13ｨｪｨ 0002ｨｪｨ 00ｨｦ1 702ｨｮ ｨｰｨ ｨｲ071 7ｨ 06ｨｦ02, ISSN

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2 13ｨｪｨ 0002ｨｪｨ 00ｨｦ1 702ｨｮ ｨｰｨ ｨｲ071 7ｨ 06ｨｦ02, ISSN ｨｦ : ｨｪ , , ｨｦ ｨｪ. ｨｰ ｨ.ｨｦ ｨｮ.ｨ. ( ) ( ) ｨ ｨｮ.ｨｦ ｨｦ.1 7. ｨｮ ｨ.ｨｦ (52), c ｰｨｪ ", , , ｨｪ , ｰｨｪ ", ｨｲ ｨｰ ｨｰ03, , : ｨｮ ｨｹ : ｨ., ｨｦ.ｨ., ｨｦ ｨ , 05.ｨｪ , , SS

3 13ｨｪ ﾁ ﾁ ﾁ (52), { ｨｮ ﾂ ﾁ. ｨｮ ﾂ ﾁ ｨｮ ﾁ ﾂ. 04. ｨｪ ﾁ. ｨｮ ﾁ ｨｦ ｨｨ ｨｦ. ｨｪ ﾁ ﾁ ﾁ ｨｮ ﾁ, ｨ. ｨｮ ﾁ09, ｨｮ. ｨｮ0809 ﾁ0103 ﾁ. ｨｬ ( ) 36 ｨｦ

4 ｨｮ ﾂ ﾁ ﾂ ﾁ0705, GH, GL, HL (G , H ﾁ , L ) ｨｦ ﾁ ﾁ ﾁ ｨｮ ﾁ ﾁ , ﾁ I ﾂ 縺 , , ﾁ ｨｬ. ｨ ﾁ0308 罟ｲ ﾂ03 ﾁ ﾁ ﾁ03: , MN ﾁ ﾂ , MN ? , ﾁ070506? ﾁ ? ﾁ ﾁ ﾁ ﾁ , ﾁ ｨｪ0401 ﾁ : , , , ﾁ ﾁ , , (04. ﾂ ﾁ, ｨｪ ﾁ, 06. ｨｨ. ｨｰ ﾁ, 縺 ﾁ ﾁ , ｨｪ.: 縺 , 1968, ) ﾁ ﾁ , ﾁ ﾁ ｨｦ , ﾁ ﾁ ｨ ﾁ , 0803 ﾁ ﾁ ﾁ ﾁ ﾁ ﾁ ﾁ ﾂ03 ﾁ : ( ﾁ ). G ( ). L ﾁ ﾁ ( ) ﾁ ﾁ ｨ ｨｮ = b ABC, a BC = a. ﾂ ﾁ ﾁ ｨｰ ﾁ ﾁ ﾁ 縺 : ﾁ ﾁ , 09 縺 ﾁ ﾁ ﾂ ﾁ ! 2

5 13ｨｮ II. ﾂ ﾂ02 ﾁ , ﾁ , , ﾁ ABC AB = c, AC = b, BC = a E BC, D AB. ｨｦ03 ﾁ CE=EB = 1 07 AD=DB = AE 07 CD ﾁ F ( ) O ﾁ ( ﾁ , ) e 1, e 2, e ﾁ , ﾁ ﾁ ﾓ1 6ﾓ1 6ﾓ1 OA, OB, OC ﾁ ﾁ ﾁ ﾁ : 6ﾓ1 OF = 1 OA e OB e OC e : (1) ｨｰ ﾁ ﾁ , OF 6ﾓ1 = OC e3 + CF. 6ﾓ1 6ﾓ ﾁ ﾁ CF, CF=F D = x ﾁ ｨｮ B 07 F ﾁ: S CEF = S 1, S EBF = S 2, S BDF = S 3, S DAF = S 4, S ACF = S ﾁ ﾁ ﾁ ﾁ ﾁ , ﾁ : 61 S 1 = S 1 ; 2 S = S 2 ; 3 S 1 + S 5 = S 2 + S 3 + S 1 ; S 4 + S 5 = S 1 + S 2 + S 2 : 3 S , 1 + S 2 = x 07 S 5 = x , x = 1(1 + 2 ) S 3 S 4 2 CF CD = x x + 1 = 1(1 + 2 ), ﾓ1 CF = 1 (1 + 2 ) 6ﾓ16ﾓ1 CD ﾁ ﾁ ﾓ , AD = 2 BD, ﾁ , AD = 2 6ﾓ16ﾓ1 BD ﾓ1 AD = OC 6ﾓ1 6ﾓ1 6ﾓ16ﾓ1 6ﾓ16ﾓ1 6ﾓ16ﾓ1 6ﾓ1 6ﾓ1 6ﾓ1 OA + CD 07 BD = 6ﾓ1CD + OB 6ﾓ1 OC: ｨｦ ﾁ ﾁ ﾓ16ﾓ1 CD: 6ﾓ1 6ﾓ1 6ﾓ1 6ﾓ16ﾓ1 OA + CD = 2 OB 6ﾓ1 (1 + 2 ) OC : 1 + 2

6 134 ﾂ ﾁ 6ﾓ1 CF = 1 6ﾓ1 6ﾓ1 6ﾓ1 OA OB 6ﾓ1 1 (1 + 2 ) OC ; ﾁ ﾁ 6ﾓ1 6ﾓ1 OA = OA e 1, OB = OB e2, OC 6ﾓ1 = OC e3, OF 6ﾓ1 = OF e3 + CF 6ﾓ (1). ｨｮ03 ﾁ ﾁ O 07 C, (1): 6ﾓ1 CF = 1 CA e CB e : (2) (2) III ﾁ ﾁ , ﾁ ﾁ ﾁ ﾁ ｨｲ ﾁ GL ｨｰ ｨｰ ｨｮ , ﾁ ﾁ A 07 C, : 1 = 2 = 1, ﾁ (2) CG 6ﾓ1 b e = 1 + a e , ﾁ ﾁ A 07 3 C ﾁ ﾁ : 1 = b=c, 2 = b=a (2) : 6ﾓ1 CL = ab( e 1 + e 2 ) a + b + c ﾓ1 GL = 6ﾓ1 CL 6ﾓ1 CG, 6ﾓ ﾓ1 (2a 6ﾓ1 b 6ﾓ1 c)b GL = 3(a + b + c) e (2b 6ﾓ1 a 6ﾓ1 c)a 1 + 3(a + b + c) e 2: (3) , GL ｬ AC 6ﾍ2 6ﾓ1 GL ｬ e 1, b 6ﾓ1 a 6ﾓ1 c = 0, b = a + c 2. ｨｦ080909, , GL , , ﾁ02 ﾁ , ﾁ ﾁ GL ﾁ ﾁ ﾁ , GL ﾁ , ﾁ ｨｮ , < b < C cos C ﾁ: ( ) c 2 = a 2 a + c 2 ( ) a + c + 6ﾓ1 2a cos C: 2 2 5a 6ﾓ1 3c ﾂ : cos C =. ｨｮ ﾁ , a > 3c , a = 3c , a < 3c ｨｦ ﾁ03 ﾁ

7 13ｨｮ ｨｰ ｨｲ ﾁ GH ｨｰ ｨｰ ｨｮ ﾂ CG 6ﾓ1 b e = 1 + a e 2 3 ﾁ ﾁ A 07 C, ﾁ ﾁ : (2) : 1 = ctg C ctg B ; ﾁ ﾁ ﾁ ﾁ , = ctg A ctg B : 6ﾓ16ﾓ1 CH = b ctg B ctg C e 1 + a ctg A ctg C e 2 : ﾁ ﾁ ﾁ ﾁ0301: ctg A ctg B + ctg A ctg C + ctg B ctg C = 1; (4) A, B, C ABC ﾓ16ﾓ1 GH = 6ﾓ16ﾓ1 CH 6ﾓ1 6ﾓ1 6ﾓ16ﾓ1 GH = b ( ctg B ctg C 6ﾓ1 1 ) e a CG, 0803 ( ctg A ctg C 6ﾓ1 1 3 ) e 2 : (5)

8 136 ﾂ ﾁ , GH ｬ AC 6ﾍ2 6ﾓ16ﾓ1 GH ｬ e 1, ctg A ctg C = 1=3, tg A tg C = 3. ｨｦ080909, , GH , , ﾁ ﾁ ﾁ, , ﾁ ﾁ , , ﾁ ﾁ , GH ﾁ ﾁ ﾁ , GH ﾁ , ﾁ ( ) tg A tg C > 0, A 07 C ﾁ ﾁ (4) ctg A ctg C = 1=3, , ctg B = 2 3(ctg A + ctg C) > 0: ｨｲ B ABC ｨｲ ﾁ LH ｨｰ LH 6ﾓ1 6ﾓ1 6ﾓ16ﾓ1 = LG + GH, 0803 ﾁ (3) 07 (5) : 6ﾓ1 LH = b ( ctg B ctg C 6ﾓ1 a 2p ) e 1 + a , LH ｬ AC 6ﾍ2 6ﾓ1 LH ｬ e1, ctg A ctg C = b=2a, ｨｦ080909, ( ctg A ctg C 6ﾓ1 b 2p ) e 2 : (6) tg A tg C = 2p b : (7) , LH , , ﾁ ﾁ ﾁ, , ﾁ , ﾁ , ﾁ ﾁ LH ﾁ ﾁ , ﾁ 0609 ﾁ ﾁ03 (7) ﾁ 0609 ﾁ , ﾁ ﾁ , LH ﾁ , ﾁ ( ) tg A tg C > 0, A 07 C ｨｲ ﾁ ﾁ (4) ctg A ctg C = b=2a, , a + c ctg B = 2p(ctg A + ctg C) > 0: ｨｲ B ABC , , ﾁ ﾁ ｨｲ ﾁ , ﾁ (7) ｨ ﾁ tg A tg C = ﾁ ﾁ0203 ﾁ ﾁ ﾁ (7) ｨｦ : 2p b = a + b + c 2R(sin A + sin B + sin C) = = b 2R sin B 2 sin A+C = 2 cos A6ﾓ1C sin A+C 2 sin A+C 2 cos A+C 2 (sin A + sin C) + sin(a + C) = sin(a + C) 2 cos A6ﾓ1C = 2 + cos A+C 2 cos A+C 2 2 cos A+C = 2 cos A 2 cos C 2 cos A 2 cos C 2 6ﾓ1 sin A 2 sin C 2 = = 2 1 6ﾓ1 tg A 2 tg C : 2

9 13ｨｮ tg A = 2 tg A 2 1 6ﾓ1 tg 2 A ; tg C = 2 tg C ﾓ1 tg 2 C tg C 2 = x, tg A 2 = t ( t ) ﾁ (7) ﾁ ﾁ (3t 2 6ﾓ1 1)x 2 6ﾓ1 2tx + (1 6ﾓ1 t 2 ) = 0: (8) ABC , < t < < x < t = 1= ﾌ 3 6ﾍ2 A = 60 71, ﾁ (8) ﾁ , x = 1= ﾌ 3 6ﾍ2 C = ﾁ ﾁ ﾁ LH t ﾙ 1= ﾌ 3, ﾁ ﾁ (8) t, ﾁ ﾁ (8) : x 1 = t + ﾌ 3t 4 6ﾓ1 3t t 2 6ﾓ1 1 ; x 1 = t 6ﾓ1 ﾌ 3t 4 6ﾓ1 3t t 2 : 6ﾓ < t < 1= ﾌ 3, ﾂ x 1 < 0; x 2 > ﾁ ﾁ03 x 2 < ﾁ ﾁ t(1 6ﾓ1 t)(1 6ﾓ1 3t 2 ) > 0, ﾁ < t < 1= ﾌ = ﾌ 3 < t < 1, ﾂ x 1 > 0; x 2 > ﾁ ﾁ03 x 1 < ﾁ ﾁ t(1 6ﾓ1 t)(1 6ﾓ1 3t 2 ) > 0, = ﾌ 3 < t < ﾁ ﾁ03 x 2 < ﾁ ﾁ t(1 6ﾓ1 t)(1 6ﾓ1 3t 2 ) < 0, ﾁ = ﾌ 3 < t < 1. ｨｮ ﾁ , ﾁ (8) x = t 6ﾓ1 ﾌ 3t 4 6ﾓ1 3t t 2 : 6ﾓ1 1 IV ﾁ 0609 ﾁ ﾁ , ﾁ ﾁ , 05 ﾁ ﾁ , ﾁ ﾁ ﾁ ﾁ GL, GH, LH ﾁ ( ) ﾁ , ﾁ ﾁ ﾁ ﾁ ｨｲ ｨｮ ﾁ ﾁ , ﾁ GL ?

10 138 ﾂ ﾁ ｨｰ ｨｰ ｨｮ , GL ﾍ BC 6ﾍ2 6ﾓ1 GL e 2 = (3) , e 1 e 2 = cos C = (a 2 + b 2 6ﾓ1 c 2 )=2ab: (b 3 6ﾓ1 c 3 ) + (3a 2 c 6ﾓ1 3a 2 b) + (2ac 2 6ﾓ1 2ab 2 ) + (b 2 c 6ﾓ1 bc 2 ) = 0; (b 6ﾓ1 c)(b 2 + bc + c 2 ) 6ﾓ1 3a 2 (b 6ﾓ1 c) 6ﾓ1 2a(b 6ﾓ1 c)(b + c) + bc(b 6ﾓ1 c) = 0; (b 6ﾓ1 c) ( (b + c) 2 6ﾓ1 2a(b + c) 6ﾓ1 3a 2) = 0; (b 6ﾓ1 c)(b + c + a)(b + c 6ﾓ1 3a) = 0: b ﾙ c, 0803 b + c = 3a. ｨｦ080909, , GL ﾁ , , , GL, ﾁ ﾁ , , b < c c ABC, ﾁ ﾁ BC = a ﾁ cos C = (a 2 + b 2 6ﾓ1 c 2 )=2ab, C a = (b + c)= cos C = (5b 6ﾓ1 4c)=3b. ｨｮ ﾁ , b > 4c , b = 4c , b < 4c ｨｦ ﾁ03 ﾁ ｨｲ ｨｮ ﾁ ﾁ , ﾁ GH ? ｨｰ , GH ﾍ ﾂｨｮ 6ﾍ2 6ﾓ16ﾓ1 GH e 2 = (5) , e 1 e 2 = cos C: ( b ctg B ctg C 6ﾓ1 1 ) ( cos C + a ctg A ctg C 6ﾓ1 1 ) = 0: ﾁ a = 2R sin A, b = 2R sin B ﾁ090301, A = ﾓ1 (B + C) ﾁ ﾂ : ( 1 sin B 3 ) ( cos B cos C 1 cos(b + C) 6ﾓ1 cos C + sin(b + C) + sin B sin C 3 sin(b + C) cos C ) = 0: sin C

11 13ｨｮ ﾁ , ﾁ , ﾁ090301, cos(b + C) = = cos B cos C 6ﾓ1 sin B sin C. ｨｦ : 1 sin C(2 sin B cos C + sin C cos B) + cos C(cos B cos C 6ﾓ1 sin B sin C 6ﾓ1 cos B cos C) = 0; 3 1 sin C(sin C cos B 6ﾓ1 sin B cos C) = 0; sin C sin(c 6ﾓ1 B) = 0: B 07 C , 0803 B = C 6ﾍ2 b = c. ｨｦ080909, ﾁ , ﾁ GH , ﾁ ｨｲ ｨｮ ﾁ ﾁ , ﾁ LH ? ｨｰ , LH ﾍ BC 6ﾍ2 LH 6ﾓ1 e2 = (6) , e 1 e 2 = cos C: ( ) ( ) cos B cos C b cos C sin B sin C 6ﾓ1 sin A cos A cos C + a sin A + sin B + sin C sin A sin C 6ﾓ1 sin B = 0: sin A + sin B + sin C ﾁ 09 = 2R sin A, b = 2R sin B. ｨｦ A = ﾓ1 (B + C): ( ) cos B cos C sin B sin(b + C) cos C 6ﾓ1 6ﾓ1 sin C sin(b + C) + sin B + sin C ﾁ : sin B sin(b + C) sin(b + C) + sin B + sin C = ﾁ : cos C ( 6ﾓ1 2 sin B 2 cos B 2 2 sin B+C cos ( cos(b + C) cos C sin C + 2 sin B+C 2 cos B+C sin B+C 2 cos B6ﾓ1C = 2 cos B+C 2 = 2 sin B 2 cos B 2 B+C = 2 sin B 2 cos B 2 cos B+C 2 + cos B6ﾓ1C 2 cos B cos C 6ﾓ1 2 sin B 2 sin C 2 cos B + C 2 6ﾓ1 2 sin B 2 sin C 2 cos B + C 2 6ﾓ1 4 sin B 2 sin C 2 cos B + C 2 6ﾓ1 4 sin B 2 sin C 2 cos B + C 2 4 sin B 2 sin C 2 cos C 2 sin B 2 sin C 2 cos C 2 sin B 2 sin C 2 cos C 2 ( ) 6ﾓ1 ( 6ﾓ1 ) sin B sin(b + C) = 0: sin(b + C) + sin B + sin C 2 cos B 2 cos C 2 cos B+C 2 = sin B 2 cos B+C 2 cos C : 2 cos(b + C) cos C + 2 sin B 2 sin C 2 cos B + C 2 (1 + cos C) + cos B cos 2 C 6ﾓ1 cos(b + C) cos C = 0; cos 2 C + cos C(cos B cos C 6ﾓ1 cos B cos C + sin B sin C) = 0; 2 cos 2 C cos C = 0; sin B 2 cos B 2 sin C 2 cos C 2 cos B 2 cos C 6ﾓ1 cos C 2 cos B + C ) = 0; 2 ( ) ( ) B B cos 2 + C + cos 2 6ﾓ1 c 6ﾓ1 cos B ( )) B 2 6ﾓ1 cos 2 + C = 0; ( ( ( ) B cos 2 6ﾓ1 C cos B 2 ) = 0; sin B 2 cos C 2 sin2 B 6ﾓ1 C B 07 C , 0803 ﾂ = C 6ﾍ2 b = c. ｨｦ080909, = 0: ) = 0;

12 1310 ﾂ ﾁ ﾁ , ﾁ LH , ﾁ V ﾁ ﾁ : ﾁ: GL, GH, LH ? ﾂ ﾁ GH 07 LH 0308 ﾁ ﾂ ﾁ ﾁ ﾁ ﾁ ｨｹ ﾁ ﾁ ｨ ｨｲ ｨｮ ﾁ , ﾁ GL ? ｨｰ ﾂ ﾁ ﾁ , GL ｬ b 6ﾍ2 a + c = 2b. ﾂ ﾁ ﾁ , GL ﾍ a 6ﾍ2 b + c = 3a. ｨｦ ﾁ : { a + c = 2b; b + c = 3a ﾁ : b = 4a=3, c = 5a=3. ｨｲ , a : b : c = 3 : 4 : 5 ( ). ｨｦ080909, ｨｰ GL , ﾁ , 4, ﾁ ﾁ ﾁ , ﾁ ﾁ ﾁ ! ﾁ ﾂ ﾁ0707, 05. ｨｰ

