76 3 B m n AB P m n AP : PB = m : n A P B P AB m : n m < n n AB Q Q m A B AQ : QB = m : n (m n) m > n m n Q AB m : n A B Q P AB Q AB 3. 3 A(1) B(3) C(

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1 A P a 1 a P a P P(a) a P(a) a P(a) a a 0 a = a a < 0 a = a a < b a > b A a b a B b B b a b A a 3.1 A() B(5) AB = 5 = 3 A(3) B(1) AB = 3 1 = A(a) B(b) AB AB = b a 3.1 (1) A(6) B(1) () A( ) B(3) 75

2 76 3 B m n AB P m n AP : PB = m : n A P B P AB m : n m < n n AB Q Q m A B AQ : QB = m : n (m n) m > n m n Q AB m : n A B Q P AB Q AB 3. 3 A(1) B(3) C(7) (1) C AB : () B AC : (3) A BC : A(a) B(b) AB m : n P p a < b AP = p a PB = b p AP : PB = m : n a < b (p a) : (b p) = m : n m n n(p a) = m(b p) A(a) P(p) B(b) na + mb p = 1 a > b m + n n m a > b 1 B(b) P(p) A(a)

3 AB m : n Q q m > n a < b B AQ (m n) : n na + (m n)q b = (m n) + n na + mb q = m n A(a) m n B(b) Q(q) m n a b A(a) B(b) AB m : n P Q na + mb P m + n Q na + mb m n a + b AB n n 3. A(3) B(6) AB : 1 P : 1 Q P + 1 = 15 3 = Q = A(4) B(8) AB (1) 3 : C () 3 : 1 D (3) : 3 E (4) M

4 A (1) A(, 3) () B(, 3) (3) C(, 3) (4) D(, 3) A(a, b) 1 A B C D a a B C D (a, b) ( a, b) ( a, b) C b D b A(a, b) A 3.5 P(, 3) (1) Q () R B

5 B A( 1, 1 ) B(, ) AB ABC AB = AC + BC B 1 AB = ( 1 ) + ( 1 ) 1 A 1 C 1 AB = ( 1 ) + ( 1 ) A( 1, 1 ) B(, ) AB AB = ( 1 ) + ( 1 ) A( 1, 1 ) A A = A(, 1) B(4, 3) AB AB = (4 ) + {3 ( 1)} = + 4 = 0 = 5 A(, 1) A A = + ( 1) = (1) A(1, ) B(4, 6) () C( 3, 1) D(, 4) (3) E(5, ) F(3, ) (4) A(, 3)

6 A(3, 1) P(, ) 5 AP = 5 AP = 5 5 ( 3) + { ( 1)} = 5 ( 3) = 16 3 = ±4 = 7, B( 5, ) Q(7, ) ABC BC M AB + AC = (AM + BM ) ABC M BC M A(a, b) B( c, 0) C(c, 0) B c M A(a, b) AB + AC = {(a + c) + b } + {(a c) + b } = (a + b + c ) AM + BM = (a + b ) + c C c AB + AC = (AM + BM )

7 P ABCD AP + CP = BP + DP C A( 1, 1 ) B(, ) AB AP : PB = m : n P A B P F B n K P CH : HD = AP : PB = m : n m EK : KF = AP : PB = m : n E A P C H D H K 1 A( 1, 1 ) B(, ) AB m : n P Q ( n1 + m P m + n, n ) ( 1 + m n1 + m Q, n ) 1 + m m + n m n m n ( 1 + AB, ) 1 + 1

8 A( 1, 6) B(4, 1) AB 3 : P ( ) ( 1) , (, 3) : Q ( ) ( 1) , 5 5 ( ) 3, A( 3, ) B(4, 5) AB (1) : 1 C () : 1 D (3) : 3 E (4) M

9 A( 1, 1 ) B(, ) C( 3, 3 ) A ABC BC M AM : 1 G G 1 B BC M ( ) , G M C G AM : , ( , 3 ) G ABC 3 A( 1, 1 ) B(, ) C( 3, 3 ) ABC ( ) , A B C ABC (1) A(1, 1) B(5, ) C(3, 4) () A(, 4) B(0, 3) C(, 1)

10 = 0 (, ) ( 1, 6) (0, 4) (, ) F (1, ) (, 0) F A (1) + 4 = 0 = + 4 () 4 3 () = 0 (3) (1) = (0, ) (3) + 3 = 0 = 3 ( 3, 0) 3.11 (1) = 0 () + 1 = 0 (3) = 0 4

