; 2. = = ; .3. , ^, , , , N 62i) = . ii) . iii) , , . iv) ^ ^ , O N. 68. kefal03.qxd22/01/2001 06:07 PMPage 63 1. A , 1. . 2. (,), 1 , 2 ,. 1 , 2 , , ^ O = 90. 3. (,R) . , . ., , . 2. (,R) , . . ^ ^ . 3. , . ^ ^ = .3.16 (, R) (, ) R . (.61). 61 (. 61). . : (i) (, ) (, R), 6163 69. kefal03.qxd22/01/2001 06:07 PMPage 64 3 R (.62). (ii) (, R) (, ) , R + (.62). (i) O , (, ) (, R), = R (.62). (ii) , , = R + (.62). . , (.63) , + , R + , . 62 63 64 O , , R R + (.62). . . . (,R) (,) .64 , . = = R, . = = . , . 6564 (, R) (, ) (.65) , R = , . , R = , = = . . 70. kefal03.qxd22/01/2001 06:07 PMPage 65 (.66). , , .66. : (i) . (ii) . (i) () . 66 ^ (1). , , (1) ^, (). (ii) . = , () = , (). = . , , .A 1. (, R) (, ) o - R, = , . . (,) (,R). . (,) (,R). . (,R) (,) . . . . . 1. R+ 2. =R+ 3. = R 4. R 5. 2= R 6. R 7. 2=R 8. RR+2. () () . i) . ii) . iii) . 1. (K, ) (, 2) i) = , 2 ii) = , iii) = 2, iv) = 3, v) = 4. 2. (, ) . -65 71. kefal03.qxd22/01/2001 06:07 PMPage 66 3 . ; 3. . (, ) . ; 1. (, R) , 2R. (, 2R). : i) (O,2R) (, ) , ii) (, R) , iii) (, R). 2. (1, R1) (2, R2) i) Na . ii) (1) , (2 ) , , , . MN AB MN, , (1 ) (2 ) .3. . ^ . = 90. 4. 12 , 5 ;12R1+R2 2R2 . . , , . (. ). 4 . , .. . . , , . .66 2.7 . : ( ), . . . , , . . , , , , . . 72. kefal03.qxd22/01/2001 06:07 PMPage 673.17 . 1 ^ xOy x. ^ xOy , x . ^ xOy (.67) 67 . A . , x . (, ) (,) . . ^ ^ xO, xOy . ^ ^ xOy xOy , (,), (,) A A . ( 2.18) , (,) (, ) . , , = : + ( ). (,) (, ). 2 . (.68). , . , 2 , . 68 , (3.16) (, ) (, ) . , + . : .67 73. kefal03.qxd22/01/2001 06:07 PMPage 68 3 : , , . 3 . , : (i) , (ii) . (i) (.69) , . ( ). (ii) (.70) . , , . , = , , . 69 70 4 . ^ xOy (.71). , , , . ( 2) . , , A ^ xOy ( 3.6). . 71 (,) . (.72) , = . , . : 137.68 72 74. kefal03.qxd22/01/2001 06:07 PMPage 693.18 (3.2-3.4) . 1 , = , = ^ = . x ^ ( 3.17) xy = (.73). x, Ay , , , 73 , = = . . ^ , , = , = = . 0180 (3.10 (ii)) . 2 , = ^ ^ = = . = , (.74) , ^ ^ , x = y = . Bx, 74 y . . ^ ^ , , = , = = . 0 + 180 (3.10 (ii)) . (4.2) + 180 x y. 3 , = , = = . = (.75) (,) (,). , . 7569 75. kefal03.qxd22/01/2001 06:07 PMPage 70 3 , , = , = (,) = (, ). , (,) (,) , (3.16) + (). (, ) (, ), , , . ,, + ( ). , +.A 1. - 1. 45. ; 2. . 2. 3. ;3. - ; 4. , , : . = + . + . + . 2(+) . . . . 4. . 5. ^ = 90, : i) = = , ii) = = , , , . 1. , ^^^^ = , = + = 2. i) AB = A, ii) . 2. , , , . = = . 3. , , (O,R), , . 4. , , , , = = . , , , , . ^ 5. 1 = 2. 70^ . 2 ^^ ^ 6. = = . 2 2 ^ =1. 7. . ^ 8. xO y . Ox Oy. , Ox, Oy , ++=++. , , + + . 76. kefal03.qxd22/01/2001 06:07 PMPage 71 , , . , , , . , , . : (). (). (). : . , . : . . . , , . : , . , . : , , . : , . , . , . , , , . : . . . : ^ ^ = , = . . , .71 77. kefal03.qxd22/01/2001 06:07 PMPage 72 3 - : : - : - : 72 78. kefal04.qxd23/01/2001 09:57 AMPage 734 . , . , , .73 79. kefal04.qxd23/01/2001 09:58 AMPage 74 4Lzl Moholy-Nagy, (, 1895-1946), - .1 1922.74 80. kefal04.qxd23/01/2001 09:58 AMPage 75 4.1 2.3, . , 1 2, , : i) (.1), ii) (.1), iii) (.1). 1 2 , : 1 2 . 1 2 , 1//2. 14.2 - 1 2 , 3. . , , , 1, 2 , , , , . 3 , 3 . , , , , . , , . 2 , . 75 81. kefal04.qxd23/01/2001 09:58 AMPage 76 4 = . 1,2 , , . ( 3.10) 1//2. 3I , , . I , , . 4 (.4) , = . 1//2. . , (.4). AB^ . // ( ). . , : . 5 ( ). . (. , . 90)76 82. kefal04.qxd23/01/2001 09:58 AMPage 77 . , . 1//2 (. 5). .. = . , x ^ xAB . x//2 . 2, . = . 5I i) , ii) . 1 2 , , 1// 2//, 1//2. 1 2 , , . 1//2. 6 1 2 , . 1 . 2, //2 2, . 2. 777 83. kefal04.qxd23/01/2001 09:58 AMPage 78 4 I , . V 2 , . 8 V . 1, 2 (. 8) + 2. 1 2 , + 2 ( . 77). 