13 ﾂ ﾁ ﾂ ﾁ ﾁ ﾁ ﾁ " ﾁ: , , 02 ﾁ ﾁ , ﾁ02 ﾁ 縺 ﾁ ﾁ 縺 , ﾁ ﾁ ﾁ0203 ﾁ , ﾁ ﾁ ﾁ , ﾁ ﾁ , ﾁ ｨｬ ﾁ , ﾁ ﾁ ﾁ030809, ﾁ ﾁ ﾁ ﾁ , ﾁ ﾁ , ﾁ02 ﾁ ﾁ , ﾁ ﾂ ﾁ ﾁ ﾁ ﾁ ﾁ02 ﾁ 縺 , ﾁ , ﾁ , ﾁ ﾁ ｨｪｨｰ05, 09 ﾁ ｨｪ 1 ﾁ ﾁ ﾁ , ｨｪｨｰ ｨｲ , ﾁ ｨｪｨｰ05, ﾁ M 1, ﾁ ﾁ ﾁ ﾁ ﾁ ﾁ ﾁ ﾁ ｨｮ ﾁ ﾁ ﾁ. ｨｰ , , ｨｲ ﾁ ｨ ﾂｨｮｨ 1 B 1 C ﾁ ｨ ﾂｨｮ, ﾁ ﾂ A ﾁ ﾁ , A 1 ｨ ﾁ ｨ ﾂ ﾁ ﾁ ﾁ ﾁ ｨｰ ﾁ ｨ ﾂｨｮ ( ), , ｨ ｨｪ, ﾂ05 07 ｨｮ06 05 ﾁ ﾁ A A ﾁ ﾁ ｨ ﾂ 07 ｨ ｨｮ, ﾁ ﾁ ﾁ ﾁ ｨ A 1 B 1 B 07 ｨ A 1 ｨｮ 1 ｨｮ , A 1 06 = A 1 05 ( , ﾁ ). ｨｹ , A 1 ｨ A 1 ｨ ﾁ- 0202, : ﾏA 1 ｨ 05 = ﾏA 1 ｨ 06 = ｨｮ ﾁ , ﾁ , A 1 06 = A 1 05 = 0;5ｨ ﾂ = 0;5 8 = 4. ｨｲ , S AA1 C 1 C = ｨ ｨｮ A 1 05 = 8 4 = 32. ｨ , S AA1 B 1 B = ｨ ﾂ A 1 06 = , ﾂ ﾂ 1 ｨｮ 1 ｨｮ S BB1 C 1 C. ﾂ , : A 1 07 ﾍ (ｨ ﾂｨｮ) 6ﾍ0 A 1 07 ﾍ ﾂｨｮ (0403 ｨｰ , ); ｨ ｨｪ ﾍ ﾂｨｮ. ｨｲ , ﾂｨｮ ﾍ (A 1 ｨ ｨｪ) (

14 ﾂ ﾁ ) 6ﾍ0 ﾂｨｮ ﾍ A 1 ｨ ( , ) , A 1 ｨ ｬ ( ﾂ 1 ｨｮ 1 ｨｮ), ｨｰｨｪ = ( ﾂ 1 ｨｮ 1 ｨｮ) ﾉ (A 1 ｨ ｨｪ) A 1 ｨ ( ﾁ , , ) : ﾂｨｮ ﾍ A 1 ｨ, A 1 ｨ ｬ ｨｮ 1 ｨｮ 6ﾍ0 ﾂｨｮ ﾍ ｨｮ 1 ｨｮ, , ﾂ ﾂ 1 ｨｮ 1 ｨｮ , 07 S BB1 C 1 C = ﾂｨｮ ｨｮ 1 ｨｮ A 1 ｨ. ﾂ 0609 ﾁ A 1 ｨ ｨ 06 = : A 1 ｨ = 4 ﾌ ｨｮ 1 ｨｮ = A 1 ｨ = 4 ﾌ 2, 0803 S BB1 C 1 C = 8 4 ﾌ 2 = 32 ﾌ , S = 2S AA1 C 1 C + S BB1 C 1 C = ﾌ 2 = 32(2 + ﾌ 2) (09 ﾁ ): V ﾂ A 1 ｨ 07: A 1 07 = ﾌ ｨ 1 ｨ 2 6ﾓ1 07ｨ 2 = ﾌ ﾌ ( 4 ﾌ 2 ) 2 6ﾓ1 ( 8 ﾌ 3 3 ) 2 = 4 ﾌ 6 3 : V = S ABC A 1 07 = 82 ﾌ ﾁ0308: 32(2 + ﾌ 2) 09 ﾁ.0302.; 64 ﾌ ﾌ 6 3 = 64 ﾌ 2 ( ): ﾁ ﾁ ﾁ , , , ﾁ ﾁ ﾁ : ﾁ , 05 ﾁ ﾁ ﾁ ﾁ , , ﾁ ﾁ ｨｰ , , ｨｲ ﾁ ｨｰ , 0609 ﾁ , ﾁ , ﾁ ﾁ ｨｰ ｨ ﾂｨｮDA 1 ﾂ 1 ｨｮ 1 D 1 ( ), ﾁ ﾏ ﾂｨ D = = 60 71, ﾏC 1 C ﾂ = ﾏC 1 CD = ; C 1 ｨｪ 07 C 1 05 ﾁ ﾁ , C ﾁ ﾁ , 0803 C 1 C = ﾂｨｮ = ﾁ 0609 ﾁ C 1 Cｨｪ 07 C 1 C : Cｨｪ = C05 = 6 ﾌ 2 2 = 3 ﾌ ｨｪ ﾍ ﾂｨｮ, 0605 ﾍ CD, ｨｪ = 0605 ( ﾁ ｨｮ 1 ｨｪ 07 ｨｮ 1 05). ｨｹ , ﾁ ﾁ C ﾂｨｮD, ｨ ｨｮ ｨ ﾂｨｮD. ﾂ ｨｮｨｪ06 (Cｨｪ ﾍ 06ｨｪ) : ｨｮ06 = CM cos = 3 ﾌ 2 : ﾌ 3 2 = 2 ﾌ 6; ﾁ ｨｮ 1 ｨｮ06 (C 1 06 ﾍ 06ｨｮ) : ｨｮ 1 06 = ﾌ ﾌ C 1 C 2 6ﾓ1 CH 2 = 6 2 ( ﾌ ) 2 ﾌ 6ﾓ1 2 6 = 2 3:

15 , 0708 ﾁ0308: V = ｨ ﾂ 2 sin ｨｮ 1 06 = 36 ﾌ ﾌ 3 = 108 ( ): ｨｮ ﾁ ﾁ 05 ﾁ , ﾁ ﾁ ﾁ , ﾁ , ﾁ ｨｦ , ﾁ ｨｮ , ﾁ0703 ﾁ ﾁ ( ), , ﾁ ﾁ (0609 ﾁ ) ﾁ ﾁ ﾁ ﾁ : 09) , ﾁ ﾁ ﾁ ﾁ ; ﾁ ﾁ , ﾁ ; 00) , ﾁ ﾁ ﾁ ﾁ ; ﾁ ﾁ ﾁ , ﾁ ﾁ ﾁ ; ﾁ) , ﾁ , ﾁ ﾁ ﾁ : 09) , ﾁ ﾁ ﾁ ; ﾁ ﾁ , ﾁ ﾁ ; 00) , 02 ﾁ ﾁ ﾁ ; ﾁ , ﾁ ﾁ ﾁ ｨｮ ﾁ ﾁ ﾁ ﾁ ﾁ ﾁ ﾁ ﾁ ( ﾁ ﾁ ) ﾁ ﾁ , , ﾁ ( ﾁ ﾁ0203 ﾁ ﾁ ), ﾁ ﾁ ﾁ ﾁ03 ﾁ , ｨｰ ﾁ ﾁ. ｨｲ ﾁ ﾁ ﾁ ﾁ ﾁ ﾁ , , ﾁ ﾁ ﾁ ﾁ ﾁ , 09 ﾁ ﾁ ﾌ ｨｰ ｨ ｨｰ = ｨ ﾂ = 18, ﾂｨｰ = 12, ｨｰ 07 = = 6 ﾌ 3 ﾁ ｨｰｨ ﾂｨｮ ( ) ﾁ ﾁ ｨｰ 05, ｨｰ ｨｪ 07 ｨｰ ﾁ ﾁ ｨｰｨ ﾂ, ｨｰ ﾂｨｮ 07 ｨｰｨ ｨｮ. ｨｦ03 ｨｰ05 ﾍ ｨ ﾂ, ｨｰｨｪ ﾍ ﾂｨｮ, ｨｰ06 ﾍ ｨ ｨｮ : 0705 ﾍ ｨ ﾂ, 07ｨｪ ﾍ ﾂｨｮ, 0706 ﾍ ｨ ｨｮ ﾏ0705ｨｰ = ﾏ07ｨｪｨｰ = ﾏ0706ｨｰ ( ﾁ ﾁ ﾁ ｨ ﾂ, ﾂｨｮ 07 ｨ ｨｮ). ｨｹ , ｨｰ, 07ｨｪｨｰ ｨｰ 0609 ﾁ0202, : 0705 = = 07ｨｪ = 0706; ｨｰ05 = ｨｰｨｪ = ｨｰ : ﾁ ﾁ ｨｰ ﾁ ﾁ ｨ ﾂｨｮ, ｨｰ , ﾁ ﾁ ｨ ﾂｨｮ = R, R

16 ﾂ ﾁ , ｨｪ ﾁ ﾁ ﾁ ｨ ﾂｨｮ , 0803 ｨ 05 = ｨ 06, ﾂ05 = ﾂｨｪ, ｨｮ06 = ｨｮｨｪ ( ﾁ ﾁ ﾁ ) ｨ ｨｮ 07 ﾂｨｮ. ﾂ ｨ ﾂｨｰ : ｨｰ05 = 2S ABP 18 = 8 ﾌ ﾁ AB = 2 ﾌ ﾌ ｨ ｨｰ : AK = ﾌ AP 2 6ﾓ1 PK 2 = ﾓ1 (8 ﾌ 2) 2 = 14. ｨｲ , ｨ 06 = ｨ 05 = ﾂ05 = ｨ ﾂ 6ﾓ1 ｨ 05 = 4 = ﾂｨｪ , ﾁ ｨｰ : 07K = ﾌ P05 2 6ﾓ1 07P 2 = ﾌ 128 6ﾓ1 108 = 2 ﾌ 5 = R ｨｮｨｪ = ｨｮ06 = 06, ｨ ﾂｨｮ 0609 ﾁ0302: ｨｰ ｨ ﾂｨｮ = 18 + ( ) + (4 + 06) = 2 ( ); S ｨ ﾂｨｮ = ﾌ ( )( ﾓ1 18)( ﾓ1 ( ))( ﾓ1 (06 + 4)) = = ﾌ ( ) = 2 ﾌ 1406( ): R = 2S ABC P = 2 2 ﾌ 14x(x + 18) ABC 2 (18 + x) = 2 ﾌ 14x(x + 18) 18 + x = 2 ﾌ 5 ﾌ ﾌ ﾌ 1406( ) = = ﾌ ﾍ0 5( ) = ﾍ0 06 = 10: ﾂｨｮ = 14, ｨ ｨｮ = V : 0708 ﾁ0308: 112 ﾌ V = 1 3 S ABC ｨｰ07 = ﾌ ( ) 6 ﾌ 3 = 112 ﾌ 15 ( ): ﾁ ﾁ , ﾁ ﾁ , , 09 ﾁ , ｨｲ ﾁ ｨｰ ｨ ﾂｨｮD 05 ﾁ ﾁ ｨ ﾂｨｮD (ｨ ﾂ ｬ ｨｮD); ﾁ ﾁ ﾁ ﾁ ﾁ ﾁ , ﾁ , ｨ ﾂ = , ｨｮD = ｨｰ ﾁ ﾁ ﾁ ｨｰ ｨ ﾂｨｮD 0609 ﾁ0202, ﾁ ﾁ ｨｰ ﾁ , ﾁ ﾁ ｨ ﾂｨｮD; ﾁ ｨｪ, ﾁ : ' = ﾏ0705ｨｰ = ﾏ07ｨｪｨｰ = ﾏ0706ｨｰ ﾁ ﾁ ﾁ , ｨｪ06 ﾍ ｨ ﾂ, 0705 ﾍ AD ( ). ｨｰ07 ﾍ (ｨ ﾂｨｮ), , ｨｰ ﾁ , ｨｪ ﾁ ﾁ ｨ ﾂｨｮD , 0803: ｨ 05 = = ｨ 06 = ｨｰ ｨ ﾂ = 16; 05D = DM = 2 1 ｨｮD = 9. ｨｲ , AD = ｨ D = = 25. ﾂ ｨ OD : 0705 = ﾌ ｨ 05 05D = ﾌ 16 9 = 12: ｨｲ , ﾁ ｨｪ ﾁ = 24, ﾁ0209 CD + AB MH = 24 = 600: 2 2

17 ｨｦ V = 1 3 S : ｨｰ07, : 3200 = ｨｰ07, ｨｰ07 = 16. ﾂ ｨｪｨｰ: tg ' = 07ｨｪ 07ｨｰ = = 4 3 6ﾍ0 cos ' = 0; ﾁ ﾁ AOB, BOC, COD 07 DOA, ﾁ ﾁ , 0803 S : = S : cos ' 6ﾍ0 S : = S : cos ' = 600 0;6 = 1000 (07012 ): 0708 ﾁ0308: , ﾁ ﾁ ｨｹ : ( ) , ( ) , ﾁ ﾁ , ﾁ ﾁ ﾁ , 縺 ﾁ , ﾁ070801, ﾁ ﾁ , ﾁ ﾁ ﾁ ﾂ ﾁ ｨｲ ﾁ ｨ ﾂｨｮA 1 ﾂ 1 ｨｮ 1 05 ﾁ ﾁ ｨ ﾂｨｮ 07 A 1 ﾂ 1 ｨｮ 1, ﾁ ﾁ ﾁ ﾁ , ﾁ ｨ ﾁ ﾁ ｨ ﾂｨｮ. ｨｰ ｨｮ ﾁ 縺 ﾁ ﾁ ｨ ﾂｨｮ, ｨｪ ﾂｨｮ, M ﾂ 1 C 1. ﾂ ｨ ﾁ ﾁ ﾁ ｨ ﾂｨｮ, , A 1 07 ﾍ ﾍ (ｨ ﾂｨｮ) ( ), A 1 O ﾍ ﾂｨｮ , ｨ ｨｪ ﾍ ﾂｨｮ ( ﾁ ｨ ﾂｨｮ). ｨｮ ﾁ , (A 1 ｨ ｨｪ) ﾍ ﾂｨｮ ﾁ ﾂｨｮ ｬ B 1 C : (A 1 ｨ ｨｪ) ﾍ B 1 C 1, A 1 07ｨｪ ｨｪ ﾂ 1 C 1, ｨ 1 ﾊ (A 1 ｨ ｨｪ) 07 A 1 M 1 ﾍ B 1 C , A 1 M 1 = 2 ﾌ ｨｪ = 2 ﾌ 3, 0803 A 1 M 1 = 07ｨｪ, , A 1 M 1 ｨｪ ｨ A 1 O ﾍ (ｨ ﾂｨｮ), 0803 A 1 M 1 ｨｪ , M 1 M ﾍ (ｨ ﾂｨｮ), ﾂ ﾂ 1 C 1 ｨｮ ｨ ﾂｨｮ A 1 ｨ ｨｪ ( ﾁ ) ﾁ ﾂ 1 ﾂ 07 ｨｮ 1 ｨｮ. ﾂ A 1 ｨ 07: M 1 ｨｪ = A 1 07 = ﾌ ｨ 1 ｨ 2 6ﾓ1 07ｨ 2 = ﾌ 8 2 6ﾓ1 (4 ﾌ 3) 2 = 4: ﾂ 1 ｨｮ 1 = 3 1 ﾂｨｮ, 0803 ﾂ 1M 1 = 1 3 ﾂｨｪ, MD = ﾂ ﾂ 1 M 1 = 2, 0803 BD = ﾂｨｪ 6ﾓ1 MD = 6 6ﾓ1 2 = ﾁ ﾂ ﾂ 1 D : ﾂ 1 ﾂ = ﾌ B 1 D 2 + BD 2 = 4 ﾌ 2. ｨ ｨｰ ﾂ ﾂ 1 ｨｮ 1 ｨｮ 0609 ﾁ , 0803 ｨｮｨｮ 1 = ﾂ 1 ﾂ = 4 ﾌ , ﾁ ﾁ0202 8, 4 ﾌ 2, 4 ﾌ ﾁ0308: 8, 4 ﾌ 2, 4 ﾌ 2. ｨｮ ﾁ , ﾁ ﾁ ﾁ 縺 , ﾁ ﾁ ﾁ 縺 ﾁ ﾁ ﾁ1 7, ﾁ ( ﾁ ).

18 ﾂ ﾁ ﾂ ( ﾁ ﾁ ﾁ ) ﾁ , ｨｲ , , ﾁ ｨｰ ｨ ﾂCDA 1 ﾂ 1 C 1 D , ｨｪ A 1 ｨｮ, , ｨｪ A 1 ｨｮ ( ). ｨｰ A 1 ｨｮ ｨ ﾂｨｮ 05 ﾁ ｨ ｨｮ ﾁ ｨ ﾂCD, ﾂD. ｨｲ , A 1 ｨｮ ﾍ ﾂD ( ). ｨ , A 1 ｨｮ ﾍ ｨｮ 1 ﾂ A 1 ｨｮ ﾍ ( ﾂC 1 D), A 1 ｨｮ ﾍ C : L = A 1 ｨｮ ﾉ ( ﾂｨｮ 1 D) = A 1 ｨｮ ﾉ C ｨｲ , A 1 ｨｮ = 6 ﾌ 3, ｨｪｨｮ = 0;5A 1 ｨｮ = 3 ﾌ 3, ｨ ｨｮ = 6 ﾌ 2, 07ｨｮ = 3 ﾌ 2. ｨｦ ﾁ 07ｨｮL 07 A 1 Cｨ : 07ｨｮ : A 1 ｨｮ = ｨｮL : Cｨ 6ﾍ0 ｨｮL = 07ｨｮ ｨｮｨ ｨ 1 ｨｮ = 3 ﾌ 2 6 ﾌ 2 6 ﾌ = 2 ﾌ 3: ML = ｨｪｨｮ 6ﾓ1 ｨｮL = 3 ﾌ 3 6ﾓ1 2 ﾌ 3 = ﾌ 3. ｨｲ , ｨｪL : LC = 1 : ﾍ A 1 ｨｮ 07 ( ﾂｨｮ 1 D) ﾍ A 1 ｨｮ, 0803 ｬ ( ﾂｨｮ 1 D), , m = ﾉ (ｨ ﾂｨｮ) BD ( ﾁ ), ｨ ｨｮ ﾁ , : 07ｨｮ = ｨｪL : LC = 1 : ﾁ , ｨ = 07ｨｮ, ﾁ02 ﾁ030209: = ﾉ ｨ ｨｮ 05 ﾁ ｨ : 00 ﾊ m, ｨ 07 m ｬ BD 6ﾍ P = m ﾉ ｨ ﾂ = ﾉ ｨ ﾂ 07 Q = m ﾉ ｨ D = ﾉ ｨ D 05 ﾁ ｨ ﾂ 07 ｨ D ( ) ｨｰ 07 Q ﾁ ﾁ , ｬ ( ﾂｨｮ 1 D), ﾁ ﾁ ﾁ , ｬ ( ﾂｨｮ 1 D), ｨ ﾂ ﾂ 1 A ｨｮDD 1 C ｨｰ ｨ ﾂ, ｨ ﾂ ﾂ 1 A 1 05 ﾁ ｨｰ05 ｬ DC 1 ｬ AB 1, ﾂ ﾂ 1 ; ﾁ ｨ , ｨ DD 1 A 1 05 ﾁ QR ｬ ｬ BC 1 ｬ ｨ D 1, R DD 1 ; R ﾁ ﾁ R06 ｬ DC 1 ( ｨｮ 1 D 1 ) 07 05N ｬ ﾂｨｮ 1 (N ﾂ 1 C 1 ) H 07 N ﾁ