11 a + b + c = 0 a b c a 0 b 0 B m = m + k 1 ( 1, 1 ) 1 = m 1 + k k ( 1, 1 ) 1 m 1 k (1) ( 1, 1 ) m 1 = m( 1 ) 3.6 (1, 3) 3 = ( 1) = (1) (, 4) 3 () ( 3, 1)

12 86 3 A( 1, 1 ) B(, ) 1 AB 1 A 1 1 = 1 1 ( 1 ) 1 B 1 A 1 1 A( 1, 1 ) B( 1, ) AB B( ( 1, 0) 1, ) A( 1, 1 ) = 1 1 () ( 1, 1 ) (, ) 1 1 = 1 1 ( 1 ) 1 = = (1, ) (3, 4) = 4 ( 1) 3 1 = (1) (3, ) (5, 6) () ( 1, 4) (, )

13 (3) (, 1) (1, 1) (4) (3, 1) (3, 4) (a, 0) (0, b) a b = A 3.8 = = = 1 = 3 = = 0

14 88 3 m l l l l l A = 1 A B C CA BC m C : BC = CA : C 1 : BC = m : 1 BC = 1 m l 1 m m 1 m 1 = m 1 + k 1 = m + k m 1 = m 1 C B 1 m l m 1 m = (1) = = 0 () + 3 = = 5

15 A(, 1) = 0 l = 0 3 l m = 0 1 A l 3 m = 1 m = 3 l 1 = 3 ( ) 3 4 = l 3( ) ( 1) = A(3, 1) = 0 1 ( 1, 1 ) a + b + c = 0 b( 1 ) a( 1 ) = 0

16 90 3 B A B l l 1 A 1 AB l B AB l l AB 3. 1 = 0 l l A(0, 4) B B (s, t) 1 s t B (s, t) 1 l AB t 4 s 0 AB l t 4 s 0 = 1 s + t 8 = 0 1 ( ) s + 0 AB, t + 4 l s + 0 t = 0 s t 6 = 0 A(0, 4) 1 s = 4 t = B (4, ) l B(s, t)

17 = 0 l l A(, 1) B

18 9 3 C A l l H H AH d A l d A l A a + b + c = 0 1 d b a = H(p, q) p = ac a + b, q = bc a + b d = H d = p + q = c (a + b ) (a + b ) = c a + b = d c a + b a + b + c = 0 d d = H(p, q) 1 l 1 c a + b = 0 d d = = 5 5 = (1) 5 = 0 () = 0

19 P( 1, 1 ) a + b + c = 0 d a + b + c = 0 l l P l P l l (, ) ( + 1, + 1 ) ( + 1, + 1 ) l 1 d l P( 1, 1 ) a( + 1 ) + b( + 1 ) + c = 0 a+b+(a 1 +b 1 +c) = 0 1 d (, ) l l a 1 + b 1 + c a + b d ( 1, 1 ) a + b + c = 0 d d = a 1 + b 1 + c a + b 3.10 (1, ) = 0 d d = = 15 5 = 15 5 = (1) (1, ) = 0 () (, 3) + 3 = 0

20 P(, 1) 10 Q A(6, ) B(, 4) 3 AB 3 A(, 1) B(, 3) C

21 A(4, 0) B(0, ) (1) AB () AB 1 ( 1, 0) (5, 0) A 3 (6, 1) 4 (1) + 4 = 0 () 3 = 0

22 C r P C P r r C A C(a, b) r P (, ) CP = r P(, ) r ( a) + ( b) = r b C ( a) + ( b) = r a 1 (a, b) r ( a) + ( b) = r r + = r 3.1 (1) () (, 3) 4 (3) (, 1) 10

23 A(3, 4) B( 1, ) C r C AB ( ) 3 + ( 1) 4 +, (1, 3) A r = CA = (3 1) + (4 3) = 5 ( 1) + ( 3) = ( 5) ( 1) + ( 3) = 5 3. A(4, 0) B(0, ) B C

24 98 3 B + + l + m + n = 0 ( + ) + ( + 3) = = 0 l m n + + l + m + n = = 0 ( 6 + 9) + ( + + 1) = ( 3) + ( + 1) = 4 (3, 1) (1) = 0 () = 0

25 C A(, 4) B(, 0) C( 1, 3) + + l + m + n = 0 A l + 4m + n = 0 B + l + n = 0 C ( 1) ( 1)l + 3m + n = 0 l + 4m + n + 0 = 0 l + n + 4 = 0 l + 3m + n + 10 = 0 l = m = 4 n = = ( 1) + ( ) A = 5 C (1, ) ABC 3 ABC B ABC ABC ABC