1 2 , , . , , . ^ ^ 1, 1=2 + 2, . 1,2 . . I , . ( 3.18 2)4.3 978 , . . ( 2.6) , ^ . = //, , . . 84. kefal04.qxd23/01/2001 09:58 AMPage 79 4.4 ^^ xy xy x//x y//y, , . x y x y . 10 . : (.11), . (.12), . (.13), . : , , , . 10 11 12 13H 1 2 . (i) . (ii) . 14 ^ ^ (i) x, y . ^ ^ ^ ^ = = . = ( ). 2 2 = . x y . x//By. ^ (ii) z , z^Ax ( ). x//By, Az^By. 79 85. kefal04.qxd23/01/2001 09:58 AMPage 80 44.5 . . . , . 15 , , , . x y , , . , =. =, . =, . , (O,OA) . . . , , . , .80 86. kefal04.qxd23/01/2001 09:58 AMPage 81 ^ ^ . I ^ ^ ^ ^ ^ ^ + = + + 2. ( 4.2 IV) 2 2 ^ , = . ^ , = . ^ . , . , . 16H . . . , , , , (.17,). , . 1781 87. kefal04.qxd23/01/2001 09:58 AMPage 82 4 ^^ x y , . x y , : ^ ^ ^ ^ + + ^ ^ x + y B = = 2 2. 2 2 , ^. , . A 1. i) - ; ; ^ ii) ^ ; 2. .3. = 120 = 60 + xx//yy. 1. - , . . ^ 2. xOy . x y , . ^ 3. xOy . y x . OA = OB. 4. (AB = A) . (,) , //. 5. , : = =. //. 6. (,) . x^OM. x//. 1. (AB = A) 4. (5) . 5. : i) , ii) , iii) , iv) . .82 . x^ = . ^ . 2. . // . = +. 3. x. //x . = . 4. , 88. kefal04.qxd23/01/2001 09:58 AMPage 83 . = +. 5. // //. . 1. , x. . (,) , (,) x , //.2. x y. , x,y , = ^ = . . 3. , . = . 4. . ^ , . : i) . ii) + = . iii) : + ) BE = = , 2 ) = = . 24.6 , (.18). . 2 . , .. , xy//. ^ ^ = (1) = (2), xy . ^ + + = 2 (3). (1), (2) (3) ^^^ + + = 2. 18 i) K . 83 89. kefal04.qxd23/01/2001 09:58 AMPage 84 4 ii) , , . iii) . iv) 60. ^ ^ ^ ^ ^ i) ++ = 2 + = 2, ^^^^^ ++ = + ^^^ = + .ii) iv) . 194.7 . ^ ^ xOy = xOy= x ^ Ox Oy ^ Oy. ^ ^ ^ ^ = =1 1 = 2 (). , = . 20i) . ii) . i) , (. 20) + =2, ~ + = 2, =~, = . . ii) , (.20) + = 2, + = 2, = . .84 90. kefal04.qxd23/01/2001 09:58 AMPage 85 4.8 - . , . 2 . (25) . ^ ^ ^ ^ ^ O1+O2+O3+O4+O5 = 4 , , : ^^^ 21^^ ++ ++ = (25 4) ., . 2 . ^ ^ ^ ^ O1+O2+O3+ ... +O = 4 : ^ ^ ^ ^ 1+2+3+...+ = (2 4) . : - 2-4 . 22 . 12... , .. 1 . -2 , , 12 1 1, . -2 2(-2)=(2-4) , : - 2-4 . 23 - 4 . ^ ^ 1 + 1 = 2 ^ ^ 2 + 2 = 2 .... .... = .... : ^ ^ + = 2^ ^ ^ ^ ^ ^ (1 + 2 + ... + ) + (1 + 2 + ... + ) = 2 ^ ^ ^ (1 + 2 + ... + ) + (2 4) = 2 ^^ 24^1 + 2 + ... + = 4. 85 91. kefal04.qxd23/01/2001 09:58 AMPage 86 4H 1 x ^ . , Ax//B. ^ ^ (i) = = = . ^ ^ ^ ^ ^ ^ = + = 2, = 1 = = . . 2 x//, . ^ ^ ^ ^ (ii) x// 1 = ( ) 2 = ( ^^^ ^ ). = ( 1 = 2), =. 2 ^ ^ . ^ (i) 90 + . 2 ^ (ii) . 2 ^ (iii) 90 . 2 A . ^ ^ ^ ^ . (i) : ^ ^ ^ ^ ^ ^ BI + 1 + 1 = 180 BI = 180 1 1 ^^^^ ^ BI = 180 BI = 90 + 90 2 2 2 2 ^ ^ ^ ^ ^ BI = 90 + ( + + = 90). (1) 2 2 2 2 (ii) . ^ ^ ^ : = 90 BI = 90 + I (2) ( ). ^^ (1) (2) I = . (3) 2 ^ ^ ^ ^ ^ (iii) = 90, I + I = 90 I = 90 I . (4) ^ ^ (3) (4) I = 90 . 2 ^86 92. kefal04.qxd23/01/2001 09:58 AMPage 87 A 5. :1. .^^2. AB = A x , - (. ).3. - , ; 4. 60 . 5. : ) 180 ) 270 ) 360 ) 540 ) . 1. 2 . 3 ( ). 2. ( = ) ^ ^ = . 2 ^ . ( 2 4.8) ^ 3. ^ ^ . = 144 . ^ 4. ( = 90) ^ ^ ^ ^ . = =. = = x = 108. ^ . ^ 6. : = 90, , ^ ^ //. 20 .7. 900. . ^ 2^ 1. = 90 + . - = . ^ ^ 2. . ^ ^ ^ ^ i) = , ^ ^ ^ ^ ^ ^ ii) = 90 , = 90 + . 2 2 ^ ^ 3. ^ ^ ^ . = . 2 ^ ^ 4. , , ^^ + ^ = . 2 5. ^. ^ ^ = E . ^ 6. ( = 90) . .87 93. kefal04.qxd23/01/2001 09:58 AMPage 88 4 ^ 7. ( = 90). -^ , . = . 1. (=) . , = . ^. ^ ^ 2. . , . ^ ^ ^ = . 2 3. = . - - = . , , . 4. ( = ) . ^, . : i) = , ii) .5. , , = = . i) =, ii) ^. ^ ^ 6. ^ ^ ^ ^ . = . 2 7. 1,2 . 1 2, . , ; ^ 1. = 60 ^^ . = . 2. ( = = = ) 1 , = = . 3 ^. ^ 3. ( = 90) . x^, . =. ^ ^ 4. = = 90 =. =. ^E. 5. . : i) = .88ii) = . iii) = . 6. 1,2,3 ,,. 123.7. , (,) (,). ,, ,, (,) (,) ,, . : i) == , == , == , ii) AZ== , iii) == , = . 94. kefal04.qxd23/01/2001 09:58 AMPage 89 1. -.i) -; ; ; ii) -. . 