19 , ﾁ ﾁ P QRHNK, ﾁ ﾁ , ｨｰ05 = 3 ﾌ ﾁ (3 ﾌ 2) 2 ﾌ 3 4 = 27 ﾌ 3 (09 ﾁ ) ﾁ0308: 27 ﾌ 3 09 ﾁ ｨｲ ﾁ ﾁ ﾁ , ﾁ , ﾁ , 09 ﾁ ﾁ0209 h. ｨｰ ｨｰ ｨ ﾂｨｮD ﾁ ( ); ｨｰ07 = h ﾁ ; E 07 F ﾂｨｮ 07 ｨ D , ﾁ ｨ D ﾂｨｮｨｰ ｨ D ｬ ﾂｨｮ, ｨ D ｬ ( ﾂｨｮｨｰ). ｨｲ , ｨｪ05, ﾂｨｮｨｰ, ｨ D ( ﾁ , , ｨｰ ) , ﾁ ADKM. ｨｦ : EF ﾍ ﾂｨｮ; P02 ﾍ ﾂｨｮ ( ﾁ ﾂｨｰｨｮ) 6ﾍ0 ﾂｨｮ ﾍ (EFP) ( ) 6ﾍ0 ( ﾂｨｮｨｰ) ﾍ (EFP) ( ) : FL = ﾉ (EFP), ｨｪ05 = ﾉ ( ﾂｨｮｨｰ) ｨ D ｬ ﾂｨｮ, ﾂｨｮ ﾍ (EFP), 0803 ｨ D ﾍ (EFP), ｨ D ﾍ FL, ｨ D ﾍ F02. ｨｲ , ﾏEFL = ﾁ , ﾁ ﾁ ﾍ ( ﾂｨｮｨｰ), (EFP) ﾍ ( ﾂｨｮｨｰ), 0803 FL ﾍ ( ﾂｨｮｨｰ) , , FL ﾍ ｨｰ02 07 FL ﾍ ｨｪ05, L = ｨｰ02 ﾉ ｨｪ05 07 MK ｬ AD ( BCP BC 07 AD). ｨｲ , ADKM 0609 ﾁ , FL ﾁ (L ｨｪ05) S ADKM ｨｪ 1, 05 1, L ﾁ ﾁ ﾁ ｨｪ, 05, L, L M ｨｰｨｮ, ｨｰ ﾂ 07 ｨｰ ﾁ ﾁ ﾁ ｨ ｨｮ, ﾂD 07 EF. ｨｮ ﾁ , ｨｪ 1 ﾊ BD, K 1 ﾊ AC, L 1 ﾊ EF , 0609 ﾁ ADK 1 M 1, ｨ DM ﾁ , 05 ﾁ ADKM ﾁ ｨｹ , S ADKM = S ADK 1 M 1 cos S ADK1 M ﾁ , L FD 0609 ﾁ , , 07L 1 = L 1 K 1, OF = FD S ADK1 M 1 = AD + K 1M 1 2 L 1 F = 2OF + 2OL 1 2 FL 1 = FL 2 1: S ADKM = FL2 1 cos L 1F. ﾂ L 1 FL: L 1 F = LF cos FL ﾍ ｨｰ02, EF ﾍ 07ｨｰ, 0803 ﾏ07P02 = = ﾏ02FL = ( ﾁ ﾁ ) ﾁ 07ｨｰ02: 0702 = 07ｨｰ tg = h tg. ｨｲ , 02F = = 2h tg ﾁ FL: FL = 02F cos = 2h tg cos = 2h sin :

20 ﾂ ﾁ ｨｮ ﾁ , L 1 F = LF cos = 2h sin cos , : S ADKM = FL2 1 cos = 4h2 sin 2 cos 2 = 4h cos 2 sin 2 cos : ｨｲ : S = AD+MK 2 F L ｨｪ ｨｮ , ｨｪK ﾁ , 05 ﾁ ﾁ MKD 1 A 1 ( ); F 1 = A 1 D 1 ﾉ PF ﾁ LF 1 = MK F 1 L. ﾂ LFF ﾏFLF 1 = (LF 1 ｬ EF); ﾏFF 1 L = , ﾏLF 1 ｨｰ = ﾓ1 ; ﾏF 1 FL = ﾏOFP 6ﾓ1 ﾏOFL = ( ﾓ1 ) 6ﾓ1 = ﾓ1 2: ﾁ LF 1 F ﾁ : LF 1 sin ( ﾓ1 2) = LF sin ( ) 6ﾍ0 LF 1 = LF cos 2 cos = 2h sin cos 2 : cos ｨｲ , ｨｪ05 = LF 1 = 2h tg cos ﾁ0905, ｨ D = 02F = 2h tg, : S = AD + MK ﾁ0308: 4h 2 sin 2 cos 09 ﾁ h tg + 2h tg cos 2 FL = 2h sin = 2 2h tg (1 + cos 2) 2h sin = = 4h 2 2 sin 2 cos (09 ﾁ ): ｨｲ , ﾁ , ﾁ [2]. ｨｹ0803, , : , , , , , , , , , , , , , , ｨｬ [1] ﾁ 02. ﾂ., ｨｲ ﾁ09 ﾁ0707 ｨｬ. ｨｦ : ﾁ ｨｪ.: , 2003{2009. [2] ﾁ 02. ﾂ., ｨｲ ﾁ09 ﾁ0707 ｨｬ. ｨｦ : ｨｲ ﾁ ｨｪ.: , 2003{ ﾁ 02 ﾁ ﾂ ﾁ0707, ﾁ ﾁ , , ﾁ ｨｰ03. [email protected]

21 13ｨｮ ﾁ ｨｮ ﾁ ﾂ. 04. ｨｪ ﾁ ｨｲ ﾁ , 0701., , [1] , ﾁ [2] , , ﾁ , ﾁ ﾂ [2], ﾁ ﾁ ﾁ ﾁ ﾁ , ﾁ , ﾁ ｨｰ : ﾁ n ﾆ j=1 a jy j + a n+1 n ﾆ j=1 b jy j + b n+1 (1) f k (y 1 ; y 2 ; :::; y n ) ﾜ B k ; (k = 1; 2; :::; m); (2) y j ﾝ 0; (j = 1; 2; :::; n): (3) ﾁ ﾁ ﾁ (2),(3) M (1) ﾁ ﾁ09 ｨｪ ﾁ 0, (1) ﾁ03 ｨｪ ﾁ , ﾂ (1), (2), (3) ﾁ , ﾁ ﾁ09 ｨｪ ﾁ 0. ｨｲ (1), (2), (3) ﾁ D ﾁ , ﾁ D 0509 ﾁ07 ﾁ , ﾁ L , ﾁ ﾁ ﾁ ﾁ09 {y 1 ; y 2 ; :::; y n }, ﾁ D ﾁ L, ﾁ ﾁ ﾁ03 ｨｪ, , ﾁ , ﾁ ﾁ09 ｨｪ. ﾂ , (2) , D ﾁ , ﾁ DL. ﾂ ﾁ ﾁ ﾁ ﾁ03 ｨｪ ﾁ09 ﾁ ﾁ03, , ﾁ , ﾁ ﾁ09 ｨｪ. ﾂ DL 0509 ﾁ07 ﾁ ﾁ , ﾁ LL , (2) ﾁ , D ﾁ ﾁ07 ﾁ ﾁ ﾁ , ﾁ D 0509 ﾁ07 ﾁ L , ﾁ ﾁ , ﾁ n ﾆ j=1 b j y j + b n+1 = 0 (4) 19

22 1320 ﾂ. 04. ｨｪ ﾁ ﾁ , ﾁ " (1), ﾁ (1) ﾁ ﾁ ﾁ ﾁ y j = (4) ( n ﾆ ﾆ n b j j=1 i= , ﾁ , 69 n ﾆ n ﾆ 61 i=1 j=1 n ﾆ j=1 d ijy i + d j;n+1 n ﾆ i=1 d iy i + d n+1 (j = 1; 2; :::; n) (5) ) ( n ) ﾆ d ij y i + d j;n+1 + b n+1 d i y i + d n+1 = 0 62 b j d ij + b n+1 d i 64 y i+ n ﾆ j=1 i=1 b j d j;n+1 + b n+1 d n+1 = 0: ﾁ ﾁ (5) (4) ﾁ ﾁ , n ﾆ j=1 b j d ij + b n+1 d i = 0; (i = 1; 2; :::; n): ﾆ 0603 ﾁ , n b jd j;n+1 + b n+1 d n+1 ﾙ 0; ﾁ , j=1 (1) ﾁ , D ﾁ L ﾁ ﾁ ﾁ ﾂ ﾁ ﾁ ﾂ ﾁ (2), (3), ﾁ (2), ﾁ (3) ﾁ ﾁ (5), (3) ﾁ ﾁ ﾁ070209, ﾁ y j ﾝ 0; (j = 1; 2; :::; n): (6) ｨｮ , ﾁ , b n+1 ﾙ 0: ﾁ , ﾁ ﾁ ﾁ070209: y j y j = n ﾆ d iy i + d ; (j = 1; 2; :::; n): (7) n+1 i= ﾁ ﾁ (4) n ﾆ b j n ﾆ j=1 y j d iy i + d 64 + b n+1 = 0; (j = 1; 2; :::; n): (8) n+1 i= ﾁ (7) ﾁ ﾁ n ﾆ i=1 d i y i + d n+1 = 0 (9)

23 13ｨｮ ﾁ ﾁ ﾁ09 M ( ﾁ ﾁ ﾁ03 ｨｪ ﾁ (7)) 0003 ﾁ (9) ﾁ ﾁ09 M ﾁ (8) ﾁ ﾁ070203: n ﾆ j=1 ( n ) ﾆ b j y j + b n+1 d i y i + d n+1 = 0 i=1 n ﾆ j=1 (b j + b n+1 d j )y j + b n+1 d n+1 = 0: ﾁ d j ; (j = 1; 2; :::; n + 1) , (4) ﾁ ﾁ (7) ﾁ d j = 6ﾓ1 b j b n+1 ; (j = 1; 2; :::; n); d n+1 = 1 (10) , ﾁ , ﾁ ﾁ y j = y j ﾆ 1 6ﾓ1 n b i bn+1 i=1 y i ﾁ (1) ﾁ ; (j = 1; 2; :::; n) (11) n ﾆ j=1 j y j + n+1 ; (12) j = b n+1a j 6ﾓ1 a n+1 b j b 2 ; (j = 1; 2; :::; n); n+1 = a n+1 : n+1 b n , ﾁ03 ｨｪ n ﾆ 1 6ﾓ1 i=1 b i b n+1 y i , (3) ﾁ (11) ﾁ ﾁ (6), ﾁ y j ﾜ 0; j = 1; 2; :::; n; y j ﾓ1y j ﾁ ﾁ (6) (2), (3) ﾁ (11) ﾁ ﾁ ' k (y 1; y 2; :::; y n) ﾜ k ; (k = 1; 2; :::; m) (13) y j ﾝ 0; (j = 1; 2; :::; n): (3 ) ﾁ , ( ) (1) ﾁ03 ｨｪ (12) ﾁ03 ｨｪ ﾁ , ﾁ b n+1 ﾙ 0, D ﾁ ﾁ (11) ﾁ ﾁ ﾁ07 ﾁ L: ﾁ (12) ﾁ03 ｨｪ, ﾁ ﾁ (13), (3 ) ﾁ , ﾁ ﾁ ﾁ (2), (3) (6ﾓ5)

24 1322 ﾂ. 04. ｨｪ ﾁ ﾁ (13), (3 ). ﾂ , (2) , (13) ﾂ (1), (2), (3) 05 ﾁ ﾁ ﾁ ﾁ (11) ﾁ ﾁ ﾁ (12), (13), (3 ) (2), (3) (12), (13), (3 ) ﾁ ﾁ (11) ﾁ ﾁ ﾁ (*) ﾁ09 M , (1) ﾁ09 ｨｪ b n ﾁ ﾁ ﾁ , ﾁ (11) ﾁ ｨｦ03 (11) ﾁ , i, j = 1; 2; :::; n y j y j = y i y i = 1 6ﾓ1 n ﾆ i=1 b i b n+1 y i: y jy i = y j y i; (i; j = 1; 2; :::; n) : (15) , 0703 (11) b n+1 y j = y j (b n+1 6ﾓ , ﾁ (15) n ﾆ i=1 b i y i ) = b n+1 y j 6ﾓ1 n ﾆ i=1 b i y j y i; (j = 1; 2; :::; n) : ｨｮ ﾁ , b n+1 y j = b n+1 y j 6ﾓ1 (b n+1 + n ﾆ i=1 n ﾆ i=1 b i y j y i; (j = 1; 2; :::; n) b i y i )y j = b n+1 y j ; (j = 1; 2; :::; n) : b y j = n+1 y j ﾆ b n+1 + n b ; (j = 1; 2; :::; n): (16) iy i i= ﾁ (16) ﾁ (*), ﾁ n ﾆ 1 6ﾓ1 i=1 b i b n+1 y i = (*) 0703 ﾁ ﾁ b n+1 b n+1 b n+1 + n ﾆ j=1 b jy j ; n ﾆ j=1 b jy j + b n+1 : ｨｮ ﾁ , (1) ﾁ03 ｨｪ b n ﾁ , (*) ﾁ03 M. ﾂ ﾁ (*) ﾁ03 M

25 13ｨｮ ﾁ ｨｮ b n+1 =0 07 ﾁ , , ﾁ ﾁ070209: y i = y i + ij ; (i = 1; 2; :::; n) ; j; (1 ﾜ j ﾜ n) ; b j ﾙ 0; 09 ij = ｨｮ ﾁ ﾁ03 ｨｪ f (y 1 ; y 2 ) = y 1 6ﾓ1 y 2 y 1 + y y 1 + y 2 6ﾓ1 2 ﾜ 0 y 1 ﾝ 0 y 2 ﾝ 0 { 0; i ﾙ j 1; i = j : ｨｮ , ﾁ ﾁ ﾁ y 1 = y 1 1 6ﾓ1 y 1 6ﾓ1 y 2 (17) y y 2 = 2 1 6ﾓ1 y 1 6ﾓ1 : (18) y ﾁ ﾁ ﾁ : ﾁ '(y 1; y 2) = y 1 6ﾓ1 y ﾁ03 M 4y 1 + 3y 2 6ﾓ1 2 ﾜ 0; y 1 ﾝ 0; y 2 ﾝ 0: ｨｬ , ﾁ y 1 = 1 2 ; y 2 = 0 07 ' ( 1 2 ; 0 ) = 1 2. ｨｹ ﾁ ﾁ ﾁ y 1 = 1; y 2 = 0; f (1; 0) = 1 2, ﾁ (17), (18). ｨｬ ｨｪ , ﾁ , ｨｪ ﾁ09: ｨｪ0706", ｨｪ ﾁ ﾂ // ｨｹ , 6 54, ｨｪ ﾁ ﾂ ﾁ0707, ﾁ ｨｪ ﾁ ﾁ ﾁ ﾁ09,

26 13ｨｦ ﾁ ﾁ ｨｨ ｨｦ. ｨｪ ﾁ ﾁ ﾂ. ｨｬ `Nova Methodus' (1684{1984)1 7, ﾁ0909. ﾂ , ｨｬ ﾁ, ﾁ ( ﾁ , ﾁ ﾁ ﾁ ) 05 ﾁ , ﾁ ｨｰ ﾁ ｨｪ ﾁ ﾁ ﾁ ", 6 52, ﾁ ﾂ ﾁ ｨｪ ﾁ ﾁ ﾂ ﾁ ﾁ ｨｬ ﾁ ｨｹ ﾁ ﾁ 縺 ﾁ 罟ｦ 縺 ｨｬ , ﾁ , ﾁ , ﾁ ﾁ , ﾁ ｨｬ , ﾁ , ｨｬ ﾁ ; ｨｮ ﾁ ﾁ ﾂ ﾁ ﾂ. ｨｬ `Nova Methodus' (1684{1984)1 7, ﾁ ﾁ ﾁ ( ) ﾁ 09 ﾁ ﾁ0902 ﾁ Studia Leibnitiana, Sonderheft 14 (Wiesbaden (Steiner Verlag), 1986), pp. 103{ G.W. Leibniz, 絲ova methodus pro maximis et minimis, itemque tangentibus, quae nec fractas, nec irrationales quantitates moratur, et singulare pro illis calculi genus1 7, Acta Eruditorum , pp. 467{473; ﾁ G. W. Leibniz Mathematische Schriften (7 vols., ed. C. I. Gerhardt), Berlin and Halle, 1849{1853 ( Hildesheim 1961{1962), vol. 5, pp ｨｮ06. H.-J. Hess 綮ur Vorgeschichte der `Nova Methodus1 7', Studi09 Leibnitiana, Sonderheft 14 (Wiesbaden (Steiner), 1986), pp. 64{102. ｨｮ ﾁ ﾁ ﾁ ｨｪ P. Dupont' C. S. Roero: Leibniz 1684, Torino (Fac. Sci. Mat. Fis. Nat., Univ. Torino; Series Quaderni di Mathematica 56), ｨｮ01. Hofmann, J. E., Die Entwicklungsgeschichte der Leibnizschen Mathematik wahrend des Aufenthaltes in Paris (1672{1676), Munchen, 1949, pp ; Hofmann, J. E., Leibniz in Paris 1672{1676 ( ﾁ ), Cambridge, 1974, pp. 187{201; Baron, M. and Bos, H. J. M., Newton and Leibniz (Unit C3 of the Open University course AM 289 History of Mathematics origins and development of the calculus), Milton Keynes (Open University Press), 197, pp. 35{46 07 Bos, H. J. M. Newton, Leibniz and the Leibnizian tradition, ( ﾁ09 II ﾁ ﾁ: From calculus to set theory, and introductory history (ed. I. Grattan Guinness), London, 1980), pp. 49{93, pp. 60{ ﾁ ﾁ ﾁ ﾁ ﾁ, ﾁ ( ﾁ ﾁ ﾁ020107) ﾁ ( ﾁ ﾁ0206, ﾁ ﾁ , 0403 ﾁ ); S.B. Engelsman 絨rthogonaltrajektorien im Prioritatsstreit zwischen Leibniz und Newton1 7 Studia Leibnitiana, Sonderheft 14 (Wiesbaden (Steiner) 1986), pp. 144{156, Engelsman, S.B. Families of curves and the origins of partial dierentiation, Amsterdam a. o. (North-Holland Publ.),

27 ﾁ ﾁ ﾁ ｨｪ ﾁ , , ﾁ ﾁ ﾁ ﾁ , ﾁ ﾁ ﾁ090107, ﾁ , ﾁ , , ﾁ ﾁ ｨｬ ﾁ ﾁ , ﾁ , ﾁ ｨｬ ﾁ ｨｬ , ﾁ ﾁ ﾁ , ﾁ ﾁ : ﾁ ﾁ02 ﾁ , ﾁ : ﾁ , ﾁ ﾁ ﾁ ﾁ , ( ), ﾁ OABC , ﾁ ﾁ ﾁ y, x, s, ﾁ0308, Q ﾁ0308, ( ﾁ 縺09 ﾁ ﾁ0308), 07 ﾁ , ﾁ 07 ﾁ ﾁ ﾁ ﾁ 07 ﾁ ﾁ ｨｬ ﾁ , ﾁ09- ｨｰ ﾁ 02 ﾁ ﾁ , 07 ﾁ ﾁ020107: ﾁ ﾁ0900, ﾁ ﾁ ﾁ , ｨ : ﾁ , ﾁ x ﾁ0209 1, ﾁ , ﾁ ﾁ ﾁ ﾁ ﾁ ﾁ0308. ﾂ ﾁ ﾁ , ﾂ ﾁ ﾁ ﾁ ｨｬ ﾁ ﾁ , ﾁ ﾁ ﾁ ﾁ ﾁ , ﾁ , ﾁ : ﾁ ﾁ ﾁ , 0102 ﾁ ﾁ ﾁ , , ﾁ0305 ﾁ ﾁ ﾁ ﾁ 絅ierentials, higher-order dierentials and the derivative in the Leibnizian calculus1 7, Archive for History of Exact Sciences, 14, 1974, pp. 1{90; ﾁ , ﾁ03 II, pp. 12{ omnes deduci posse ex generali quodam meo dimentiendorum curvilineo-rum principio, quod gura curvilinea censenda sit aequipollere Polygono innitorum laterum1 7. (Leibniz, Mathematische Schriften ( ) vol. 5, p. 126) , ﾁ ﾁ Acta Eruditorum ﾁ ; ｨｬ , ﾁ ｨｦ. ｨｨ ﾁ ﾁ

28 1326 ｨｨ ｨｦ. ｨｪ ﾁ , ﾁ ( a 1, ) ﾁ : a 1, a 2, a 3, a 4,..., ﾁ : ﾁ : s 1, s 2, s 3, s 4,... d 1, d 2, d 3, d 4, s 1 = a 1, d 1 = a 2 6ﾓ1 a 1, s 2 = a 1 + a 2, d 2 = a 3 6ﾓ1 a 2, s 3 = a 1 + a 2 + a 3, d 3 = a 4 6ﾓ1 a 3, s 3 = a 1 + a 2 + a 3 + a 4, d 4 = a 5 6ﾓ1 a 4, ｨｦ ﾁ ﾁ ﾁ ﾁ ﾁ ﾁ , ｨｬ ﾁ09 ﾁ ﾁ ﾁ02 ﾁ030209: 6ｦ1 ｨｦ ﾁ ﾁ ﾁ ﾁ , ; ﾁ ﾁ , 09 ﾁ ﾁ ﾁ ﾁ ﾁ , ﾁ ﾁ ﾁ ｦ ﾁ , ﾁ ﾁ ﾁ ｨｹ , ﾁ ﾁ. ｨｦ , ﾁ ( ), ﾁ ﾁ ﾁ ｨｹ , , ﾁ , ｨｬ ﾁ0900 ﾁ ﾁ ﾁ ﾁ02: 6ｦ ﾁ ﾁ , ﾁ , 07 ﾁ ﾁ020107, ﾁ , ﾁ02 ﾁ ﾁ ﾁ ｦ1 ｨｹ ﾁ ( , ) ｦ1 ｨｹ ﾁ , ( ﾁ d), ﾁ ﾁ ﾁ ﾁ ﾁ ﾁ , ( ﾁ ﾆ ), ﾁ ﾁ ﾁ ﾁ ﾁ ﾁ ﾁ , ﾁ ﾁ ｦ d 07 ﾆ ﾁ , ｨｬ ﾁ ﾂ ﾂ :

29 ﾁ ﾁ ｨｰ ﾁ ﾁ ﾁ ﾁ , , ﾁ ﾂ ｨｬ : ﾁ : ｨｰ , ﾁ ﾁ , ﾁ ﾁ ﾁ : ﾆ dx = x; ﾁ : d ﾆ x = x. 9 ｨｹ ﾁ ﾁ ﾁ ﾁ , ﾁ , ｨｬ ﾁ ﾁ ﾁ ( ﾁ ｨｬ ), , , ﾁ0908, 0703 ﾁ ﾁ ﾁ ( ), : ﾁ ﾁ , ﾁ ﾁ , ﾁ ﾁ ﾁ , ﾁ ( ). ﾂ ﾁ ( ) ﾁ , 07 ﾁ , ﾁ ﾁ , ﾁ ﾁ ｨｹ ﾁ ﾁ ﾁ090801; ﾁ ﾁ ﾁ ﾁ ﾁ 0703 ﾁ ｨｰ , ﾁ ﾁ ｨｬ ﾁ , ﾁ ﾁ ﾂ ﾁ , , ﾁ020403, ﾁ ｨｬ ｨｰ , , ﾁ ﾁ ﾁ ; ﾁ ﾁ , ﾁ ﾁ ﾁ ﾁ ; , , ﾁ ｨｬ , ﾁ , ﾁ , ｨｬ 絎ihi considerationern dierentiarum et summarum in seriebus numerorum primam lucern auderat, cum animadverterem dierentias tangentibus, et summas quadraturis respondere1 7. ( ｨｬ ﾂ , Leibniz, Mathematische Schriften ( ) vol. 4, p. 25). 9 紮undamentum calculi: Dierentiae et summae sibi reciprocae sunt, hoc est summa dierentiarum seriei est seriei terminus, et dierentia summarum seriei est ipse seriei terminus, quorum illud ita enuntio: P dx aequ. x; hoc ita; d P x aeq. x1 7 ( ｨｬ Elementa calculi novi pro dierentiis el summis, tangentibus et quadraturis, ﾁ C.I. Gerhardt'0301 ﾁ Die Geschichte der h01heren Analysis, erste Ableilung, die Entdeckung der hoheren Analysis, Halle 1855, pp ; p. 153).

30 1328 ｨｨ ｨｦ. ｨｪ ﾁ ﾁ 07 ﾁ ｨｮ , ﾁ ﾁ ｨｹ , , ｨｬ , ﾁ ｨｦ , ﾁ , ﾁ , ﾁ , ﾁ , , ﾁ ﾁ ﾁ ﾁ ﾁ ﾁ , ﾁ ｨｦ ﾁ ﾁ ﾁ0308, ﾁ03 ﾁ , ﾁ ﾁ ﾁ ; ﾁ , ﾁ ﾁ ﾁ ﾁ , ﾁ ﾁ ﾁ d 07 ﾆ, ﾁ ﾁ ﾁ ﾁ ﾁ ﾁ020107, d 07 ﾆ ﾁ , ﾁ ﾁ d ﾁ x, y, s, Q ﾁ , ﾁ dx, dy, ds, dq ｨｹ , , ﾁ ﾁ ﾁ ﾁ x, y, s 07 Q. ｨ , ﾆ ﾁ0708 ﾁ ﾁ ﾁ x, y, s, Q , 0203 ﾁ ﾆ x, ﾆ y, ﾆ s, ﾆ Q, 05 ﾁ , ﾁ , ﾁ ﾁ ﾁ ﾁ , 05 ﾁ ﾁ ﾁ ﾁ , ﾁ02 ﾁ , , ﾁ ﾁ d ﾁ , ｨｹ ﾁ ﾁ ﾁ: ｨｰ dx d 6ﾓ1 ddx ds d 6ﾓ1 dds ｨｹ ﾁ ﾁ ﾁ ﾁ ﾁ : ddx = dx I 6ﾓ1 dx; ( ), dx I 07 dx ﾁ ﾁ dx ﾁ ﾁ0203 ﾁ01 05 ﾁ ; ﾁ ﾁ ｨ , ﾆ ﾁ , ﾁ ﾁ ﾆ ﾆ x, ﾆ ﾆ y, ﾂ ﾁ ﾁ ﾁ , ﾆ ﾁ ﾁ , ﾁ

31 ﾁ ﾁ 縺 ﾁ d, d(x + y) = dx + dy, ｨｮ ﾁ , , ﾁ ﾁ , ﾁ ｨｰ ﾁ ﾁ , ｨｬ , ﾁ ﾁ , ﾁ ﾁ : ｨｦ ﾁ ﾁ ﾁ ? ﾁ , ﾁ0308, ﾁ , ﾁ0908. ｨｪ ( ) ﾁ , ﾁ , ﾁ , ﾁ ﾁ ﾁ ｨｬ , ﾁ0908, ﾁ ﾁ ( , 09), ; ﾁ , , ﾁ02, ﾁ ﾁ ﾁ X Y ( , ﾁ), ﾁ , ﾁ ﾁ ｨｹ , 縺 ﾁ ﾁ ; a priori, ﾁ ; , ﾁ ﾁ0308. ﾂ , ﾁ0703 ds 0609 ﾁ , ﾁ0703 dds ﾁ , 09 dx 07 dy , ddx 07 ddy ﾁ ( ﾁ ﾁ ﾁ ) , ﾁ0703 dx 0609 ﾁ , dds ﾙ 0 07 ddx = 0, ﾁ ﾁ ﾁ ﾁ ﾁ ﾁ ﾁ ﾁ0906 ﾁ : 6ｦ ﾁ0308, 07 6ｦ , 0403 ﾁ ﾁ ﾁ , ｨｬ ﾁ0900 縺 ｨｹ0803 ﾁ ﾁ , , , dx ﾁ090203, ﾁ ﾁ ｨｰ ﾁ , ds , ﾁ ﾁ ﾁ ﾁ ﾁ , ﾁ09 ﾁ ﾁ ｨｪ , ﾁ ﾁ ﾁ ﾁ ﾁ , ﾁ ﾁ , ﾁ , 09

32 1330 ｨｨ ｨｦ. ｨｪ dx x, ﾁ ﾁ , , ﾁ ﾁ , ｨｬ ﾁ ﾁ ﾁ , ﾂ ﾁ ﾁ ﾁ ﾁ ﾁ ; 0203 ﾁ , ﾁ ﾁ ﾁ x, y ｨｬ ﾁ ﾁ : ﾁ ﾁ03 ﾁ , dx dy ddx = 0, ﾁ x y ( y ) , ｨｬ ﾁ ; ﾁ ﾁ ﾁ ﾁ090107, ﾁ ﾁ ﾁ , ﾁ ﾁ ; ﾁ , ﾁ ﾁ ﾁ ﾂ , ﾁ , ﾁ ﾁ ｨｬ ﾁ ﾁ ﾁ , ﾁ ﾁ ﾁ ﾁ, ﾁ0206, ｨｹ , ﾁ ﾁ ﾁ ﾁ ｨｬ ; , , ﾁ ﾁ ﾁ ; ﾁ ｨｰ , ﾁ ay = x 2 : ｨｮ ﾁ ﾁ ﾁ x 07 y ﾁ ﾁ ﾁ ﾁ ｨｹ ﾁ ﾁ 0309 ﾁ , dx , ﾁ ﾁ0702: ady = 2xdx addy = 2(dx) 2 ad 3 y = 0 ad 4 y = 絋s ist ganz nicht notig dass die dx oder dy constantes und die ddx = 0 seyen, sondern man assumiert die progression der x oder y (welches man pro abscissa halten will) wie man es gut ndet1 7. ( ｨｬ , Leibniz, Mathematische Schriften ( ), vol. 7, p. 387). 11 絅ierentials1 7 ( ) pp. 5{ {77.

33 ﾁ ﾁ dy, ﾁ : ady = 2xdx 0 = 2(dx) 2 + 2xddx 0 = 6dxddx + 2xd 3 x 0 = 6(ddx) 2 + 8dxd 3 x + 2xd 4 x ﾁ ﾁ ( , ds ydx) ﾁ02 ﾁ ﾁ ﾁ ﾁ ﾁ ﾁ ﾂ x 07 y ﾁ02: ady = 2xdx addy = 2(dx) 2 + 2xddx ad 3 y = 6dxddx + 2xd 3 x ad 4 y = 6(ddx) 2 + 8dxd 3 x + 2xd 4 x , ﾁ ﾁ ﾁ ﾁ , ﾁ ﾁ ﾁ090308, ﾁ ﾁ ﾁ ﾁ. ｨｹ ﾁ03 07 ﾁ ﾁ , ﾁ ﾁ ｨｬ ﾁ ﾁ ｨｰ ﾁ ｨｪ0308 ﾁ ﾁ ｨｰ ( ) ﾁ0908 ｨｮ ﾁ P 07 P, R ﾁ0308, ﾁ ﾁ ﾁ P 07 P. ｨｰ ﾁ ﾁ P PR P P. (ｨｦ0007, ｨｬ : ﾁ PP R , ﾁ ﾁ ﾁ P 07 P; RP ﾁ ) ｨｬ ﾁ ﾁ ﾁ ｨｹ ﾁ ﾁ; ﾁ ﾂ ( ): r = ds dx = const; dxddy r = ds dy = const; dyddx r = dxds ds = const: ddy ｨｰ ﾂ , ﾁ ﾁ , : 12 絅ierentials1 7, pp. 35{42. r = dyds 2 dsddx 6ﾓ1 dxdds :

34 1332 ｨｨ ｨｦ. ｨｪ ｨｹ ; ﾁ, ﾁ ﾁ , ﾁ ﾁ ﾁ ﾁ r y = f(x) ﾁ (x; y): r = [1 + (f (x)) 2 ] 3=2 : f 茖 (x) d2 x dx d2 x dx ﾁ ﾁ 0703 ﾁ ﾁ ﾁ ; y = f(x), ﾁ f (x) = dy dx ; f 茖 (x) = d2 y dx 2; : : : f (n) (x) = dn y dx n: ﾁ ﾁ ﾁ ﾁ 05 ﾁ ﾁ , ﾁ ﾁ d 2, d 3, ﾁ ﾁ ﾁ ﾁ ﾁ ﾁ ﾁ070203, , 05 ﾁ ; ﾁ ｨｦ ﾁ ﾁ ﾁ ﾁ ﾁ , dx ; ﾁ ｨｬ , ﾁ , y = f(x) = x2 a d 2 y dx 2 = 2 a = f 茖 (x) dx = const; d 2 y dx2 = dy = const; d 2 y dx 2 = 2a a 2 + 4x ds = const ( ﾁ ). ｨｹ ﾁ ﾁ ﾁ , ﾁ , ( dx), ﾁ d2 x dx ﾁ ﾁ ｨｪ , ﾁ ﾂ 07 ﾁ , ﾁ ﾁ ﾁ ﾁ ｨｹ ﾁ : ﾁ , ﾁ ﾁ , ﾁ ﾁ ﾁ , ﾁ , ﾁ ｨｹ , , ﾁ , 絅ierentials1 7, pp. 47{53. dv :: v:

35 ﾁ ﾁ ｨｹ ﾁ ﾁ ﾁ ﾁ , ﾁ , dv ｨｹ ﾁ , , ﾁ : ﾁ ( ) ﾁ , ﾁ , ｨｹ , ｨｬ ﾁ ; ﾁ ﾁ ﾁ ﾁ , ﾁ dv, dv ﾁ ﾁ , ﾁ ﾁ v = ce 6ﾓ1t ; c dv ﾁ , ﾁ , ﾁ ds (070007, , vdt , ds = vdt), ﾁ ﾁ v = c t : ｨｲ09 ﾁ ﾁ ﾁ ﾁ ｨｬ , ﾁ ﾁ ﾁ , ﾁ0203 ﾁ ﾁ ﾁ , ﾁ , ﾁ ﾁ ﾁ ﾁ ﾁ , ﾁ ﾁ ﾁ ﾁ ﾁ ﾁ ﾁ ﾁ ﾁ ﾁ 0703 ﾁ ﾁ ﾁ , ﾁ ﾁ ﾁ, ﾁ , ( ). ﾂ ﾁ ﾁ ( , ﾁ ﾁ ﾁ ), ﾁ ﾁ ﾁ ﾁ , ﾁ ﾁ ﾁ, ﾁ , ; ﾁ ﾁ ﾁ ﾁ ; , ﾁ ﾁ ﾁ 07 ﾁ ﾁ ﾁ ｨｹ ﾁ 07 ﾁ ﾁ ﾁ , , ﾁ ﾁ ﾁ ｨｰ ﾁ ﾁ ﾁ ; , ﾁ ﾁ ﾁ ﾁ ﾂ ﾁ , 07 ﾁ ﾁ; ﾁ ﾁ , ﾁ ﾁ ﾁ ﾁ , ﾁ 絅ierentials1 7, pp. 66{77.

36 1334 ｨｨ ｨｦ. ｨｪ ｨｰ ﾁ ﾁ ﾁ ﾁ ﾁ ﾁ ｨｰ ﾁ ﾁ ﾁ0008 ﾁ ﾁ , 02 ﾁ ﾁ ﾁ ; , , ｨｬ ﾁ ﾁ , ﾁ ｨｹ , ﾁ020403, ﾁ , ﾁ ﾁ ﾁ ｨｮ03 ﾁ , ﾁ, ﾂ03- ﾁ , ﾁ ｨｬ ﾁ ﾁ ﾁ0708 ﾁ ﾁ ﾁ ﾁ0908, , ﾁ ｨｮ03 ﾁ ﾁ0708 ﾁ ﾁ ﾁ ﾁ , ﾁ ﾁ ; ｨｬ ﾁ ﾆ ( ﾁ 縺 ﾁ , 縺 ﾁ , ﾁ ﾁ ) ﾁ0708 ﾁ ﾁ ﾁ ﾁ0908, ｨｮ03 ﾁ ﾁ0708 ﾁ ﾁ ﾁ ﾁ , ﾁ , ﾁ ﾁ ﾁ, ﾁ ﾁ ﾁ 0703 ﾁ ﾁ090308, ﾁ ﾁ ﾁ ﾁ ｨｬ02ｨｺ0906ｨｦ0002 ﾂ07 ｨｦｨｮ06ｨｦｨｮｨｬ0206ｨｦ ﾁ: ﾁ ｨｮ07 ﾂｨｰ02ｨｪ020606ｨｴｨｺ ｨ 06ｨ ｨｬｨｦｨｲ ﾂ ﾁ : x dx f f dy ddy ｨｮ ﾁ : x ﾆ x ydx ﾆ ydx ﾁ, ﾁ ｨｦ ﾁ : f F ( F(x) = ﾒ x a f(t)dt) ﾁ ﾁ ﾁ ｨｹ ﾁ ﾁ090308, ﾁ ﾁ ﾁ , ﾁ ﾂ ﾁ ( ) Baron and Bos 絲ewton and Leibniz1 7 pp. 54{57, 絅ierentials1 7 pp. 34{35, 絲ewton, Leibniz and the Leibnizian tradition1 7, pp. 92{93.

37 ﾁ ﾁ ﾁ; ﾁ ﾁ ﾁ , ﾁ , ﾁ , ﾁ ﾁ , , ﾁ ﾁ ﾁ ｨｲ ﾂ , ﾁ0007 ﾁ ﾁ ﾁ , ﾁ ﾁ ﾁ , : 6ｦ1 ｨｬ ﾁ ﾁ ﾁ ｦ1 ｨｦ ﾁ ﾁ , ﾁ ﾁ ｦ1 ｨｹ ﾁ , ﾁ ﾁ , ﾁ ﾁ , ﾁ , ﾁ , ﾁ ﾁ ﾁ : ﾁ ? ﾁ ﾁ , ｨｬ ﾁ , , , ｨｨ ｨｦ.ｨｪ , ｨｮ ﾁ ﾁ , ﾁ : ｨ. ｨｦ ﾁ.