2. (3 ). (, .) . (=), . : . : i) . ii) .89 95. kefal04.qxd23/01/2001 09:58 AMPage 90 4 . ( 23). , , : , ' ( V). : (1) , . (2) . (3) . (4) , . (5) , . (6) , . (7) , . (8) . (9) . (10) . ( ). -90 . . E . . , (3). ( ) (petitio principi), . [, ] , . , , , - ( 993) , . (836-901), . , , , . , . (6). , , . 96. kefal04.qxd23/01/2001 09:58 AMPage 91 (1 . ..). , , (6). . -T , - -. , . - (). , (908-946): . -. -, , , . . (2) , . , - - (1201-1274) -, . 13 . . -, -, - (973-. 1050) , . - ( -, 1263). . . 13- 14 . - (1236-1311), -. . . , . 13 18 . 13 . (Vitelo, 1225-1280) (1572). -. , . 14 . (Levi ben Gerson Gersonides, 1288-1344) , (1270-1346). 16 . . . (1472-1538), 1507. 1574 (Clavius (Schlussel), 1537-1612) . (6). -, . 17 . , . 1603 91 97. kefal04.qxd23/01/2001 09:58 AMPage 92 4 . (1548-1626), 1658 .. (1608-1679) , 1680 (1633-1711). 1693 . (J. Wallis, 1616-1703) VI , -, , (9), . - (G.G. Saccheri, 1667-1733). - . , , . . , ( ) . (J.H. Lambert, 1728-1777). 92 , , , , , . . . . , , . , (.. , , ..) . , . (L. Bertrand, 1731-1812), , 1778, .. (1752-1833), , .. (1764-1813), (Farkas Bolyai, 1775-1856), , . 98. kefal04.qxd23/01/2001 09:58 AMPage 93 I 2 . 1 . .u //. u . ( ) .u 1 2 . : : . . . ^1 ^2u 1//2 .. 1 2 : . . .93 99. kefal04.qxd23/01/2001 09:58 AMPage 94 4 . , .u : , . ( ) u ( ) ( ) 2 , : 2 , .u - 2 4 .94 . . 60. 100. kefal05.qxd23/01/2001 10:00 AMPage 955 - , . , .95 101. kefal05.qxd23/01/2001 10:00 AMPage 96 5Josef Alberts (, 1888-1976). : , , 1959. Solomon R. Guggenheim, .96 102. kefal05.qxd23/01/2001 10:00 AMPage 97 5.1 2.20, . (.1) , . , , . (.2), (.3). 1 25.2 . 3 , // //. : (i) . (ii) . (iii) . i), ii) , (. 5). : ^ ^ B1 = 1 = ( ). . ^ ^ B2 = 2 = ( ). , , = ^ ^ ^ ^ = . = = = + . 4 5 iii) , . : 97 103. kefal05.qxd23/01/2001 10:00 AMPage 98 5 = ^ ^ B1 = 1 = ( ). ^^1 = 1 = ( ). , , , = =. 6I . . I (.7). 7 8 (.8) , . , (.9). () : : (i) . (ii) . (iii) . (iv) . 9 1098 . , , . (i) = = (.10). , , ^ ^ = , = . B1 = 1 = ^ ^ B2 = 2 = , // //, . 104. kefal05.qxd23/01/2001 10:00 AMPage 99 (ii) // = (.10). ^ ^ , = , B1 = 1 = . , (i), . ^ ^ ^ ^ ^ ^ ^ ^ (iii) = = B = = (.11) +B++= 4 2 + 2 = 4 + = 2. , ^ ^ ^ ^ + = 2, // + B = 2, // , . (iv) = = (.12). , . , (i), // //, . 11 12A 1. -, ;5. : i) . ii) . iii) . iv) . ( x ). 1. . ^2. ; 3. .4. . . = . 2. . , = , . 3. , , . : i) . ii) , . 4. . . , = . 1. (AB = A) - . // ( 99 105. kefal05.qxd23/01/2001 10:00 AMPage 100 5 ) // ( ). + = . 2. . // ( ). //. 3. . = = . , . 4. . , = = . i) = , ii) , . 5. , . 1. , ,, = = . i) , ii) , , . 2. = , = . ^ = 90. 3. . = = . , . 4. ( = ) . = . . 5. , , . , ; , , - , . : , .5.3 . , ( ), . 13 .100 106. kefal05.qxd23/01/2001 10:00 AMPage 101 . (.14). ^ ^ ( = = 90, , = ), = . , : 14(i) . (ii) . (iii) . (iv) . (i) . ii) = . ( = , = , ), ^ ^ ^ ^ ^ ^ = . + = 2, = = 1. , . (iii) , 4. (iv) , . 155.4 . . 16 (i) . ii) . . , ^ ^ . ^ . ^ ^ ^ B . 17101 107. kefal05.qxd23/01/2001 10:00 AMPage 102 5 , : (i) . (ii) . (iii) o . (iv) . 18 (i) (ii) . (iii) ^. , . E, , ^. , = . . ^ (iv) . ( ), .5.5 . . , : (i) . (ii) . (iii) . (iv) , , . 19 , .102 108. kefal05.qxd23/01/2001 10:00 AMPage 103 , : (i) . (ii) . (iii) . (iv) . (v) . (vi) . A 1. i) , ii) , ; i)iii) ,5. x : i) . ii) . iii) .ii)iv) . 1. AE^ iii)2. : i) ii) 3. ; i) ii) iii) 4. , : i) ii) iii) ^AB. . 2. B = 2A. , , . 3. , . 4. , . 5. . , = = = . . 6. . , , , , , = = = . .103 109. kefal05.qxd23/01/2001 10:00 AMPage 104 5 ^ 1. , 1. = 45. . , . .2. , , = . i) = , ii) ^. 3. , . , . 4. , . ( ). . 2. ^. ^ , = . 3. : i) ( ), ii) , , ( ). 5.6 I . 20 , , (.20). // = . 2 EZ = . , . = // , = // , = . , : (i) // // (ii) = 2 = = . 2 I , .104 110. kefal05.qxd23/01/2001 10:00 AMPage 105 (.21). . . Z , , // . , , , . . 21 I () , . 1, 2, 3 1 , , , (.22). 2 1, 2, 3 , , , = . // . , = (1) = (2). // . , = (3). (1), (2) (3) = . 22 1 2 = , 1 2. 1 2, 1 2 , M = = . 2 , 1 2, + = = , M = = . 2 (// = , // = ), // 1, 2. , . 23105 111. kefal05.qxd23/01/2001 10:00 AMPage 106 5 : 1 2 1 2, . 1 2. 1 . , , , , , , . . (.24). , = // (1). 2 24 = // (2). 2 (1) (2) = // , . (.24). 24 2 (.25). x , , . , , . , . 105, = = .106 25 112. kefal05.qxd23/01/2001 10:00 AMPage 107 5.7 2 . 3 . . ^ ^ ^ ^ 1 + 1 + 2, . , i) 2 , = ii) = . 3 i) = . , E = // (1). 2 = // (2). 2 (1) (2) // // , (3). , = . , , ( ) . ii) = = , = = 2. 2 2 (1) (3) = = = 2. 2 2 (2) (3) = 2. , , . = + = 2 + = 3. 1 2 = , = . 3 3 2 2 = = . 3 3 26 . .107 113. kefal05.qxd23/01/2001 10:00 AMPage 108 5 : 2 . 35.8 , , , . , , , (.27). , : = = , = = = = . , , . 27 28 , , .108 . , . , , (.28). , , , . , , , ( , ) . , , . . , , , , , . 114. kefal05.qxd23/01/2001 10:00 AMPage 109 .. , . , , .5.9 29 . ^ ( = 90) (.30). = . 2 . , // . ^, ^. , , = , = . 2 , : 30 , . (.31). ^ = , . 2 ^ ^ = = , 1 = (1) 2 ^ ^ = , 2 = (2). ^^^^ (1) (2) 1 + 2 = + , ^^^^^^^^ 31 = + . + + = 2, 2 = 2 = 1.109 115. kefal05.qxd23/01/2001 10:00 AMPage 110 5 30, . 32 33 ^ ^ ( = 90) = 30 (.32). = . 2 ^ ^ = 30, = 90 30 = 60. ^ = = . 2 = = 60, 2 . = = . 2 , = 2 ^ (.33), = 30. , = = = ( 2 = ). , 2 ^ ^ = 60. = 90 60 = 30. A 2. .1. x y.3. ; 4. = .110 116. kefal05.qxd23/01/2001 10:00 AMPage 111 ^5. ( = 90) ; . 1. , . 2. . , , , . 3. . , M = . ^ 4. ( = 90) ^ = 30. , , =. 5. = , = . ^6. ( = 90). - . ^, . ^. ^ ^ 7. ( = 90) = 30 , . = . . ^1. ( = 90) . i) , , ^ ^ = = 90. ii) , = . 4 2. . , = . 4 ^ ^ ^ 3. ( = 90) . ^ ^ ^ = . 4. , , , . 5. , , , . 6. , . , = . 2 7. = . , i) = , ii) = . 2 ^ 8. = 30 . : i) = , ii) = . 3 9. , . , , ^. 10. . (), . ; ; ^^1. . , ^ ^ ^ E Z = . ^ 2. ( = 90) ^ . = 15, = 4 . (: ). 3. , , . //. ^ ^ 4. = 290 . = . . 5. , . , : i) // , ii) = , 2 ^ ^ iii) = . 2 6. , . =. , .111 117. kefal05.qxd23/01/2001 10:00 AMPage 112 5 7. - ^ , : i) . ii) . ^ 8. ( = 90) . , , : i) = , ii) ^, iii) . 5.10 . (.34) . . . 34 . 35112, , : + i) // , ii) = . 2 ( // ) (.35), . . : //, = (1). 2 //, = (2). 2 i) // , ( ). 118. kefal05.qxd23/01/2001 10:00 AMPage 113 ii) (1) (2) + + = + = . 2 2 2 . (.35). , (.36), // , = (3). 2 , , // , . (1) (3) : = = ( ). 2 2 2 365.11 . 37 , : (i) . (ii) O . (i) (// =). . ^ ^ ^ ^ ( = = 90, = = = ), = . ^ ^ ^ ^ + = 180 + = 180 ( ^ ^ ), = . (ii) (. 39) ( = , ^ ^ = ), = . 38 39113 119. kefal05.qxd23/01/2001 10:00 AMPage 114 5 , . (i) . (ii) . : (i) , (ii) . (i) (//) . , ^ ^ ^ ^ 1 = 1 = ( ). (ii) , . //. , , . 40A 1. x, y, .2. ; 3. ; ; 4. : A = 5x, ^ = 3x A = 60. :i) 10x ii) 11x iii) 12x iv)13x v)14x . 1. (//) 114 120. kefal05.qxd23/01/2001 10:00 AMPage 115 . , , , , , , . 2. (AB = A), . 3. (//) . , , , , , , , . 4. . , , . 5. (//) . = = . 2 6. . = . 2 1. (//) . ^ = 90. 2. (AB = A) . , = . ^ ^ ^ 3. = = 90 = 120. = 2 = , . 4. , . ^ , = 90. 5. (//) . ^. 6. . , , , , - . 7. , . 8. (//) = 3AB , . . ; 3 9. (//) = . 2 , , , , . , = . 10. , , , , , + + + = 4. 1. (//) = + . ^ ^ . ^ ^ 2. = = 90 = 2. , ^ ^ = 3 . 3. . , , , = ( ). ^ 4. ( = 90) , . . 3 ^ = , . 8 5. (//) , . i) = , = , 3 ii) , = . 4115 121. kefal05.qxd23/01/2001 10:00 AMPage 116 55.12 (, .) ( 4.2). (4.5 5.7 - 5.8) , , . , : . (,) . . , . . . . . ENIKE 1. (^) = 60, , , , . = . 2. 1 , 2 1. ^2. 2 , , 1 , = 2, ^ ^ = 3 . 