38 ﾁ ｨｬ ｨｮ ﾁ, ｨ. ｨｮ ﾁ09, ｨｮ. ｨｮ0809 ﾁ0103 ﾁ , ﾂ ﾁ ﾁ ﾂ ﾁ ﾁ ﾂ 罍ﾂ ﾁ ﾁ, ﾁ , , ﾁ ﾁ ﾁ ﾁ ﾁ 罟 ﾂ ﾁ ﾁ ﾁ 0703 ﾁ 罟 , ﾁ ﾁ ﾁ ｨ ﾁ , , ﾁ ﾁ 07 ﾁ , ﾁ ﾁ ﾁ ﾁ03 ﾁ ﾁ0202 ﾁ ｨ. ｨｹ ﾁ , ﾁ , ﾁ , ﾁ , , ﾁ ﾁ , ﾁ , , , ﾂ 07 ﾁ ﾁ ﾁ : , ｨｹ , , ﾁ , , , ﾁ ﾁ, ﾁ ﾁ ! ﾁ ｨｲ ﾁ02 ﾁ ﾁ , ﾁ ﾁ , ﾁ , ﾁ , ﾁ ﾁ 縺 ﾁ 縺 ﾁ ﾁ ﾁ ﾁ , ﾁ , , ﾁ ﾁ Oxy, ﾁ , ﾁ ﾁ ( ﾁ070003, ﾁ x 07 y): F(x; y) = 0; F(x; y) ﾁ , 2x + 3y 6ﾓ1 2 = 0; x 2 + y 2 6ﾓ1 4 = 0; sin x = cos y 36

39 13ｨｬ ( ﾁ , ), ﾁ ﾁ , ﾁ03 ﾁ , ﾁ ﾁ ﾁ ｨｰ ﾁ ﾁ ﾁ x 6ﾓ1 2y = 0. ｨｬ : (0; 0), (2; 1), (20; 10),..., 07 ﾁ , (x; y) ﾁ ﾁ ﾁ , x ﾁ 02 ﾁ y 罍ﾁ , ﾁ ﾁ ﾁ , 0803, ﾁ , ( ) ﾁ ﾁ , ﾁ ﾁ ﾁ , ﾁ ﾁ02 ﾁ ﾁ ﾁ , ﾁ ﾁ ﾁ , ﾁ ﾁ ﾁ : ) ﾁ ﾁ0705, ﾁ M , ﾁ ﾁ , 2) ﾁ ﾁ0705 ﾁ ﾁ ﾁ ﾁ02 ﾁ ﾁ ｨｰ , ﾁ ﾁ x 6ﾓ1 2y = ﾁ ﾁ : ﾁ ﾁ l , A(x 0 ; y 0 ) ﾁ ﾓ1 a (p; q) ( ) , ﾁ ﾓ1 a ﾁ ﾁ ﾁ l. ｨｮ , ﾁ ﾁ M(x; y) 07 ﾁ , ﾁ l ﾁ ﾁ ﾁ , A 07 M ﾁ , , M l , ﾁ ﾓ1 a 07 6ﾓ16ﾓ1 AM ｨｹ ﾁ0703, , ﾁ , ﾁ ﾁ , ﾁ0309- ｨｰ ﾁ02 ﾁ ﾁ ﾓ1 a 07 6ﾓ16ﾓ1 AM , ﾁ ﾓ1 a (p; q) 0703 ﾁ ﾁ0708, ﾁ ﾓ16ﾓ1 AM ﾁ ﾓ16ﾓ1 AM(x 6ﾓ1 x 0 ; y 6ﾓ1 y 0 ) ﾁ : ｨｦ080909, , x 6ﾓ1 x 0 p = y 6ﾓ1 y 0 : (7.1) q M (x; y) l, A(x 0 ; y 0 ) ﾁ ﾓ1 a (p; q), , (x; y) ﾁ ﾁ ﾁ0708 (7.1). ｨｦ (7.1) 05 ﾁ ﾁ , ﾁ ﾁ ﾂ p 07 q ﾁ ﾁ , 09 (x 0 ; y 0 )

40 1338 ｨｮ ﾁ, ｨ. ｨｮ ﾁ09, ｨｮ. ｨｮ0809 ﾁ0103 ﾁ ｨｰ ﾁ ﾁ ﾁ ﾁ ﾂ ﾁ ﾁ , ﾁ ｨｦ ﾁ02 ﾁ ﾁ ﾁ ｨｮ ﾁ ﾁ ｨｦ ﾁ , ﾁ ﾁ , ﾁ { x = x(t); y = y(t); ﾁ (x; y) ﾁ 0309 ﾁ t ( , ﾁ ). ｨｮ ﾁ (7.1), ﾁ ﾁ ﾁ : x 6ﾓ1 x 0 p = y 6ﾓ1 y 0 : q t ﾁ ﾁ ﾁ : 61 x 6ﾓ1 x = t; p 6463 y 6ﾓ1 y 0 = t: q ﾁ ﾁ ﾁ x 0 07 y 0 ﾁ ﾁ , ﾁ : { x = x0 + pt; (7.2) y = y 0 + qt: ﾂ (p; q) ﾁ ﾁ , 09 (x 0 ; y 0 ) , , t ﾁ , ﾁ 縺 ﾁ t = (x; y) 0703 ﾁ A(x 0 ; y 0 ) A ﾁ ﾁ020308, , ﾁ ﾁ ﾁ , ﾁ ﾁ ﾁ , ﾁ , ﾁ07 ﾁ ﾁ ﾁ ﾁ , ﾁ , ﾁ l , A(x 0 ; y 0 ) 07 B(x 1 ; y 1 ) ( ). ｨｰ , ﾁ , ﾁ ﾁ ﾁ ﾁ ﾁ ｨｦ03 ﾁ ﾁ A 07 B, ﾁ ﾁ ﾁ AB, 6ﾓ , (AB) ﾁ ﾁ ﾁ ﾁ ﾁ AB 6ﾓ1 ﾁ : 6ﾓ1 AB(x 1 6ﾓ1 x 0 ; y 1 6ﾓ1 y 0 ): ﾁ ﾁ (7.1). ﾂ ﾁ , ﾁ : x 6ﾓ1 x l: 0 = y 6ﾓ1 y 0 : (7.3) x 1 6ﾓ1 x 0 y 1 6ﾓ1 y 0

41 13ｨｬ ｨｲ (x 0 ; y 0 ), (x 1 ; y 1 ) , ﾁ ﾁ ﾁ ﾁ , ﾁ ﾓ1 n (A; B) M 0 (x 0 ; y 0 ) ﾁ l, M ﾁ ﾓ1 n ( ) , ﾁ ﾁ M(x; y), ﾁ ﾓ16ﾓ16ﾓ1 M 0 M : ﾁ M l? , , ﾁ ﾓ16ﾓ16ﾓ1 M 0 M ﾁ ﾓ1 n. ﾂ ﾁ ﾁ , ﾁ ﾓ16ﾓ16ﾓ1 M 0 M (x 6ﾓ1 x 0 ; y 6ﾓ1 y 0 ) ﾁ ﾁ , ' ﾁ , ﾁ ﾂ ﾁ , : cos ' = (6ﾓ1 n ; 6ﾓ16ﾓ16ﾓ1 M 0 M) 6ﾓ1 n 6ﾓ16ﾓ16ﾓ1 M : 0 M M M n ｨｰ , ﾁ , ﾁ ﾁ , ﾁ ﾁ ﾁ , ﾁ ﾁ : ( 6ﾓ1 n ; 6ﾓ16ﾓ16ﾓ1 M 0 M) = A(x 6ﾓ1 x 0 ) + B(y 6ﾓ1 y 0 ): ｨｮ ﾁ , ﾁ l ﾁ : l: A(x 6ﾓ1 x 0 ) + B(y 6ﾓ1 y 0 ) = 0: (7.4) A 07 B ﾁ ﾓ1 n, l, 09 (x 0 ; y 0 ) ﾂ ﾓ1 n (A; B) ﾁ l ﾁ ｨｰ ﾁ ﾁ ﾁ (7.4), Ax + By + (6ﾓ1Ax 0 6ﾓ1 By 0 ) = 0: ﾂ ﾁ , C ﾁ ﾁ ﾁ Ax + By + C = 0: (7.5) ｨｪ , ﾁ ﾁ ﾁ ﾁ (7.4), ﾁ ｨｦ ﾁ090107, , ﾁ ﾁ ﾁ , ﾂ ﾁ : 09 ﾁ ﾁ (7.5) ﾁ ﾁ , ﾁ : A = B = 0, ﾁ ﾁ C = C , ﾁ03 ( C = 0), ﾁ ﾁ , 縺 ﾁ ﾁ ﾁ , ﾁ ﾁ03 ( C ﾙ 0), ﾁ ｨｮ , A 07 B ﾁ ﾁ , ﾁ (7.5) ﾁ ﾓ1 n (A; B). ｨｹ ﾁ ﾁ ﾁ

42 1340 ｨｮ ﾁ, ｨ. ｨｮ ﾁ09, ｨｮ. ｨｮ0809 ﾁ0103 ﾁ ｨｦ080909, ﾁ (7.5), ﾁ A, B , , , , A ﾙ y = ﾁ ﾁ ﾁ070203: Ax = 6ﾓ1C; x = 6ﾓ1C=A. ｨｲ , (6ﾓ1C=A; 0) 05 ﾁ ﾁ ﾁ , ﾁ ﾁ (7.5), ﾁ l M (6ﾓ1C=A; 0) ﾁ09 l , ﾁ , , ﾁ x + 5y + 9 = ﾁ y = 0, x = 6ﾓ19, x = 6ﾓ , (6ﾓ13; 0) ﾁ , ﾁ ﾁ ﾁ ｨｮ ﾁ , M 0 (6ﾓ13; 0) ﾁ09 l ( ﾁ , ) ﾁ (7.5) : Ax + By + C = 0 6ﾍ2 A(x + C=A) + B(y 6ﾓ1 0) = 0 07 ﾁ ﾁ ﾁ ﾁ: 6ﾓ1 n (A; B) 07 6ﾓ16ﾓ16ﾓ16ﾓ1 M 0 M(x + C=A; y 6ﾓ1 0) M 0 (6ﾓ1C=A; 0) ﾁ09 l, M (x; y) ﾁ ｨｮ A(x + C=A) + B(y 6ﾓ1 0) = ﾁ n , M(x; y) ﾁ ﾁ03 l, ﾁ ﾁ ﾁ 6ﾓ1 n 07 6ﾓ16ﾓ16ﾓ1 M 0 M 0609 ﾁ , ( ) , ﾁ03 l ﾁ , M 0 (6ﾓ1C=A; 0) ﾁ ｨｰ ﾓ1 n (A; B) ( ), ﾁ ﾁ : 3x + 5y + 9 = 0 6ﾍ2 3(x + 3) + 5(y 6ﾓ1 0) = 0: ｨｲ ﾓ1 n (3; 5), M 0 (6ﾓ13; 0), M(x; y) 07 6ﾓ16ﾓ16ﾓ1 M 0 M(x + 3; y 6ﾓ1 0) ﾁ ﾁ ﾁ ( 6ﾓ1 n ; 6ﾓ16ﾓ16ﾓ1 M 0 M), ﾁ ﾁ ﾁ ﾁ ﾁ ﾁ ｨｰ ﾁ (7.5) y: l: Ax + By + C = 0 6ﾍ2 y = 6ﾓ1 A B x 6ﾓ1 C B : ﾁ , , B ﾙ 0 ( ﾁ ﾁ y ﾁ ﾁ ). ﾂ ﾁ : k = 6ﾓ1A=B, b = 6ﾓ1C=B ﾁ : y = kx + b: (7.6) ﾁ ﾁ ﾁ , ﾁ ﾁ ｨｦ

43 13ｨｬ ｨｰ ﾁ ﾁ ﾁ , A(0; b) 07 B(1; k +b) ﾁ02 ﾁ0302: b ﾁ ﾁ , ﾁ ﾓ Oy , AB(1; k) ﾁ ﾁ l , ﾁ ﾁ AC 07 BC: k , AC = 1; BC = k: tg ' = BC AC ; l ｨｦ ﾁ (7.6) ﾁ ﾁ ﾁ ﾁ ﾁ ﾁ ﾁ ﾁ ﾁ , ﾁ , , ﾁ ﾁ l, Ox ﾁ M(a; 0), Oy ﾁ N(0; b) ( ) ﾁ ﾁ Ax + By + C = , ﾁ l. ｨｪ ﾁ l ﾁ ﾁ : { Aa + B0 = 6ﾓ1C; A0 + Bb = 6ﾓ1C: ﾁ ﾁ ﾁ ﾁ : A, B 07 C. ｨｲ , ﾁ ﾁ C = 0, , ﾁ ﾁ , , ﾁ C ﾙ ﾁ : { AC a + B C 0 = 6ﾓ11; A C 0 + B C b = 6ﾓ11; A C = 6ﾓ11 a ; B C = 6ﾓ11 b ﾁ ﾁ ﾓ1 1 a x 6ﾓ1 1 b y + 1 = 0 ｨｰ ,

44 1342 ｨｮ ﾁ, ｨ. ｨｮ ﾁ09, ｨｮ. ｨｮ0809 ﾁ0103 ﾁ x a + y = 1: (7.7) b ﾁ ﾁ ﾁ ﾁ ﾁ ﾁ ｨｰ l, ﾁ x 6ﾓ1 3y + 4 = ﾁ , ﾁ x 6ﾓ1 12y + 16 = 0, ﾁ , ﾁ ﾁ ﾁ , ﾁ , ｨｹ0803, , ｨｨ , ﾁ ﾁ03 ﾁ ﾁ ﾁ ﾁ ﾁ ﾁ x + y 6ﾓ1 = 0; = ｨｪ ﾁ ﾁ ﾁ ﾁ ﾁ ﾁ0703. ｨｦ ﾁ090107, ﾁ ﾁ , ﾁ090308, ﾁ ﾁ0701 ﾁ ﾁ ( ﾓ5 ). ｨｮ ﾁ ﾁ ｨ ﾁ ﾁ ﾁ x + y 6ﾓ1 = ﾁ l ) ﾓ1 n (; ) l 0609 ﾁ0209 1; 2) ﾓ1 n ﾁ l, , 09; 3) l 0609 ﾁ0203. M n n ｨｰ ﾁ ｧﾑ) ｧﾒ) ﾁ03. ｨｲ , ﾁ ﾓ1 n (; ) ﾁ ﾁ l ( ), ﾁ ﾓ1 n (; ) ﾁ ﾓ1 n = ﾌ ﾁ ﾁ ﾁ ﾓ1 n = ﾁ ﾁ ) 07 3) l ﾁ M 0 (x 0 ; y 0 ) ( , 00) ﾁ ﾓ16ﾓ16ﾓ1 OM 0 (x 0 ; y 0 ) ﾁ ﾓ1 n (; ). ｨｲ , ｨｦ ﾁ090107, t, x 0 = t, y 0 = t t 1 7 0, ﾁ ﾁ ( ) 07 t < 0, ﾁ ﾁ ｨｮ , M 0 (x 0 ; y 0 ) l, (x 0 ; y 0 ) ﾁ ﾁ ﾁ , x 0 + y 0 6ﾓ1 = 0:

45 13ｨｬ ﾁ0701 ﾁ x 0 = t, y 0 = t: 07 ﾁ t : t + t 6ﾓ1 = 0 t( ) 6ﾓ1 = 0: ﾁ = 1, ﾁ ﾁ03: t =. ｨｦ080909, ﾁ , , 07 t 1 7 0, ﾁ ﾓ1 n 07 6ﾓ16ﾓ16ﾓ1 OM ﾁ ﾓ1 n ﾁ ﾁ ｨｰ l ﾁ ﾁ ﾓ16ﾓ16ﾓ1 OM 0, ﾁ : 6ﾓ16ﾓ16ﾓ1 OM 0 = ﾌ (t) 2 + (t) 2 = ﾌ t 2 ( ) = t 1 = : 7ﾁ3 ｨｮ ﾁ ﾁ ｨｮ ﾁ ﾁ ﾁ ﾁ , ﾁ ﾁ ﾁ ( ) ｨｮ ﾁ x + y 6ﾓ1 = ﾁ l. ｨｰ (A; l) A(x 0 ; y 0 ) l ﾁ : (A; l) = x 0 + y 0 6ﾓ1 : ﾁ03. ｨｲ , ﾁ , A (0; 0) ( ) ﾁ ﾁ ﾁ Auv ( ), , ﾁ A. ﾂ , ﾁ ﾁ ｨｲ , A ﾁ (x 0 ; y 0 ), 09 ﾁ 0203 ﾁ0308 (0; 0), ﾁ090107, x = x 0, 0803 u = 0, y = y 0, v = , ﾁ ﾁ : u = x 6ﾓ1 x 0 ; v = y 6ﾓ1 y 0 : ﾂ x 07 y: ｨｰ ｨｮ Auv ﾁ Oxy ﾁ0701 ﾁ ﾁ l: x = u + x 0 ; y = v + y 0 (u + x 0 ) + (v + y 0 ) 6ﾓ1 = 0: ｨｪ ﾁ l ﾁ 0203 ﾁ ｨｰ ﾁ , u + v + (x 0 + y 0 6ﾓ1 ) = 0: ﾁ , = ﾁ ﾁ 0203 ﾁ ( ﾁ ﾁ ) ﾁ , ﾁ ( A) l 0609 ﾁ0203 x 0 + y 0 6ﾓ1. 7ﾁ3

46 1344 ｨｮ ﾁ, ｨ. ｨｮ ﾁ09, ｨｮ. ｨｮ0809 ﾁ0103 ﾁ 44. ﾂ , 02 ﾁ , , ﾁ , ﾁ ｨｮ ﾁ ﾁ , ﾁ l 1 07 l ﾁ ﾁ : l 1 : A 1 x + B 1 y + C 1 = 0; l 2 : A 2 x + B 2 y + C 2 = 0: ﾁ ﾁ , 0803, ﾁ , l 1 = l 2. ﾂ ﾁ ﾁ : A 1 A 2 = B 1 B 2 = C 1 C 2 : , ( ), , ﾓ1 n 1 (A 1 ; B 1 ), 09 6ﾓ1 n 2 (A 2 ; B 2 ), ﾁ ﾁ A 1 = B 1 : A 2 B ﾁ , ｨｰ A 1 = B 1 ﾙ C 1 ; A 2 B 2 C , ﾁ , , , , A 1 A 2 ﾙ B 1 B 2 : ｨｮ0003 ﾁ03 縺 ﾁ , ﾁ ﾁ , , , , 罍ﾁ ﾂ 罍ﾁ ( ). ｨｰ ﾁ : ﾁ , ( , 00), ﾁ , ( , 09). ﾂ ﾁ ﾁ ﾁ ﾁ ｨｮ ﾁ02 ﾁ ｧﾑ) ｧﾒ) ｨｰ : 09) ; 00) ﾁ ｨｲ ﾁ ﾓ1 a (p 0 ; q 0 ) ﾁ ﾁ ﾁ ｨｲ ﾁ ﾁ ﾁ

47 13ｨｬ (7.1) , , , x 0 =, y 0 =. ﾂ : x 6ﾓ1 p 0 = y 6ﾓ1 q 0 : , ﾁ ﾁ , , 6ﾓ1 a (2; 3) ﾁ , ﾁ , ﾁ x 6ﾓ1 2 = y 6ﾓ1 : ﾁ ﾁ ﾁ ﾁ070203: q 0 x 6ﾓ1 p 0 y + (p 0 6ﾓ1 q 0 ) = 0: , (p 0 ; q 0 ) ﾁ ﾁ ﾁ ﾁ , ﾁ ﾁ ﾁ ﾁ0703 ﾁ p 0 6ﾓ1 q ﾂ0305 ﾁ , ﾁ ﾁ : q 0 x 6ﾓ1 p 0 y + = 0: (7.8) , ﾁ ﾁ ﾁ , ﾁ , ﾁ ﾁ ﾁ , ﾁ ｨｮ , , 05 ﾁ ﾁ , ﾁ ｨｦ080909, M 0 (x 0 ; y 0 ) ﾁ , ﾁ ﾁ ﾓ1 a (p; q): x 6ﾓ1 x 0 p = y 6ﾓ1 y 0 ; q q(x 6ﾓ1 x 0 ) 6ﾓ1 p(y 6ﾓ1 y 0 ) = 0: (7.9) ｨｹ ﾁ ﾁ ﾁ ﾁ py 0 6ﾓ1 qx ﾁ , ﾁ ! ﾁ (7.9) ﾁ ﾁ ( ). 46 6ﾓ ﾁ ﾂ 0703 ﾁ ( , ) ﾁ , 縺 ﾁ , ﾁ ﾁ ﾁ ﾂ ﾁ , 縺 ﾁ , ﾁ09 ﾁ ﾁ, ﾁ07 ﾁ ﾁ ﾁ , ﾁ ﾁ , ﾁ ﾁ ﾁ090801

48 1346 ｨｮ ﾁ, ｨ. ｨｮ ﾁ09, ｨｮ. ｨｮ0809 ﾁ0103 ﾁ , ﾁ ﾁ ( , ﾁ ﾁ ﾁ 0703 ﾁ ) , ﾁ03 ﾁ ﾁ ﾁ ! ｨｮ , ﾁ ﾁ ﾁ ﾁ09 ( ) ﾁ ﾁ03, (7.8) ﾁ ﾁ l, ( ) l , ﾁ ｨｰ ｨｪ ﾁ0702: q 0 x 6ﾓ1 p 0 y + = 0; l. ｨｲ , , l ﾁ , ﾁ ( l) (7.9), ﾁ ﾁ ( ) ﾁ qx 6ﾓ1 py = 0: (7.10) , ﾁ ﾁ03, ﾁ ﾁ l, ( ) ( , l 0 ) l 0603 ﾁ0203 ﾁ x ﾞ, ﾁ ﾁ l 0! ﾂ ﾁ , ﾁ l , ﾁ ﾁ P 1. ｨｰ ｨｰ ﾁ ｨｮ , ﾁ : ( ) ﾁ ( ) , x ﾞ ﾁ , ﾁ ﾁ l 1 ; l 2 ; : : :, l 0 ( ) l ﾁ , ﾁ ﾁ03 ﾁ ｨｪ ﾁ l 6ﾓ11 ; l 6ﾓ12 ; : : : ( ), l ｨｮ ﾁ ﾁ ﾁ l ﾁ0003 ﾁ03 ﾁ ﾁ ﾁ , ﾁ x ﾞ. ｨｲ , x ﾞ l, ﾁ ﾁ ﾁ0908 縺 l ( ). ﾂ , 縺

49 13ｨｬ ｨｰ ﾁ ﾁ , ﾁ , ﾁ ﾁ (7.10), ﾁ ﾁ ﾁ ﾁ p 07 q ﾁ0208 ﾁ (p; q) ﾁ ﾁ , (p; q), (2p; 2q), (6ﾓ12p; 6ﾓ12q),... ﾁ ﾁ ﾁ ﾁ , ﾁ (p : q) ﾁ ﾂ ﾁ l, A(2; 3) ﾁ ﾓ1 a (6ﾓ11; 4). ｨｰ ﾁ , ﾁ ﾁ (7.1), ﾁ x 0, y 0, p 07 q ﾁ ﾁ ﾁ ﾁ x 0, y , ﾂ x 0 = 2, y 0 = p 07 q ﾁ ﾁ ; p = 6ﾓ11, q = ﾁ ﾁ ﾁ (7.1), ﾁ0308: x 6ﾓ1 2 l: 6ﾓ11 = y 6ﾓ1 3 4 : ﾂ , 縺 ! ﾁ , ﾁ ﾁ ﾁ ﾁ ﾁ , ﾁ , ﾁ ﾁ ﾁ ﾁ 縺 , ﾁ ﾁ , ﾁ ﾁ ﾁ02 ﾁ ﾁ , ﾁ ﾁ ﾁ ﾁ ( ), , A(2; 3), ﾁ ﾓ1 a (6ﾓ11; 4) M(x; y): l 07 ﾁ ﾁ , ｨｮ03 ﾁ ﾁ , ﾁ ﾁ0900 ﾁ ｨｦ , M ﾊ l , ﾁ ﾓ1 a 07 6ﾓ16ﾓ1 AM , ﾁ ﾁ ﾁ ﾁ, ｨｰ ｨｰ ( ) ﾁ ﾁ ﾓ16ﾓ1 AM(x 6ﾓ1 2; y 6ﾓ1 3) ﾁ ﾁ ﾓ1 a (6ﾓ11; 4): l: x 6ﾓ1 2 6ﾓ11 = y 6ﾓ1 3 4 :

50 1348 ｨｮ ﾁ, ｨ. ｨｮ ﾁ09, ｨｮ. ｨｮ0809 ﾁ0103 ﾁ ﾁ , , ! ﾂ ﾁ l, A(2; 3) 07 B(2; 7). ｨｰ ｨｪ ﾁ ﾁ ﾁ02 ﾁ ﾁ (7.3), ﾁ02 ﾁ ﾁ ( ) A(2; 3) 07 B(2; 7), ﾁ , ﾁ AB(0; 6ﾓ1 4). ﾂ ﾁ M(x; y) ﾁ ﾓ16ﾓ1 AM(x 6ﾓ1 2; y 6ﾓ1 3) , M l , ﾁ ﾓ16ﾓ1 AM ﾁ AB, 6ﾓ ﾂ ﾁ ﾁ ﾁ , ﾁ0308: ｨｰ ｨｰ x 6ﾓ1 2 l: = y 6ﾓ : ! , ﾁ , , 09 ﾁ ﾁ ﾁ ｨ 縺 ! , 0907 ﾁ ﾁ ? ﾁ ﾁ ﾁ ? ﾁ ﾁ ﾁ ﾁ ﾁ , , , 0203 ﾁ ﾁ ﾁ ﾁ ! ﾂ ﾁ 縺0609 ﾁ ｨ ﾁ 縺 , ﾁ ﾁ ﾁ : x 6ﾓ1 2 = y 6ﾓ ﾁ ﾁ 6ﾍ ﾁ ﾁ ﾍ2 4 (x 6ﾓ1 2) = 0 (y 6ﾓ1 3) 6ﾍ2 x 6ﾓ1 2 = 0: , ﾁ x = ﾁ l, M 0 (6ﾓ11; 0) ﾁ ﾓ1 n (5; 2). ｨｰ ( ), ﾁ ﾁ M(x; y) ﾁ ﾁ , M , ﾁ ﾓ16ﾓ16ﾓ1 M 0 M(x + 1; y) ﾁ ﾓ1 n ﾁ ﾁ ﾁ ﾁ : ｨｰ ｨｰ ｨｲ , ﾁ ﾁ0702: 6ﾓ16ﾓ16ﾓ1 M 0 M ﾍ 6ﾓ1 n 6ﾍ2 ( 6ﾓ16ﾓ16ﾓ1 M 0 M; 6ﾓ1 n ) = 0: ｨｮ ﾁ ﾓ1 n (5; 2) 07 6ﾓ16ﾓ16ﾓ1 M 0 M(x+1; y) ﾁ : 5(x + 1) + 2y = 0: ( 6ﾓ16ﾓ16ﾓ1 M 0 M; 6ﾓ1 n ) = 5(x + 1) + 2y: l: 2x 6ﾓ1 y + 4 = 0. ｨｰ ﾁ ﾁ ﾁ ﾁ ﾁ ｨｦ ﾁ ﾓ1 N (2; 6ﾓ11) ﾓ1 N = ﾌ = ﾌ 5.

51 13ｨｬ ﾁ ﾓ1 N , ﾁ ﾓ1 n ( 2 ﾌ5 ; 6ﾓ11 ﾌ 5 ), ﾁ ﾓ1 N ﾁ ﾁ l 0209 ﾌ 5, ﾁ ﾌ 5 x 6ﾓ1 1 ﾌ 5 y + 4 ﾌ 5 = 0; ﾁ , ﾁ0702, ( ) 2 2 ( ) 6ﾓ11 2 ﾌ5 + ﾌ = 1: ﾁ ﾁ , ﾁ ﾁ ﾁ ﾓ1, , ﾌ ﾁ ﾁ ﾁ (6ﾓ11), ﾁ : ﾁ ﾌ 5. 6ﾓ1 2 ﾌ 5 x + 1 ﾌ 5 y 6ﾓ1 4 ﾌ 5 = A(1; 2) l: 2x 6ﾓ1 y + 4 = 0. ｨｰ ﾁ ﾁ ﾁ ﾁ070209, ﾁ ﾁ A 07 ﾁ ﾁ ﾁ ﾁ ﾁ : 6ﾓ1 ﾌ 2 5 x + ﾌ 1 5 y 6ﾓ1 ﾌ 4 5 = ﾁ0701 ﾁ A ﾁ0308: ｨO (A; l) = ｨO 6ﾓ1 ﾌ ﾌ 1 2 6ﾓ1 4 ｨO ｨOｨOｨO ﾌ = ﾌ 4 : ﾂ : l 1 : 2x 6ﾓ1 3y + 4 = 0; l 2 : 2x 6ﾓ1 3y + 2 = 0: ｨｰ ﾂ ﾁ ｨ ﾁ ﾁ ﾁ ( ). O = ｧﾑ) ｧﾒ) ｨｰ ﾂ ﾁ , ﾁ ﾁ 0309 ﾁ ( ) ﾁ ﾁ ﾁ O(0; 0) ,

52 1350 ｨｮ ﾁ, ｨ. ｨｮ ﾁ09, ｨｮ. ｨｮ0809 ﾁ0103 ﾁ ｨｮ ﾁ ﾁ ﾁ070209, ﾁ (6ﾓ1 ﾌ 13): l 1 : 6ﾓ1 2x ﾌ y ﾌ 13 6ﾓ1 4 ﾌ 13 = 0; l 2 : 6ﾓ1 2x ﾌ y ﾌ 13 6ﾓ1 2 ﾌ 13 = 0: (O; l 1 ) = ﾌ 4 13, (O; l 2 ) = ﾌ , ﾁ , : ﾁ ﾁ , , ﾁ ﾁ , ﾁ ! ﾂ ﾓ1 n 1 = 6ﾓ1 n 2 (6ﾓ1 ﾌ 2 13 ; ﾌ3 13 ) , , 09. ｨｮ ﾁ , (l 1 ; l 2 ) = (O; l 1 ) 6ﾓ1 (O; l 2 ) = 4 ﾌ 13 6ﾓ1 2 ﾌ 13 = 2 ﾌ 13 : ﾂ l 1 : x + 3y 6ﾓ1 4 = 0; l 2 : 2x 6ﾓ1 y + 7 = 0: ｨｰ ﾁ ( ) , ﾁ ', ﾁ ﾁ = ﾁ ( ﾁ ﾓ1 n 1 (1; 3) 07 6ﾓ1 n 2 (2; 6ﾓ11)). ｨｪ , , , ﾁ ' + = 07 cos = 6ﾓ1 cos ' < ﾁ , ﾁ ﾁ ﾂ ﾁ ﾓ1 n ﾓ1 n 2. ｨｹ ﾁ : cos = (6ﾓ1 n 1 ; 6ﾓ1 n 2 ) ｨｰ ﾓ1 n 1 6ﾓ1 n 2 : ｨｮ ﾁ ﾁ ﾁ, , ﾁ ﾁ : ( 6ﾓ1 n 1 ; 6ﾓ1 n 2 ) = (6ﾓ11) = 2 6ﾓ1 3 = 6ﾓ11; 6ﾓ1 n 1 = ﾌ = ﾌ 10; 6ﾓ1 n 2 = ﾌ (6ﾓ11) 2 = ﾌ 5: ｨｦ080909, cos = 5 6ﾓ11 ﾌ cos , ' = 6ﾓ1 07 cos ' = cos( ) = 5 ﾌ1 2. ｨｮ ﾁ , ' = arccos 1 5 ﾌ ﾁ , ﾁ l 1 : x + 3y 6ﾓ1 4 = 0; l 2 : 2x 6ﾓ1 y + 7 = 0: ｨｰ ｨｲ ﾁ ﾁ ﾁ ﾁ , : , 0609 ﾁ ｨｦ ﾁ090107, M(x; y) ﾁ ﾁ ﾁ03: (M; l 1 ) = (M; l 2 ).

53 13ｨｬ ｨｰ ﾁ ﾁ ﾁ : l 1 : x ﾌ + ﾌ 3y 6ﾓ1 ﾌ 4 = 0; l 2 : 6ﾓ1 ﾌ 2x + ﾌ y 6ﾓ1 ﾌ 7 = 0: (M; l 1 ) = ｨO x ﾌ + ﾌ 3y 6ﾓ1 4 ｨO ｨOｨOｨO ﾌ ; (M; l 2 ) = ｨO 6ﾓ1 2x ﾌ 5 + y ﾌ 5 6ﾓ1 7 ﾌ 5 ｨO ｨOｨOｨO : ﾁ : ｨO x ﾌ + ﾌ 3y 6ﾓ1 4 ｨO ｨOｨOｨO ﾌ = ｨO 6ﾓ1 2x ﾌ 5 + y ﾌ 5 6ﾓ1 7 ﾌ 5 ｨO ｨOｨOｨO : ﾁ ﾁ M(x; y) , ﾁ ﾁ ﾁ ﾁ ﾁ ﾁ ﾁ , ｨｦ ﾁ , a = b , a = ﾀb: x ﾌ + ﾌ 3y 6ﾓ1 ﾌ 4 = 6ﾓ1 ﾌ 2x + ﾌ y 6ﾓ1 ﾌ 7 ; x ﾌ + ﾌ 3y 6ﾓ1 ﾌ 4 = ﾌ 2x 6ﾓ1 ﾌ y + ﾌ 7 : ﾁ ﾁ , ﾂ ﾁ ﾁ ﾁ ﾁ , , ﾁ , ﾁ ﾁ 0703 ﾁ ｨｮ ﾁ , ﾁ 0308 ﾁ ﾁ ﾁ b 1 07 b , ﾁ : b 1 : (1 + 2 ﾌ 2)x + (3 6ﾓ1 ﾌ 2)y 6ﾓ ﾌ 2 = 0; b 2 : (1 6ﾓ1 2 ﾌ 2)x + (3 + ﾌ 2)y 6ﾓ1 4 6ﾓ1 7 ﾌ 2 = 0: ﾁ ｨｮ ﾁ ﾁ ｨｲ ﾁ ﾁ ﾁ, x 07 y? 7.3. ｨｮ ﾁ ﾁ , , ﾁ ﾁ ｨｮ ﾁ ﾁ , , 07 ﾁ ｨｲ ﾁ ﾁ k 07 b? 7.6. ｨｲ ﾁ ﾁ ﾁ ﾁ ? 7.7. ｨｲ ﾁ , M 0 (x 0 ; y 0 ) , M 0, ﾁ ﾁ ? ﾁ ﾁ ｨｮ ﾁ ｨｲ ｨｬ M 1 (2; 6ﾓ13), M 2 (6ﾓ12; 6ﾓ13), M 3 (2; 3) x 6ﾓ1 3y + 7 = 0?

54 1352 ｨｮ ﾁ, ｨ. ｨｮ ﾁ09, ｨｮ. ｨｮ0809 ﾁ0103 ﾁ ｨｲ ﾁ , ﾁ ﾁ Ox : 09) 45 o ; 00) 90 o ; ﾁ) 120 o ; 05) 135 o : A(2; 0) o ﾁ Ox ﾁ ｨｮ ﾁ ﾁ , A(2; 4) 07 B(6ﾓ11; 3) ? x + 2y 6ﾓ1 6 = 0; y = 6ﾓ12x + 1; x 3 + y 4 = 1; 2x + 5 = 0, y + 3 = ﾁ ﾁ ﾁ A(8; 6ﾓ12), B(2; 5), C(0; 0). ｨｮ ﾁ ﾁ ﾁ Ox M(1; 3) x 6ﾓ1 4y + 1 = 0. ｨｲ ﾁ ? ｨｮ ﾁ ﾁ , P(6ﾓ12; 1) x + 3y + 5 = x 6ﾓ1 3y 6ﾓ1 1 = x 6ﾓ1 y + 3 = ｨｮ , ﾁ x 6ﾓ1y 6ﾓ12 = 0; y = x 6ﾓ12, x 6ﾓ1 2 = ﾁ A(2; 0) 07 B(0; 4) C(1; 1) ﾁ , ﾁ ﾁ A(1; 6ﾓ11) 07 B(3; 1) C(3; 2) ﾁ , ﾁ ｨｮ ﾁ ﾁ Ox 07 Oy a = 2 07 b = ﾁ ﾁ ｨｮ ﾁ ﾁ ﾁ ﾁ , A(6ﾓ13; 3), B(5; 5) ﾂ ﾁ , A(1; 6ﾓ11) ﾁ ﾁ ﾁ ﾓ1 a (2; 7) ｨｦ03 ﾁ , O(0; 0) 07 A(4; 0) 05 ﾁ ﾁ , ﾁ B(0; 2). ｨｮ ﾁ ﾁ M(6ﾓ16; 12) x + 7y + 5 = P(6ﾓ13; 6) , A(7; 6ﾓ19) 07 B(0; 6ﾓ15).

55 13ｨｬ ﾁ , A(4; 3) S = ﾁ ﾁ M(1; 6), x + 2y = ﾁ ﾁ ﾁ ﾁ A(6ﾓ13; 4); B(5; 2), C(8; 2) ﾁ , ﾁ , ﾁ ﾁ B AC ﾁ ABC, A(3; 0), B(1; 6ﾓ13), C(1; 3) , : 09) l 1 : 3x 6ﾓ1 y + 11 = 0; l 2 : x + 3y 6ﾓ1 7 = 0. 00) l 1 : 3x + 2y 6ﾓ1 1 = 0; l 2 : A(1; 4); B(6ﾓ15; 6ﾓ111). ﾁ) l 1 : A(0; 1); B(6; 6ﾓ13); l 2 : M(3; 8); N(6ﾓ11; 2) ﾁ ﾁ ABC, ﾁ A(1; 6ﾓ13), B(0; 0), C(3; 9) A(2; 6ﾓ15) x 6ﾓ1 y 6ﾓ1 5 = M(6ﾓ18; 12) (AB); A(2; 6ﾓ13); B(6ﾓ15; 1) ﾂ x 6ﾓ1 2y = x 6ﾓ1 4y = 7, ﾁ, ﾂ ﾁ , : 09) 3x 6ﾓ1 4y + 5 = 0; 00) y = 3 x + 2; ﾁ) x = 3; 05) y = 6ﾓ11: ﾁ , A(2; 0) B(6; 0) ｨｮ ﾁ ﾁ , ﾁ l 1 07 l ﾁ , l 1 : 2x 6ﾓ1 5y + 7 = 0, l 2 : 4x 6ﾓ1 10y + 9 = ﾂ ABC 0703 ﾁ ﾁ A(5; 6ﾓ13) ﾁ ﾁ0906 ﾁ : 2x 6ﾓ1 y = 0, 3x 6ﾓ1 7y + 16 = ﾁ ﾂ ABC 0703 ﾁ ﾁ A(2; 6), ﾁ ﾁ x + 2y 6ﾓ1 27 = x + 5y 6ﾓ1 8 = 0, ﾁ ﾁ ｨｮ ﾁ ﾁ ﾂ ABC 0703 ﾁ ﾁ A(1; 3) ﾁ ﾁ x 6ﾓ12y +1 = 0, y 6ﾓ1 1 = 0. ｨｮ ﾁ ﾁ P(0; 10) ﾁ , , x 6ﾓ1 3y + 10 = x + y 6ﾓ1 8 = ﾁ P ﾂ ABC 0703 ﾁ ﾁ A(2; 3), B(1; 0) ﾁ ﾁ ﾁ ABC: A(6ﾓ18; 8), B(2; 6ﾓ17), C(6ﾓ15; 6ﾓ13) ﾁ , ﾁ ﾁ A , ﾁ ﾁ B ﾓ , ﾁ090308, , ﾁ , ﾁ ﾓ , ﾁ l 1 07 l 2, ﾁ y = k 1 x + b 1 07 y = k 2 y + b 2, ﾁ tg ' = k 2 6ﾓ1 k k 1 k 2 :