3. , , = . 2 ^ , . : i) = . 2 ii) . 4. =2, ^ 60 (AE ^B). 116, , : i) , ^ ii) HE, iii) , ^ ^ iv) = 3. 5. . , , , ,, , + + = 3. 6. ( = ) . ^. , : i) //, ii) AH^BE, . 7. . . , . 122. kefal05.qxd23/01/2001 10:00 AMPage 117 8. . , = . 4 , a . : a i) OZ = , 3 ii) . 9. . , , : i) , , , ii) = ( ). 2 iii) , , . 10. , . , . 1. , . ;2. 4 . , - ; . 3. , . . i) iv) ii) v) iii) vi) 4. . , (: ). , . , ;117 123. kefal05.qxd23/01/2001 10:00 AMPage 118 5 . , . , ( 1): , . , , , ( )( ) ( ) ( ) ( ) ( ) 1: () , , () , () , . . .. ( 2).( )( )( )( ) 2: 118 124. kefal05.qxd23/01/2001 10:00 AMPage 119 . , . , , , .. -. . , ( 3). 16 . (1569) (Petrus Ramus Pierre de la Rame ( 4). e) . , . . : , ( ), . , ( ) . , , , . , , . . 3: 4: 119 125. kefal05.qxd23/01/2001 10:00 AMPage 120 5120 126. kefal06.qxd22/01/2001 06:16 PMPage 1216E . , . , .121 127. kefal06.qxd22/01/2001 06:16 PMPage 122 6 Leonardo da Vinci (1452-1519), Architectura de Vitruve, 1492.122 128. kefal06.qxd22/01/2001 06:16 PMPage 123 6.1 - ^ xAy (,R). : (i) (.1), , 2.18. (ii) (.2) , . ^ A . (iii) , (.3), . 1 26.2 : 3 . (O,R) A B. ^ A B. ^ ^ = 2. (i) ^ (.4). . ^ ^ , = . ^ , ^ ^ ^ ^ = 2 = 2. 4123 129. kefal06.qxd22/01/2001 06:16 PMPage 124 6 (iii) (.5) . ^ ^ = 2 . (ii) ^ (.4). ^ ^ ^ , = 2. (iii) (.4). I (i) . (ii) . (iii) . 56.3 : . 6124 ^ xy ^ (. 6) ^ ^ . = . 2 ^ ^ ^ , = = = 2 ^ ^ ^ . xy = . ^ ^ xy = . , . 130. kefal06.qxd22/01/2001 06:16 PMPage 125 1 , . ^ (i) xAy, (.7). x, Ay ^ 1,1 2,2 . , xAy ^ xAy , : ^ ^ ^ xAy = 12 + 12.^ (ii) xAy (.8). x, Ay 1,2 1,2 ^ . xAy ^ , xAy, ^ ^ ^ ^ ^ xAy = B2B 12 12 1, B2B 1212 1. ^ (i) xAy 12, ^ ^ ^ , xAy = 12 + 12. 1 1 2 2 . xAy = + 2 2 ^ 7 ^(ii) 212 12, ^^^^^ ^ 212 = xAy + 21 xAy = 212 121. 2 2 1 1 ^ xAy = . 2 2 8125 131. kefal06.qxd22/01/2001 06:16 PMPage 126 6 2 . (i) . . (ii) , . , , . (i) 1 2 , . 12 12 ^ ^ 12 = 12 = (.9). 9 . ^ ^ ^ xx = 2 ( 1x = 2x = 1) ^ ^ ^ yy = 2 . , xx = yy. (ii) A , = 1 (.9), ^ ^ 12 + 1x =2, x 2 .6.4 ^ (.10). ) . ) , . . 10126 132. kefal06.qxd22/01/2001 06:16 PMPage 127 . . 1 ^ , = , (. (iii), 6.2). , , , , ^ = (.11). x ^ ^ x = = ( 11 ) x , , . OB^Bx, x . , . ^ Bx , x = . Bx , . (,) (.11) ( ) . ^ ^ ^ = x = , x ^ , ^ ^ ^ = x = 2 x = 2 , Bx Bx. , ^ , x = 0. 12 1 , , (.12) ( ). = 1 , (.13). , , . 13127 133. kefal06.qxd22/01/2001 06:16 PMPage 128 6 1 ^ (.14). = , I ^ ^ . 14 15 1 . 2 (.15). : , , ,, , . : , . , : , , . : . . , , ( 3.17). : , , - .128 134. kefal06.qxd22/01/2001 06:16 PMPage 129 A 3. x y. 1. ; 2. , : . = , . = 2, . = 2, . = 90+, . . . 3. : ................................... 4. 1 1; 1. x y.^ 4. = 25, . ^ 5. = = 70, .^ 2. = 40, .6. , ; i) x y + z = 0, ii) x 2y + z = 0, iii) x y + z = 0,129 135. kefal06.qxd22/01/2001 06:16 PMPage 130 6 iv) x + y = 2z, v) . . , , . ^ ^ ^ = 100, = 125, = 135.7. - , , . ; . a . 1. ( ) 1. () . 2. . , . 3. , () . . 4. , , . //. 2. . , . ^ . 3. (), . . ^ , =. 6.5 , . . (,R) : (i) . (ii) K . 130 136. kefal06.qxd22/01/2001 06:16 PMPage 131 ^ ^ (i) H , , ^ ^ + = 4(.16). + = 2. (ii) (.. , ) ^ ^ ( ), , (.17). 16I .6.6 17 . ( ) . , , .. , , , . , , () . 18 , : (i) . (ii) . (iii) . 131 137. kefal06.qxd22/01/2001 06:16 PMPage 132 6 19 ^ ^ (i) + = 2. , , . , , ^ , (.19) , ^ . , , . (ii) ^ ^ = = . 20 , . (. 6.4) . , , , , , , . ^ ^ ^ ^ (iii) x = (. 21), + = 2, , (i). 21 1 , , , . () : (i) , . (ii) . () : (i) . (ii) . () (. .22). 22132 138. kefal06.qxd22/01/2001 06:16 PMPage 133 () (i) , , , , , , (. 23). ^ , , =. ==, , , , (,) . 23 (ii) + = + (1). ^ ^ , , , ^, H^ ^ (.24). = = (,) . . (,) . , + = + (2). 24 (1) (2) : = = + , , . 2 . ^ ^ (.25) = , ^ ^ ^ ^ ^ ^ = + = + = 2. ^ ^ + = 2, . 25 3 . 1111 26 (. 26). ^ ^ ^ ^ + + ^ ^ ^ ^ 1 11 = 1 = 2 . 1 11 = 1 = 2 2 2 ^ ^ ^ ^ + + + ^ ^ 1 11 + 1 11 = 4 = 2, 2 .133 139. kefal06.qxd22/01/2001 06:16 PMPage 134 6 A 1. :i) . ii) . () () . 2. : ^ ^ ^ ^ ^ ^ . = . = . + = 2^ ^ ^ ^ . = . = . 3. ; 4. i) ; ii) , ; . 5. ; 6. : , , , ; ^ 1. = 120^ ^ ^ ^ = 80. , -. 2. , . 3. o . 4. , .134 1. . . //. 2. () , , . . 3. , , . 4. . 1. (,R). OA^E ( Nagel). 2. (O,R) 1 2 . - , 1 2 . = . 3. ( Simson). 4. . , = . 140. kefal06.qxd22/01/2001 06:16 PMPage 135 : i) , ii) , iii) , .6.7 , , , , , . . . 1 ^ B=, = = , , . . (.27) ^ ^ = , = = . = , 1 , . , , 2 . 1 2. 27135 141. kefal06.qxd22/01/2001 06:16 PMPage 136 6 . = . = , // . 1, 2, . . , B = , = ^ = , .. . . 1 2 . , A 1 1, ( ), . =, , , , , . , . , 1, 2 . , , = , . : (i) (ii) , . , . , , 1, 2 , , 1, 2 . . : i) ii) iii) . . 136 142. kefal06.qxd22/01/2001 06:16 PMPage 137 2 = = , . . ^ = 1, = = (.28). . : (i) , (,), (ii) , . 28 . (.28) A . . ^ . = 1, , = (,). . , . , . 3 (,R) . . . , (.29). ^ , () . (O,R) (). , . 29 . (), (O,R) . . ^ ^ . = = 1, () . (O,R).137 143. kefal06.qxd22/01/2001 06:16 PMPage 138 6 . , () (O,R) () (O,R). : . 30 (K1) (2). . (.30) , 1, , 2, . . 4 (1,), (2,R) R 12R . . . (O1,), (2,R), , (.31). 1, 2 . 1 , 2 , 1 . 1^2 2 2 = 2 = 2 1 = R 1 . 31 . 2 R 1 1 1 . 2, 2 (O2,R) , 1, 1 (O1,) 2, 2 . .138 144. kefal06.qxd22/01/2001 06:16 PMPage 139 . 1 , , 1// = = 2 2 = R (R ) = 1, 1=//, .^ , 1 (2, R ) 1 . 1 2 . . . . 1 1 1 (2, R ). 1 (2, R ), 12R . , 12R (1,) (2,R). (1,) (2,R) O1O2R + . . .A 2. 1. : i) , ii) , iii) , iv) , v) , vi) , vii) .i) .ii) .iii) .iv) . v) . () () . 1. :i) ,139 145. kefal06.qxd22/01/2001 06:16 PMPage 140 6 ii) 10km . 2. : i) , ii) . 3. 4m . . 4. . 1. . 2. . ^ 3. xOy . ^ (A = 1) Oy Ox -. . 4. ^ (A = 1) : i) = . ii) = = . 1. . . 2. : ^ i) =, A = = . ^ ii) =, A = = . 3. , , ^ , , , , A . 1. . , x y . , y x . : i) ,, , ii) , iii) . 2. , . ^ A, . 3. ^ ^ = , = . 4. (,R) . , , . 1405. . , , , , . . ( Miquel). 6. , , , . , . 7. , , . , , . 8. . 1 , 2 , 3 , , , 1 , 2 , 3 1 , 2 , 3 , , , : i) 1123 , ii) 1112 , iii) Mi , Hi , Zi , i = 1, 2, 3 ( 9 uler). 146. kefal06.qxd22/01/2001 06:16 PMPage 141 i) . ii) . i) . ii) . iii) . , : i) . ii) . iii) . i) , . ii) . : i) . ii) . 141 147. kefal06.qxd22/01/2001 06:16 PMPage 142 6142 148. kefal07.qxd22/01/2001 06:17 PMPage 1437 . , . .143 149. kefal07.qxd22/01/2001 06:17 PMPage 144 7 , (, 1866 - 1944), 1923.144 150. kefal07.qxd22/01/2001 06:17 PMPage 1457.1 . . , , , , . , . .7.2 , ( 2). x, 1 = 12 = ... = 1. 1, 2, ..., 1 1, 2, ..., 1 . , , 5.6, . 1, 12, ..., 1 . , , . 17.3 . 2.8, AB= , , =. = . , , = 2145 151. kefal07.qxd22/01/2001 06:17 PMPage 146 7 1 . ( ) = . 1 = = . 3 = q AB. . q = , 1 . = q AB. q = 0 . , = . , . . , : = = . . , q = , = q . q , q = . H . () .146 152. kefal07.qxd22/01/2001 06:17 PMPage 147 . . . .7.4 - , , , : = (1). , = = . (1) , , . . . , , , . , . = . . . . , , . :O = 2 = = = , H = = = = , = = + +L+ = =L= = + +L+ , (.. ), .147 153. kefal07.qxd22/01/2001 06:18 PMPage 148 77.5 H . , . , , . , . . 2 . . , . : , . , , .7.6 4 7.2 . . xy . xy, . : 1) . , = . . .148 154. kefal07.qxd22/01/2001 06:18 PMPage 149 , =, : = , . =, = = = 1 + + 1 +1 1 = = = . +1 +1 2) , . , = . , . (i) =1, . =1, . (ii) 1, 1 , , (.5). : = = = = 1 1 1 5 = = = 1 1 1 = . 1 149 155. kefal07.qxd22/01/2001 06:18 PMPage 150 7 (iii) 1 1 , , (.6). = 1 6 , , MA MB = =. MB MA1 = . 1(iv) ) , , . ) , , . ) , , . , = 1 1. = , . . . 