56 ｨｮ ﾁ ﾁ ﾁ ﾁ , 0907 ﾁ0307 ﾁ , ﾁ , ﾁ ｨｰ , ﾁ M 0 (x 0 ; y 0 ; z 0 ) 07 ﾁ ﾓ1 n (A; B; C) ﾁ , M ﾁ ﾓ1 n ( ) ﾁ , ﾁ ﾁ M(x; y; z) 07 ﾁ , ﾁ ﾁ ﾁ ﾓ1 n, , , ﾂ , (M 0 M) ( , 07 ﾁ ﾓ16ﾓ16ﾓ1 M 0 M) ﾓ1 n , M ( ) ﾁ ﾓ16ﾓ16ﾓ1 M 0 M ﾁ : 6ﾓ16ﾓ16ﾓ1 M 0 M(x 6ﾓ1 x 0 ; y 6ﾓ1 y 0 ; z 6ﾓ1 z 0 ), ﾁ ﾁ 0609 ﾁ , ﾁ ﾁ : ( 6ﾓ1 n ; 6ﾓ16ﾓ16ﾓ1 M 0 M) = , ﾁ ﾁ0702: : A(x 6ﾓ1 x 0 ) + B(y 6ﾓ1 y 0 ) + C(z 6ﾓ1 z 0 ) = 0: (8.1) ﾂ ﾁ ﾓ1 n (A; B; C) ﾁ , , ﾁ , 09 (x 0 ; y 0 ; z 0 ) , ﾁ ﾁ ﾁ (8.1) ﾁ , Ax + By + Cz 6ﾓ1 (Ax 0 + By 0 + Cz 0 ) = 0: ﾁ (6ﾓ1(Ax 0 +By 0 +Cz 0 )) , D. ﾂ ﾁ ﾁ0702: Ax + By + Cz + D = 0: (8.2) ﾁ ﾁ ﾁ ｨｪ , (A; B; C) , ﾁ ﾁ : ﾁ ﾁ ? ﾁ , (A; B; C) = (0; 0; 0), ﾁ ( D ﾙ 0), ﾁ ﾁ ﾁ09 ( D = 0) , A; B; C ﾁ ﾁ (A; B; C) ﾓ1 n , ﾁ ﾁ ﾁ (8.2). 54

57 13ｨｬ ﾓ1 n (A; B; C) ﾙ 0, , A ﾙ ﾁ ﾁ ﾁ y = z = 0, x: Ax + D = 0 6ﾍ2 x = 6ﾓ1D=A: , M 0 (6ﾓ1D=A; 0; 0) ﾁ ﾁ (8.2) ﾁ (8.2) ﾁ ﾁ A(x + D=A) + By + Cz = 0; ﾁ ﾁ ﾁ ﾁ ﾁ 6ﾓ1 n (A; B; C) 07 6ﾓ16ﾓ16ﾓ1 M 0 M(x + D=A; y; z); ﾁ ﾁ ﾁ ﾁ , ﾁ ﾁ 6ﾓ1 n (A; B; C) 07 6ﾓ16ﾓ16ﾓ1 M 0 M(x + D=A; y; z). ｨｲ , (x; y; z) M ﾁ ﾁ ﾁ (8.2) , ﾓ1 n (A; B; C) ﾍ 6ﾓ16ﾓ16ﾓ1 M 0 M(x + D=A; y; z): , M(x; y; z) , M ﾁ ﾓ1 n (A; B; C). ｨｦ080909, , ﾓ1 n (A; B; C) ﾙ 0, ﾁ (8.2) ﾁ , ﾓ1 n (A; B; C) ! ﾁ , ﾁ ﾁ , ﾁ , M 0 (x 0 ; y 0 ; z 0 ), ﾁ ﾓ1 a (a x ; a y ; a z ) 07 6ﾓ1 b (b x ; b y ; b z ). ｨｲ , ﾁ , ﾁ ﾁ ! ﾁ ﾁ09 M(x; y; z). ｨｲ , ﾁ M ? , ｨｰ ﾁ ﾓ1 a, 6ﾓ1 b 07 6ﾓ16ﾓ16ﾓ1 M0M , ﾁ ﾓ1 a, 6ﾓ1 b 07 6ﾓ16ﾓ16ﾓ1 M 0 M ﾁ ﾁ , ﾁ ﾁ ﾁ ﾁ ﾁ : : ｴ 6ﾓ16ﾓ16ﾓ1 M 0 M; 6ﾓ1 a ; 6ﾓ1 b ｵ = ｨO x 6ﾓ1 x 0 y 6ﾓ1 y 0 z 6ﾓ1 z 0 a x a y a z b x b y b z ｨO = 0: (8.3) , ﾁ , ﾁ ﾁ ﾁ070209: ｨO ｨO x 6ﾓ1 x 0 y 6ﾓ1 y 0 z 6ﾓ1 z 0 a x a y a z b x b y b = z ｨO = (x 6ﾓ1 x 0 )(a y b z 6ﾓ1 a z b y ) 6ﾓ1 (y 6ﾓ1 y 0 )(a x b z 6ﾓ1 a z b x ) + (z 6ﾓ1 z 0 )(a x b y 6ﾓ1 a x b y ) = 0: ｨｬ ﾁ02 ﾁ ﾁ (8.3) , 0308 ﾁ ﾁ ﾁ ﾁ , ﾁ

58 1356 ｨｮ ﾁ, ｨ. ｨｮ ﾁ09, ｨｮ. ｨｮ0809 ﾁ0103 ﾁ ﾁ (8.3), ﾁ , ﾁ ﾁ ﾁ070209, ﾁ ﾁ ( , ﾁ ) ﾁ ﾁ : , ﾁ , ﾁ (8.3) ﾁ ﾁ ﾁ ﾁ ﾁ ﾁ , M 0 (x 0 ; y 0 ; z 0 ) , ﾁ07 ﾁ ﾁ (x; y; z) ﾁ ﾁ ﾁ ｨO x 0 6ﾓ1 x 0 y 0 6ﾓ1 y 0 z 0 6ﾓ1 z 0 a x a y a z b x b y b z , , 0609 ﾁ ｨｲ , M ﾁ ﾁ ﾁ (8.3), M 0 ﾊ. ﾂ ﾁ M 1 (x 1 ; y 1 ; z 1 ) ﾁ ﾁ 0003 ﾁ ﾁ (8.3): ｨO x 1 6ﾓ1 x 0 y 1 6ﾓ1 y 0 z 1 6ﾓ1 z 0 a x a y a z b x b y b z ﾁ ﾁ , ﾁ ﾓ16ﾓ16ﾓ16ﾓ1 M 0 M 1, 6ﾓ1 a 07 6ﾓ1 b, ﾁ , ﾁ , , ｨｹ , M 1 ﾊ ﾁ , ﾁ ﾁ ﾁ , , ﾁ ﾁ , M 0 (x 0 ; y 0 ; z 0 ) ﾁ ﾓ1 a (a x ; a y ; a z ) 07 6ﾓ1 b (b x ; b y ; b z ). ｨｪ ﾁ , M , ﾁ ﾓ1 a, 6ﾓ1 b 07 6ﾓ16ﾓ16ﾓ1 M 0 M , ﾁ , ﾁ ｨｮ , ﾁ ﾓ1 a 07 6ﾓ1 b , ﾁ ﾁ ﾁ ﾁ 6ﾓ1 a, 6ﾓ1 b 07 6ﾓ16ﾓ16ﾓ1 M 0 M ﾁ , ﾁ ﾓ16ﾓ16ﾓ1 M 0 M ﾁ ﾓ1 a 07 6ﾓ1 b : 6ﾓ16ﾓ16ﾓ1 M 0 M = t 6ﾓ1 6ﾓ1 1 a + t 2 b ; t 1 07 t ﾁ ﾁ ﾁ0701 ﾁ , , ﾁ ﾁ ﾁ ﾁ ﾁ , ﾁ ﾁ , ﾁ : (x 6ﾓ1 x 0 ; y 6ﾓ1 y 0 ; z 6ﾓ1 z 0 ) = t 1 (a x ; a y ; a z ) + t 2 (b x ; b y ; b z ) 6ﾍ2 6ﾍ2 (x 6ﾓ1 x 0 ; y 6ﾓ1 y 0 ; z 6ﾓ1 z 0 ) = (t 1 a x ; t 1 a y ; t 1 a z ) + (t 2 b x ; t 2 b y ; t 2 b z ) 6ﾍ2 6ﾍ2 (x 6ﾓ1 x 0 ; y 6ﾓ1 y 0 ; z 6ﾓ1 z 0 ) = (t 1 a x + t 2 b x ; t 1 a y + t 2 b y ; t 1 a z + t 2 b z ): ﾁ ﾁ ﾁ : : ; ｨO : ｨO x = x 0 + t 1 a x + t 2 b x ; y = y 0 + t 1 a y + t 2 b y ; z = z 0 + t 1 a z + t 2 b z : ｨｲ (x 0 ; y 0 ; z 0 ) , , (a x ; a y ; a z ) 07 (b x ; b y ; b z ) ﾁ ﾁ, , 09 t 1 07 t (8.4)

59 13ｨｬ ﾁ , ﾁ ﾁ , ﾁ ﾁ , M 0 (x 0 ; y 0 ; z 0 ), M 1 (x 1 ; y 1 ; z 1 ) ﾁ ﾓ1 a (a x ; a y ; a z ). ｨｰ ﾁ ﾁ ( , 09), ﾁ ﾁ09 M(x; y; z) M 0 07 M , 0803 ﾁ ﾓ16ﾓ16ﾓ16ﾓ1 M 0 M ( , 00) , ﾁ ﾁ ﾁ , M ﾁ0901 ﾁ ﾓ1 a 07 6ﾓ16ﾓ16ﾓ16ﾓ1 M 0 M 1, ( ). ｨｰ , ﾁ ﾁ ｨｮ , , ﾁ ｨｦ , 00, , M , ﾁ ﾓ1 a, 6ﾓ16ﾓ16ﾓ16ﾓ1 M 0 M ﾓ16ﾓ16ﾓ1 M 0 M , ﾁ ﾁ ﾁ ﾁ ﾁ ﾁ 6ﾓ1 a (a x ; a y ; a z ), 6ﾓ16ﾓ16ﾓ1 M 0 M(x 6ﾓ1 x 0 ; y 6ﾓ1 y 0 ; z 6ﾓ1 z 0 ) 07 6ﾓ16ﾓ16ﾓ16ﾓ1 M 0 M 1 (x 1 6ﾓ1 x 0 ; y 1 6ﾓ1 y 0 ; z 1 6ﾓ1 z 0 ). ｨｮ ﾁ ﾁ : ｴ 6ﾓ16ﾓ16ﾓ1 M 0 M; 6ﾓ16ﾓ16ﾓ16ﾓ1 M 0 M 1 ; 6ﾓ1 a ｵ = ｨO x 6ﾓ1 x 0 y 6ﾓ1 y 0 z 6ﾓ1 z 0 x 1 6ﾓ1 x 0 y 1 6ﾓ1 y 0 z 1 6ﾓ1 z 0 a x a y a z ﾁ ﾁ : : ｨO x 6ﾓ1 x 0 y 6ﾓ1 y 0 z 6ﾓ1 z 0 x 1 6ﾓ1 x 0 y 1 6ﾓ1 y 0 z 1 6ﾓ1 z 0 a x a y a z ｨO : ｨO = 0: (8.5) ﾁ , (8.5) ﾁ ﾁ ﾁ , , ﾁ ﾁ , ﾂ ﾁ ﾁ , ﾁ , M 0 (x 0 ; y 0 ; z 0 ), M 1 (x 1 ; y 1 ; z 1 ) 07 M 2 (x 2 ; y 2 ; z 2 ). ﾂ ﾁ ﾁ ( ) , ﾁ ﾁ ﾁ , ﾁ ﾁ ﾁ , ﾁ ﾁ , M 0 07 M 1, M 0 07 M ﾁ ﾁ ﾁ : 6ﾓ16ﾓ16ﾓ16ﾓ1 M 0 M 1 (x 1 6ﾓ1 x 0 ; y 1 6ﾓ1 y 0 ; z 1 6ﾓ1 z 0 ); 6ﾓ16ﾓ16ﾓ16ﾓ1 M 0 M 2 (x 2 6ﾓ1 x 0 ; y 2 6ﾓ1 y 0 ; z 2 6ﾓ1 z 0 ): ﾂ ﾁ ﾁ ﾁ09 M(x; y; z) ﾁ ﾓ16ﾓ16ﾓ1 M 0 M(x 6ﾓ1 x 0 ; y 6ﾓ1 y 0 ; z 6ﾓ1 z 0 ):

60 1358 ｨｮ ﾁ, ｨ. ｨｮ ﾁ09, ｨｮ. ｨｮ0809 ﾁ0103 ﾁ ﾁ ﾁ : : ｨO x 6ﾓ1 x 0 y 6ﾓ1 y 0 z 6ﾓ1 z 0 x 1 6ﾓ1 x 0 y 1 6ﾓ1 y 0 z 1 6ﾓ1 z 0 x 2 6ﾓ1 x 0 y 2 6ﾓ1 y 0 z 2 6ﾓ1 z ﾁ , ﾁ ﾁ ﾁ ! ｨO = 0: (8.6) 53. ﾂ ｨｦ ﾁ , ﾁ ﾁ03 ﾁ ﾁ ｨｦ ﾁ030605, ﾁ : 1) 0703 ﾁ : 1 = 2 ; 2) , ﾁ : 1 2, 1 ﾙ 2 ; 3) l: 1 ﾉ 2 = l. ｨｲ ﾁ , ﾁ ﾁ (8.2): 1 : A 1 x + B 1 y + C 1 z + D 1 = 0; 2 : A 2 x + B 2 y + C 2 z + D 2 = 0: ｨｲ , ﾁ ﾁ ﾁ ﾁ , ﾁ ﾁ , ﾁ ﾁ ｨｲ , ﾁ ﾁ ﾁ ｨｹ : , ﾁ ﾁ : A 1 A 2 = B 1 B 2 = C 1 C 2 = D 1 D 2 : (8.7) A 1 A 2 = B 1 B 2 = C 1 C 2 ﾙ D 1 D 2 : (8.8) A 1 = B 1 = C 1 ; A 2 B 2 C ﾁ ﾁ 6ﾓ1 n 1 (A 1 ; B 1 ; C 1 ) 07 6ﾓ1 n 2 (A 2 ; B 2 ; C 2 ) , , ﾁ ﾓ1 n ( ), 09 6ﾓ1 n ( ). ｨｲ , , ﾁ0703 (8.8) ﾁ , , ｨｦ ﾁ030605, ﾁ ﾁ ｨｰ ﾁ , ﾁ ﾁ ﾁ

61 13ｨｬ ﾁ ﾁ ﾁ ﾁ (8.2) 07 A ﾙ 0, B ﾙ 0, C ﾙ 0 07 D ﾙ ﾁ : Ax + By + Cz + D = 0 6ﾍ2 Ax + By + Cz = 6ﾓ1D 6ﾍ2 6ﾍ2 6ﾓ1 A D x 6ﾓ1 B D y 6ﾓ1 C D z = 1 6ﾍ2 x 6ﾓ1D=A + y 6ﾓ1D=B + z 6ﾓ1D=C = 1: ﾂ ﾁ a = 6ﾓ1D=A, b = 6ﾓ1D=B, c = 6ﾓ1D=C ﾁ ﾁ ﾁ070203: x a + y b + z = 1: (8.9) c ﾁ ﾁ ﾁ ﾁ ﾂ ﾁ (8.9) a; b; c , ﾁ ﾁ ﾁ ﾁ ﾁ a; b; c ﾁ , y = 0, z = ﾁ (8.9) P(a; 0; 0) Ox ( ). ｨ Q(0; b; 0) Oy, 09 R(0; 0; c) Oz ﾁ , a, b 07 c ﾁ, , ﾁ (8.9) ﾁ ﾁ ﾁ ﾁ ﾁ ﾂ ﾁ ﾁ , , , , ﾁ ﾁ ﾁ0901 ﾁ , ﾁ ﾁ , ( , ﾁ ﾁ ) ﾁ , ﾁ ﾁ ﾁ , 0803, ﾁ , ﾁ , P; Q; R. ｨｲ , ﾁ (P Q), (QR), (P R), ﾁ02 ﾁ ﾁ (8.9) , A ﾙ 0, B ﾙ 0, C ﾙ 0 07 D ﾙ 0. ｨｰ ﾁ , ﾁ 0703 A; B; C 0609 ﾁ A = 0, ﾁ (8.2) ﾁ0702 By + Cz + D = 0; ｨｰ ﾁ ﾁ ﾁ ﾁ y b + z c = 1: ﾁ x, Ox.

62 1360 ｨｮ ﾁ, ｨ. ｨｮ ﾁ09, ｨｮ. ｨｮ0809 ﾁ0103 ﾁ ｨｰ ﾁ , Ox, Q(0; b; 0) 07 R(0; 0; c) ﾁ ﾁ Oy 07 Oz ( , 09). ｨｮ (QR) QM 1 07 RM 2, Q 07 R Ox. ﾂ , Ox, ( , 09) A = 0 07 B = ﾁ (8.2) 05 ﾁ Cz + D = 0; ﾁ ﾁ : z c = 1: ﾁ x 07 y, Ox, Oy, Oxy ( , 00) , Oxy, R(0; 0; c) Oz (RM 1 ) 07 (RM 2 ), Oy 07 Oz ( ﾁ ﾁ ﾁ Oyz 07 Oxz), , 00. ｨｰ , , D = A ﾙ 0, B ﾙ 0, C ﾙ ﾁ (8.2) ﾁ0702 Ax + By + Cz = 0: D = ﾁ ﾁ ﾁ ﾁ ｨｰ , O(0; 0; 0) ﾁ ﾁ ﾁ , ﾂ ﾂ02, 0209 ﾁ , ﾁ , , ﾂ , ﾂ Oyz x = 0, ﾁ Oyz ﾁ0702 { x = 0; By + Cz = 0: ｨｮ , ( ).