7A 1. :i) , ii) , iii) , , iv) . 2. = 10 = 2. :150i) , ii) , iii) , iv) . 3. 1 = . 23 : i) 2 ii) 3 iii) 2 v) . ( ).2 iv) 3 156. kefal07.qxd22/01/2001 06:18 PMPage 151 4. AB = 12 cm . , . 1. 4, 3, 2. . 2. 1 . . 3 3. 6, 3, 4. 65cm, . 1. 2, 3 4. .2. , , , = 6cm, = 12cm, = 2cm. , . 3. , AB = , 3 . 47.7 5.6 , . . , . : 1//2//3 , = = (.8). 1//2//3 , = = (.8). 8151 157. kefal07.qxd22/01/2001 06:18 PMPage 152 7 9 (i) (.9) , , = = (1), , . . 1, 2. 1 , 2 , . = = (2). A B (1) (2) : = = B = = , = = (3). (3) : + = = + = = . (ii) , . . . 1 2 1 2 , , . 1 2 , = , 1 2 (.9). .152 158. kefal07.qxd22/01/2001 06:18 PMPage 153 // (.10). ////, = . . 10 , . . // (.12). : = = . //, (1). = 11 , , = (2). //, BZ = B = (1) (3) (3). . = = 12 . .153 159. kefal07.qxd22/01/2001 06:18 PMPage 154 7 1 ( ) , , x, = . x zOy. z .^ = , = y = . , y . = x OA O = AB . = x 13 x x x = = = , x x . 2 ( ) , , , , . x, =. x ==. . , , : = = . , :154 14 160. kefal07.qxd22/01/2001 06:18 PMPage 155A . = = =, , , . 15 , , (.16). , . = 16 , = = . (,) (,) . . , , .155 161. kefal07.qxd22/01/2001 06:18 PMPage 156 7 A 1. x y.4. = . ; 5. = 4, = 2. , (,) (,).1//2 1. //, // //. = . 2. // //// .2. , . i) = , ii) 2 = . 3. , . . . 4. , . 3. : = i) ii) // () () .156 = . 5. . , , . //. 6. , . 2 . //. 7. , = 162. kefal07.qxd22/01/2001 06:18 PMPage 157 , . = . 8. (//) . , , , . 9. =12m h. 15m , . 10m. h ; 1. , , - , . 2. AB = a x, y, , 4x = 6y = 3. 3. (,R) . , //. 4. 1 , = . 5 , = 3. 5. , . = 1. 6. , = = . . : i) . ii) //. 7. (//) , . , . = . 1. , = . , , , , . 2. , . = . 3. , , , , . , , . , , , = . 4. , . 5. , , . , , ^AB.157 163. kefal07.qxd22/01/2001 06:18 PMPage 158 77.8 , , . ( ) . , , . = 17 (.18). , . = (1). , = . : ^^^^^^1 = 1 ( ), 2 = ( ), 181 = 2 ( ), ^^ 1 = = (2). (1) (2) = . , , : = ^ .158 164. kefal07.qxd22/01/2001 06:18 PMPage 159 , ,,. .18 . : = = = + + + = . = . + + ( ) , ., , : EB . = (.20). , . = (1). , = . : ^ ^ 1 = 1 ( ), ^ 19 20^2 = 1( ), ^ ^ 1 = 2 ( ),^^ 1 = 1 = (2). (1) (2) = . 159 165. kefal07.qxd22/01/2001 06:18 PMPage 160 7 . ^ ^ ^ = 0 . $ $ A1 + B 2180 o . $ $ $ A + $ $ $ = 2 = 180 - , A1 = 2 2 ^ ^ ^ ^ ^ ^ ^ 1 + 2 = 180 + + = 180 = 180 180. 2 2 2 2 = , . ( 1 4.8) , : = , ^ . ,,. .20 . : EB = = - - EB , = - = . = . ^ , , , = = , = . .160 21 166. kefal07.qxd22/01/2001 06:18 PMPage 1617.9 1. , = (1). . = = (2) = = (3). 22, , ^ . = 90, ^ ^ . . $ : . = 90 . NA . = NB //, = (2) = (4). //, = (3) = (5). (4) (5) NA N , =, = NE NZ . $ $ = 90 //, //, EBZ = 90 , $ B , = = (6).161 167. kefal07.qxd22/01/2001 06:18 PMPage 162 7NA = . NB . (4) (6) , , 2, 7.7 . . = 1, = 1 = . , . A , . . , , , , , , .A 4. 1. , , ^ , , . 2. = , 2 + = 2. 3. ; ; ;162 . (,) (,) ; .5. =1 : i) ii) iii) iv) . ( ). 168. kefal07.qxd22/01/2001 06:18 PMPage 163 1. AE A . = 2 . EM 2. =6, =10, =9. , ^ , . ^ 3. = 90 ^ . AMB , = . 4. ^ ^ AMB AM , //. 5. , , : = 1. . 6. ( = ) (,R). , = . 7. , , . , , . 8. 20m 36m. , , , 12m. . ^ ^ ^ 1. xOy = yOz = zOt =45 , x, t , OA = O. , y, z , 2 = . 2. , , , . = . 3. , . i) , , , . ii) + = 2 , : + ) // ) = , , 3 , . ^ ^ 4. , , + 2. 5. (//) $ . , , : i) = , ii) = . 1. (,R). , , , , , . 2. , = . . 3. B (,R). MA B, = , , MB . 4. , , , = . , ^ . 5. =, = = , , . 163 169. kefal07.qxd22/01/2001 06:18 PMPage 164 7 1. (,R) (,) - . . 2R = . R+ 2. , , , . + = 1 . 3. i) . , , , , . = 1. ii) Ceva. , , , , , . , , : = 1. , . 4. . ^ , = 1. 5. . , = .6. (,) (,) - , , : i) 2 = , . 2 1 1 ii) = + . 7. x ^ , , , x , = , , . ^ 8. xOy . , , : i) , 2 ii) AB = 3 iii) = , , . 9. (,R) (,) . , , : 3 i) = , ii) = . 4 10. , , , + + 6. 1. . 2. , . , . 3. . . , , , . . ( ).164 170. kefal07.qxd22/01/2001 06:18 PMPage 165 , . , , , , , .. , , , .. . . . , . . 1/2, 1/3 2/3 , . (1/2, 1/3, 1/4 2/3), , , 1/, . . , . . . - , . () . . . . , .. , : , . . , , . , , , , . , , , A = E, . , , . = , B = . . , . ( ), VII . , . , . 1, 1. , 1. , 1 , 2. 165 171. kefal07.qxd22/01/2001 06:18 PMPage 166 7 . . . 2 . . . , (. ) . . V . , (, , ). . , , , : 1) 2) = = , 3) . , , , . , , . , . , ( ) . , ..., 166 (, , 1). , , , . : 1. . 2. 1/3 . 3. 1/3 . , , , .. . ,. , (, V, 4). (, , ). . , . .. , . . a, , , , . 172. kefal07.qxd22/01/2001 06:18 PMPage 167 . . , , = . q = (0, 0), . q = , = q, q = . = = . . . , , , = = , , , , . . : - . - , , . , , = , . - , . - . - : 1//2//3 , = = . : 1//2 = , //1//2. 167 173. kefal07.qxd22/01/2001 06:18 PMPage 168 7- A A //, = = = . A : = , //. , . = = , = + + . = = , = ^ A, , . , . 168 174. kefal08.qxd22/01/2001 06:19 PMPage 1698 . . , .169 175. kefal08.qxd22/01/2001 06:19 PMPage 170 8 Khajuraho , 10 - 11 .170 176. kefal08.qxd22/01/2001 06:19 PMPage 1718.1 , , . , AB B 1 = = = = . AB 2 . , , AB= 3. , . , = = = 3. , , ( ) . . 1 2 , . , . . . ( 171 177. kefal08.qxd22/01/2001 06:19 PMPage 172 8 ). = = = = = , , , : ++++EA = . ++++EA8.2 3 , ( 3) . I (1o ) , . ^ ^ ^ ^ = , = , ^ ^ = . X , , , = . . , //, = = ^ ^ ^ ^ ^ , = = . , . . 4I (i) , . (ii) . (iii) , , .172 178. kefal08.qxd22/01/2001 06:19 PMPage 173 II (2o ) , . ^ ^ (.4) , = = . , , = . . , . = = . , = . = = . , . II (3o ) , . (. 4), = = . , , = . , . = = = = . . , . . . , .173 179. kefal08.qxd22/01/2001 06:19 PMPage 174 8 = = . = = , = = . AB . H 1 , M , , = . M = = = . M , , = , = , //. o, = . . , = = = = . 5 = = . = = , . 174 180. kefal08.qxd22/01/2001 06:19 PMPage 175 I (i) . (ii) . (iii) . E 2 ; 6 ^ ^ (. 6). (. ) , ^ ^ , .. = 19 = 43. K , .. =12 . , 1,8 . . AB=6 cm. ^ ^ xy = 19 xBz = 43. O Ay Bz . . 3,3 cm. O = =200. = = 6,6 . , 8,4 . .175 181. kefal08.qxd22/01/2001 06:19 PMPage 176 8 , . , , () , ( ) , . 7 , . 8 (. 8). : 1. , . , . 3 . , . 9 . , , , , , , , , . ^^ . , 1 = 1 ^ ^ ^ ^ = . 2 = 2 ( = ) = . .176 182. kefal08.qxd22/01/2001 06:19 PMPage 177 4 =. = 2R, R . . ^ ^ ^ ^ , = = 1 = . = = 2R. 10 ^ = 1 , = = 2R .A 4. . 1. i) , ; ii) , ; 2. ; 3. = 3. . 5. 3cm, 4cm 5cm. 24cm. ; 6. , ; ;177 183. kefal08.qxd22/01/2001 06:19 PMPage 178 8 7. 1,70m. ( ; . E ^ 1. ( = 1). ^. : i) , ii) = . 2. 1 , = 3 2 = . : 3 i) , ii) = 3. 3. ,,. 1 . 15 ; 4. 24m. , , 2m 3m. . 5. . : i) 2 = , ii) 2 = , iii) = . 6. (O,R) o x y . = . 1. . , =3m, =2m 178 ).2. :i) , , ii) . 3. (1,R1) (O2,R2) . 1 2 , 2 = 1 2 .4. , = = . 5. , . = . ^ ^ 6. ( = = 1) . . 1. , . ^ 2. xOy . x, Oy . MB d = , d, d M d x, Oy , . ^ 3. ( = 1) ^ . A 184. kefal08.qxd22/01/2001 06:19 PMPage 179 ^ . //. ^ ^ 4. = 1 . 2 = .5. . : i) AB A = , ii) 2 = .E 1. (,R) 5. (,R), 2. (,R) . = . 3. A , , ( ) , , . 4. .ii) d 2 d = , 2R . . 2 = . 2 : i) d d = , 2Riii) d, d1, d2 , , d 2 = d1d2 .6. : . 1. , .2. . , . 2.179 185. kefal08.qxd22/01/2001 06:19 PMPage 180 8 O . .180 186. kefal09.qxd22/01/2001 06:20 PMPage 1819 , . - , . . .181 187. kefal09.qxd22/01/2001 06:20 PMPage 182 9 (, 1878 - 1935), , 1915.182 188. kefal09.qxd22/01/2001 06:20 PMPage 183 9.1 . . , .. , . , , , , .9.2 1 , . . 2 = 2 = . = , , ^ ^ ^ = = 1 . A2 = . 2 = A2 = : I 2 , . () , . 183 189. kefal09.qxd22/01/2001 06:20 PMPage 184 9 (.2) AB2 + A2 = 2 2 = 2 + 2 : AB 2 = B 2 = . : 2 + 2 = + = ( + ) = = 2 . ( ) 2 + 2 = 2 , ^ = 1. ^ x, Oy xOy = =.