63 13ｨｬ Oxz: { y = 0; Ax + Cz = 0: ( ) ﾁ ﾁ , ﾁ , ﾁ ｨｲ ( ) ﾁ x + y + z 6ﾓ1 = ﾁ , ) = 1; 2) ﾁ , ﾁ ﾁ x + y + z 6ﾓ1 = ) ﾓ1 n = 1; 2) ﾓ1 n ﾁ ; 3) ﾁ ﾁ ﾁ ﾁ ﾁ ﾁ ﾁ, ﾓ1 n (; ; ) ( ), ﾁ ﾁ ﾁ ﾁ ﾁ M M 0 ﾊ ( ) M 0 (x 0 ; y 0 ; z 0 ) ﾓ16ﾓ16ﾓ1 OM , ﾓ16ﾓ16ﾓ1 OM 0 6ﾓ1 n. ｨｲ , ｨｦ ﾁ090107, , x 0 =, y 0 =, z 0 = ( ﾁ ﾓ16ﾓ16ﾓ1 OM ﾁ0202 (x 0 ; y 0 ; z 0 )) ﾁ , , 0803 ﾁ ﾓ16ﾓ16ﾓ1 OM ﾓ1 n ﾁ ( ), ﾁ ﾁ ﾁ ﾁ , ﾁ ﾁ ﾓ16ﾓ16ﾓ1 OM 0 ( ﾓ16ﾓ16ﾓ1 OM 0 = 6ﾓ1 n 07 6ﾓ16ﾓ16ﾓ1 OM 0 = 6ﾓ1 n = 1 = ) M 0 ﾊ, (; ; ) ﾁ ﾁ ﾁ , () + () + () 6ﾓ1 = 0 05 ﾁ ﾁ ﾁ : ｨｰ ﾁ () + () + () 6ﾓ1 = 0 6ﾍ2 ( ) 6ﾓ1 = 0 6ﾍ2 = : ｨｦ080909, , = , , ﾁ ﾁ0206, ﾁ0203, 09, ﾁ03- ﾁ , ﾓ1 n ﾁ , ( ?). 7ﾁ3

64 1362 ｨｮ ﾁ, ｨ. ｨｮ ﾁ09, ｨｮ. ｨｮ0809 ﾁ0103 ﾁ , ﾁ , ﾁ ﾁ , ﾁ ﾁ x + y + z 6ﾓ1 = 0, M 0 (x 0 ; y 0 ; z 0 ) ﾁ0203 ｨO ｨOx 0 + y 0 + z 0 6ﾓ1 ｨO ｨO: ﾁ03. ｨｪ , ﾁ ﾁ ﾁ ﾁ ﾁ M 0 uvw, ﾁ M 0 ( ). ｨｦ ﾁ , u = x6ﾓ1x 0, v = y6ﾓ1y 0 07 w = z6ﾓ1z 0. ﾂ x, y 07 z ﾁ ﾁ ﾁ , ﾁ ﾁ 0203 ﾁ : ｨｰ ﾁ : : (x 0 + u) + (y 0 + v) + (z 0 + w) 6ﾓ1 = 0: : u + v + w 6ﾓ1 ( 6ﾓ1 x 0 6ﾓ1 y 0 6ﾓ1 z 0 ) = 0: (8.10) ｨｮ ﾁ , ﾁ ﾁ Oxyz 05 ﾁ , = , (6ﾓ1x 0 6ﾓ1y 0 6ﾓ1z 0 ) 1 7 0, ﾁ (8.10), ﾁ ﾁ M 0 uvw, ( M 0 ) ﾁ0203 6ﾓ1 x 0 6ﾓ1 y 0 6ﾓ1 z 0 ( ) ( 6ﾓ1 x 0 6ﾓ1 y 0 6ﾓ1 z 0 ) < 0, ﾁ : 6ﾓ1 u 6ﾓ1 v 6ﾓ1 w 6ﾓ1 (6ﾓ1 + x 0 + y 0 + z 0 ) = ﾁ0203 (6ﾓ1 + x 0 + y 0 + z 0 ). ｨｰ ｨｮ M0uvw ﾁ Oxyz , ﾁ ﾁ , M 0 (x 0 ; y 0 ; z 0 ) , ﾁ x + y + z 6ﾓ1 = 0, 0609 ﾁ0203 (M 0 ; ) = ｨO ｨOx 0 + y 0 + z 0 6ﾓ1 ｨO ｨO: 7ﾁ3 (8.11)

65 13ｨｬ ﾁ ﾁ , A(1; 2; 6ﾓ13) ﾁ ﾓ1 n = 6ﾓ13 6ﾓ1 { + 5 6ﾓ ﾓ1 k. ｨｰ M(x; y; z) ﾁ ﾁ ﾓ16ﾓ1 AM: 6ﾓ16ﾓ1 AM(x 6ﾓ1 1; y 6ﾓ1 2; z + 3): ﾁ ﾓ16ﾓ1 AM 07 6ﾓ1 n , ﾁ ﾁ , ﾁ : 6ﾓ13(x 6ﾓ1 1) + 5(y 6ﾓ1 2) + 2(z + 3) = 0: ﾁ : 3x 6ﾓ1 5y 6ﾓ1 2z + 1 = 0: M(0; 1; 2) ﾁ ﾓ1 l = 6ﾓ1 6ﾓ1 { +2 6ﾓ ﾓ1 k 07 6ﾓ1 m = 6ﾓ1 6ﾓ1 + 6ﾓ1 k ﾁ ｨｰ ｨｦ ﾁ ﾁ ﾁ ﾁ ﾁ ﾓ1 n = [ 6ﾓ1 l ; 6ﾓ1 m] = ｨO 6ﾓ1 k ｨO 6ﾓ1 { 6ﾓ1 6ﾓ = 0 6ﾓ11 1 ｨO 56ﾓ1 { + 6ﾓ1 + 6ﾓ1 k : ﾁ ﾁ ﾁ070501, ﾁ ﾁ : 5 (x 6ﾓ1 0) + 1 (y 6ﾓ1 1) + 1 (z 6ﾓ1 2) = 0 6ﾍ2 5x + y + z 6ﾓ1 3 = 0: ﾁ , , ﾁ , ﾁ ﾁ ﾁ0906 ﾁ ﾁ, ﾁ ﾁ ﾓ16ﾓ1 AM, ﾁ ﾁ ﾁ ﾁ 6ﾓ16ﾓ1 AM(x; y 6ﾓ1 1; z 6ﾓ1 2), 6ﾓ1 l (6ﾓ11; 2; 3) 07 6ﾓ1 m(0; 6ﾓ11; 1), ﾁ ( , ). ｨｲ ﾁ ﾁ ﾁ: ｴ 6ﾓ16ﾓ1 AM; 6ﾓ1 l ; 6ﾓ1 m ｵ = 0 6ﾍ2 ｨO x y 6ﾓ1 1 z 6ﾓ1 2 6ﾓ ﾓ11 1 ｨO ｨO = 0 6ﾍ2 6ﾍ2 x (2 1 6ﾓ1 3 (6ﾓ11) ) 6ﾓ1 (y 6ﾓ1 1) ( 6ﾓ11 1 6ﾓ1 3 0 ) + (z 6ﾓ1 2) ( (6ﾓ11) (6ﾓ11) 6ﾓ1 2 0 ) = 0 6ﾍ2 6ﾍ2 5x + y 6ﾓ1 1 + z 6ﾓ1 2 = 0 6ﾍ2 5x + y + z 6ﾓ1 3 = 0: ｨｦ080909, ﾁ ﾁ , A(0; 1; 2) 07 B(6ﾓ12; 1; 0) ﾁ ﾓ1 s = 6ﾓ1 { 6ﾓ1 2 6ﾓ1 + 6ﾓ1 k. ｨｰ ﾂ ﾁ M(x; y; z), ｨｲ ﾁ ﾁ: 6ﾓ16ﾓ1 AM(x; y 6ﾓ1 1; z 6ﾓ1 2); 6ﾓ1 AB(6ﾓ12; 0; 6ﾓ12):

66 1364 ｨｮ ﾁ, ｨ. ｨｮ ﾁ09, ｨｮ. ｨｮ0809 ﾁ0103 ﾁ ﾂ ﾓ16ﾓ1 6ﾓ1 AM, AB 07 6ﾓ1 s , , ﾁ ﾁ , : ｨO x y 6ﾓ1 1 z 6ﾓ1 2 6ﾓ12 0 6ﾓ12 1 6ﾓ12 1 ｨO ｨO = 0 6ﾍ2 x 6ﾓ1 z + 2 = 0: ﾁ , ﾁ ﾁ ﾁ y, Oy ﾁ , A(1; 2; 6ﾓ11), B(2; 1; 3), C(0; 6ﾓ12; 4). ｨｰ M(x; y; z) ﾁ M, A, B, C ﾁ ｨｲ ﾁ ﾁ , , ﾓ16ﾓ1 6ﾓ1 6ﾓ1 AM(x 6ﾓ1 1; y 6ﾓ1 2; z + 1); AB(1; 6ﾓ11; 4); AC(6ﾓ11; 6ﾓ14; 5): ﾁ , ﾁ ﾁ , , ﾁ ﾁ ﾁ 0609 ﾁ : ｨｲ ｨO x 6ﾓ1 1 y 6ﾓ1 2 z ﾓ11 4 6ﾓ11 6ﾓ14 5 ｨO = 0 6ﾍ2 11x 6ﾓ1 9y 6ﾓ1 5z + 2 = 0: ﾂ M(5; 6ﾓ12; 0) x + y 6ﾓ1 2z + 4 = 0. ｨｰ ｨｹ , , , , ﾁ ﾁ ﾁ ﾁ ﾁ , ﾁ ﾁ , ﾁ ﾁ ﾁ0701, ﾁ ﾁ ﾁ ﾁ Ax + By + Cz + D = 0; 0209 ﾁ ﾌ A 2 + B 2 + C 2, D < 0, ﾁ ﾓ1 ﾌ A 2 + B 2 + C 2, D ﾂ ﾁ Ax + By + Cz + D ﾌ A 2 + B 2 + C 2 = Ax + By + Cz + D 6ﾓ1 ﾌ A 2 + B 2 + C 2 = 0 05 ﾁ ｨｮ ﾁ ﾁ ﾁ ﾁ ﾁ M(x 0 ; y 0 ; z 0 ) ﾁ (M 0 ; ) = Ax 0 + By 0 + Cz 0 + D ﾌ A 2 + B 2 + C 2 : (8.12) (8.12) , (6ﾓ12) 6ﾓ = ﾌ (6ﾓ12) 2 = 12 3 = 4:

67 13ｨｬ : x 6ﾓ1 y + 2z 6ﾓ1 3 = : 2x 6ﾓ1 2y + 4z 6ﾓ1 5 = 0. ﾂ , , , ｨｰ ｨｮ ﾁ ｨｮ (8.8), , ﾁ ﾁ x, y 07 z : 1 2 = 6ﾓ11 6ﾓ12 = 2 4 : , ﾁ , ﾁ0703 (8.7) 0203 ﾁ : D 1 D 2 = 6ﾓ13 6ﾓ15 = 3 5 ﾙ 1 2 : , ﾁ ﾁ ( ) ﾁ ﾁ M 0 (x 0 ; y 0 ; z 0 ), ﾁ ﾁ03 ｨｰ ｨｰ ﾁ ﾁ03 x 0 6ﾓ1 y 0 + 2z 0 6ﾓ1 3 = 0 6ﾍ2 x 0 6ﾓ1 y 0 + 2z 0 = 3: ｨｰ M (8.12): = 2x 0 6ﾓ1 2y 0 + 4z 0 6ﾓ1 5 ﾌ (6ﾓ12) = 2(x 0 6ﾓ1 y 0 + 2z 0 ) 6ﾓ1 5 ﾌ 24 : ﾁ07 ﾁ ﾁ ﾁ ﾁ03 x 0 6ﾓ1 y 0 + 2z 0 = 3 ﾁ , ﾌ 2 3 6ﾓ1 5 6 = ﾌ = : ﾁ , ﾁ ( ). ｨｰ : x + y + 2z 6ﾓ1 3 = : 2x 6ﾓ1 y + 4z 6ﾓ1 5 = ﾁ ﾁ ｨｰ ﾁ ﾁ ﾁ03. ｨｦ06 ﾁ ' ﾁ0202, '+ = ﾁ ﾁ ﾁ ' 07. ﾂ ﾓ1 n 1 (1; 1; 2) 07 6ﾓ1 n 2 (2; 6ﾓ11; 4) ﾁ =2, ﾁ ( ?). ｨ , ﾁ ﾁ0708. ﾂ ﾁ cos ' = ｨO cos(6ﾓ1 n 0ｼ86ﾓ1 1 ; n 2 ) ｨO : ﾁ ﾁ ﾁ : cos ' = ｨO ｨO (6ﾓ12) ｨOｨOｨOｨO ﾌ ﾌ (6ﾓ11) = ﾌ 4 : 126

68 1366 ｨｮ ﾁ, ｨ. ｨｮ ﾁ09, ｨｮ. ｨｮ0809 ﾁ0103 ﾁ ﾁ ｨｲ ﾁ ﾁ ﾁ ﾁ ﾁ, ﾁ ﾁ x, y, z? 8.2. ｨｲ ﾁ , , 07 ﾁ ﾁ ﾁ , ? ﾁ ｨｲ ﾁ ﾁ ﾁ ﾁ ﾁ ? ﾁ ﾁ ? ? 8.7. ｨｮ ﾁ ? ｨｲ : 09) 2x + 3y + z = 6; 00) x 6ﾓ1 4y + z 6ﾓ1 5 = 0; ﾁ) x 6ﾓ1 y 6ﾓ1 z = 0; 05) x + 2y 6ﾓ1 3z = 0; 02) 4x + y = 8; 03) x + z 6ﾓ1 2 = 0; 00) x 6ﾓ1 3 = 0; 03) 2y = 5: ﾂ ? ｨｲ ﾁ , M(1; 6ﾓ12; 6ﾓ11) ﾁ ﾓ1 n = 2 6ﾓ1 { 6ﾓ1 6ﾓ1 +5 6ﾓ1 k ｨｮ ﾁ ﾁ , A(1; 1; 0) ﾁ ﾓ1 a = 6ﾓ1 { + 2 6ﾓ1 6ﾓ1 5 6ﾓ1 k 07 6ﾓ1 b = 6ﾓ12 6ﾓ ﾓ1 k A(6ﾓ12; 3; 4) 07 B(1; 0; 1) ﾁ ﾓ1 l (1; 6ﾓ11; 2) ﾁ ｨｮ ﾁ ﾁ , A(0; 6ﾓ11; 6), B(1; 1; 5) 07 C(1; 4; 5) ﾁ ﾁ ﾁ070203: 09) 2x 6ﾓ1 2y + z 6ﾓ1 3 = 0; 00) x + 2y 6ﾓ1 z + 1 = 0; ﾁ) 4x 6ﾓ1 4y 6ﾓ1 2z + 1 = 0; 05) 3y 6ﾓ1 z + 2 = 0; 02) 3x + 4z 6ﾓ1 1 = 0; 03) 6ﾓ1 x + 3 = 0: 8.7. ｨｲ ﾁ , A(2; 3; 0) 07 B(6ﾓ12; 0; 1) Ox Oy C(1; 3; 2). ｨｲ ﾁ ｨｲ ﾁ , D(6ﾓ12; 6ﾓ13; 1) Oyz P(4; 6ﾓ17; 2) 07 Q(5; 6ﾓ13; 1) P ﾁ QP. 6ﾓ ﾁ ﾁ , A(0; 1; 6ﾓ12) x 6ﾓ1 y 6ﾓ1 4z + 3 = ｨｮ ﾁ ﾁ , M(6ﾓ12; 3; 1) 07 K(1; 0; 3) x + 2y + z 6ﾓ1 1 = 0.

69 13ｨｬ ﾁ : 09) x 6ﾓ1 2y + z + 3 = 0; 00) 4x + 2y 6ﾓ1 1 = 0; ﾁ) 2x 6ﾓ1 y + 7 = 0; 05) 3x 6ﾓ1 6y + 3z + 1 = 0; 02) 3x 6ﾓ1 1 = 0; 03) z = 2: ｨｦ ﾁ Ox, Oy, Oz ; 5; ﾁ ﾁ ｨｮ ﾁ ﾁ Ox Oy 07 Oz ﾁ ﾁ ｨｲ ﾁ : 09) x + y + 1 = 0; y 6ﾓ1 z + 1 = 0; 00) x 6ﾓ1 3y 6ﾓ1 2z + 1 = 0; x + y 6ﾓ1 z + 2 = 0; ﾁ) x 6ﾓ1 y + 1 = 0; 6ﾓ14x + 4y + 3 = 0; 05) z = 1; z = x; 02) 3x 6ﾓ1 y = 0; 2x + y + 2 = 0; 03) 4x + 3z + 7 = 0; x + 2y + 2z + 3 = 0: M(5; 6ﾓ12; 0) x + y 6ﾓ1 2z + 4 = x 6ﾓ1 2y + 2z + 1 = x 6ﾓ1 8y + +8z 6ﾓ14 = 0. ｨｮ ﾁ ﾁ , 0609 ﾁ ﾁ , x6ﾓ12y+3z+1 = , 0609 ﾁ x 6ﾓ1 y + z = 2; 3x + 2y + 2z = 6ﾓ12; x 6ﾓ1 2y + z = ｨｮ ﾁ ﾁ , x6ﾓ12y+ +z = 4, 2x + 3y 6ﾓ1 z = A(0; 1; 2) ﾁ ﾁ ﾁ : 09) 2x + y 6ﾓ1 2z + 3 = 0; 00) 3x 6ﾓ1 y + z + 2 = 0; ﾁ) x 6ﾓ1 y + 2z 6ﾓ1 3 = 0; 05) 4x + 2y 6ﾓ1 z + 2 = 0: ﾓ A(2; 6ﾓ11; 2) x 6ﾓ1 y + z + 2 = ﾓ5. ﾂ Q, P(2; 1; 1) x 6ﾓ1 2y 6ﾓ1 z + 3 = ﾓ ﾁ , ﾁ , ﾁ : 09) x + y = 0; x + z = 1; 00) 2x + 2y + z + 9 = 0; 4y + 3z 6ﾓ1 1 = 0: : , ﾁ , ﾁ ﾓ x 6ﾓ1 y + 2z = 0 05 ﾁ ﾁ , ﾁ : 2x + 4y 6ﾓ1 4z 6ﾓ1 5 = ﾁ ﾓ x 6ﾓ14y 6ﾓ12z 6ﾓ18 = ﾓ16x 6ﾓ13y +z 6ﾓ16 = 0, ﾁ A(0; 6ﾓ12; 4) ﾓ , ﾁ ﾁ , ﾁ A(6ﾓ16; 4; 4), B(6ﾓ11; 0; 6ﾓ13), C(6ﾓ13; 6ﾓ12; 6ﾓ12) 07 D(1; 6ﾓ14; 2).

70 1368 ｨｮ ﾁ, ｨ. ｨｮ ﾁ09, ｨｮ. ｨｮ0809 ﾁ0103 ﾁ ﾁ ｨｮ ｨ ﾁ0707, , ﾂ " ﾂ ﾁ ｨ ﾁ ｨｮ ﾁ09 ｨ ﾁ0209, , ｨｮ0809 ﾁ0103 ﾁ ｨｮ ﾁ ｨｬ ﾁ0707, ,

71 13ｨｦ ｨｹ ｨ. 04. ｨｪ ﾁ ﾁ ｨ. 04. ｨｪ ﾁ : ﾂ ", ﾁ : ｨｹ ﾁ ｨ. ｨ ｨｹ ﾁ ｨ. ｨ , ﾁ ﾁ (50), , ｨｲ ﾁ (50), ( ﾁ ), : : "; : " , ﾁ

72 ﾁ ﾁ ﾁ ﾁ ﾁ ﾁ03 ﾁ ﾁ ﾁ ｨｰ , ﾁ03 ﾁ ﾁ ﾁ ﾁ ﾁ ﾁ ﾁ ﾁ02, ﾁ ﾁ ﾁ ﾁ ﾁ ﾁ0901 ﾁ ﾁ , ﾁ ﾁ , ﾁ ﾁ , ﾁ ﾁ ﾁ 0803 ﾁ : (495) ｨ : ﾁ ｨｪ ﾁ , ﾁ0803 ﾁ ｨｮ ﾁ ( ﾁ ) { ﾁ ﾁ ﾁ , ﾁ ｨｮ , ﾁ ﾂ , ﾁ ｨｪ ﾁ ", , ﾁ ﾁ. ｨｰ0309 ﾁ : : ｨｦ ﾁ ﾁ ｨｰ : 06/ ﾁ 07ｨ ｨｰ0903 ﾁ ｨｮ ", 05. ｨｪ ﾁ09, 09/ , 09ｨｦ , ﾂ ﾁ ﾁ ﾁ ﾂ ﾁ ﾁ ( ﾁ ). ﾂ ( ﾁ ) , ﾁ ﾁ ﾁ , ｨｮ ( ) ｨｰ ﾁ ﾁ ﾁ ﾁ ﾁ070203, ｨｰ ﾁ0303 ﾁ ﾁ ﾁ ﾁ ﾁ. ｨ ﾁ ﾁ ﾁ ﾁ ﾁ ﾁ ｨｪ ﾁ ﾁ ﾁ ﾁ.

73 13Mathematical Education 6 54(52), 2009 Contents V. Drozdov. Seven Geometric Problems 2 Let G be the mass center of a triangle, H be the orthocenter, and L be the center of the inscribed circle. The conditions for GH, GL, HL to be parallel or/and orthogonal to a side of the triangle are analyzed. The exposition is based on vector approach. E. Potoskuev. On Necessity of Argument while Solving Stereometric Problems 11 Argument is still necessary if some stereometric fact seems to be intuitively clear" or evident". This is illustrated by numerous examples. V. Motanov. Reduction of Optimization Problem with a Homographic Functional to that with a Linear Functional 19 It is shown that a problem of optimization with a homographic functional can be reduced to that with a linear functional by a projective transformation which conserves nonnegativity of variables. H.J.M. Bos. Basic Concepts of Leibnitz Calculus 24 It is shown that Leibnitz calculus based on the concept of variable" and progression of a variable" (in opposition to the modern concept of function) is a self-consistent, beautiful and applicable theory. S. Kuleshov, A. Salimova, S. Stavtsev. Lectures on Analytic Geometry (continued) 36 Themes 7 and 8, equation of a straight line in plane and equation of a plane in space are considered. Information 69