26 A B C (1) A(1, 1) B(, 1) C( 1, 0) () A(, 3) B(, 1) C(, 3)

27 A = 5 (1) = 1 () = + 5 (1) { + = 5 1 = ( 1) = 5 = 0 = 1, 3 + = 5 = 1 = = = 1 ( 1, ) (, 1) () { + = 5 1 = ( + 5) = = 0 = 3 = 1 (, 1)

28 (1) + = 5 = + 1 () + = 8 + = 4

29 B a + b + c = 0 D = b 4ac D D > 0 D = 0 D < 0 a + b + c = 0 ( 1 ) = 8 = + m m + = 8 = + m + m + (m 8) = 0 D = (m) 4 (m 8) = 4(m 16) D 0 m m 4 (m+4)(m 4)50

30 = 5 = + m (1) m () m

31 r l l d d r d < r d = r d > r l l r r d d l d r 3.8 r + = r = 0 r = 0 d r + = r d = = = 10 r = d r = 10 d r 3.8 r 10 0 < r < r + = r = 0 r

32 106 3 C + = 5 P(3, 4) l l l P 4 P 3 l 3 4 l P(3, 4) P(3, 4) 5 + = 5 4 = 3 ( 3) = =5 + = r P(a, b) a + b = r P = 5 (3, 4) 3 + ( 4) = = P (1) + = 10 P(3, 1) () + = 13 P(, 3) (3) + = 16 P(4, 0)

33 A(1, 3) + = 5 + = 5 P(a, b) A(1, 3) a b P(a, b) P a + b = 5 1 P A(1, 3) a + b = 5 5 A(1, 3) 5 a + 3b = 5 + = 5 1 a b 3b + = 0 b = 1, b = 1 a =, b = a = 1 + = 5 (, 1) + = 5 ( 1, ) a+b=5 (a, b)

34 A(, 1) + = 1

35 = 4 = + 1 A B B + 4 = = 0 k A = = 4 ( + 4) + k( + 1) = A B + = 4 = + 1 A B 3 1 = 0 = 0 ( 4) + k = 0 k = 4 1 B = 0 A ( + ) + ( ) = 8

36 A(, 1) B(1, 4) C(0, 5) ABC

37 (1) + = c = 0 c () + = = 0 A B A B ( 1, 3) 6 (1) 5 < c < 5 () + 3 = 0

38 C CP = P P P C C A 3.13 A(0, ) B(4, 0) AP = BP P P (, ) AP = + ( ) BP = ( 4) + AP = BP + ( ) = ( 4) + A P(, ) B 4 + ( ) = ( 4) + 3 = 0 P 3 = 0 P(, ) AP = BP AP = BP P 3 = P AB

39 P 1 P (, ) P 1 P 3.30 A( 1, 0) B(1, 0) AP BP = 8 P

40 A(3, 0) : 1 P P (, ) P P : AP = : 1 AP = P 4AP = P A 3 1 P(, ) AP = ( 3) +, P = + 4{( 3) + } = = 0 ( 4) + = P ( 4) + = P(, ) (4, 0) 3.31 A( 3, 0) B(, 0) 3 : P

41 B P P Q P 3.4 Q + = A(4, 0) Q AQ P Q(s, t) P(, ) Q Q (s, t) Q + = Q(s, t) s + t = 1 P (, ) = s + 4, = t s = 4 t = 1 ( 4) + () = ( ) + = 1 P(, ) A 4 P ( ) + = 1 P(, ) (, 0) = + l Q l A(1, 5) Q AQ P

42 = + (, ) 1 > + (, ) A (, ) = + l P( 1, 1 ) > > 1 + l Q( 1, ) = > P l P l 1 l l l 1 P( 1, 1 ) Q( 1, ) 1 > + 1 < + l

43 l = m + k l 1 > m + k l > m + k < m + k l m + k m + k l < m + k = > (, ) = 3.33 (1) () + 4 > 0 (3) 1

44 118 3 B 3 + < 3 1 P(, ) P < 3 P < 3 P(, ) = 3 r r + > r 1 + < r + = r + > r + = r + r + r + = r 3 r + < r ( 1) 1 + ( 1) = 1 1 r 3 r 3.34 (1) + < 4 () + 5 (3) ( ) + 4

45 C { + > < > 0, < { + < 5 + < + = 5 + = 5 5

46 { { + 1 > (1) () + 1 > (3) { + < > 0 (4) {

47 ( )( + 1) < 0 { { A > 0 A < 0 AB < 0 B < 0 B > 0 ( )( + 1) < 0 { > < 0 { < > { { AB > 0 A > 0 B > 0 A < 0 B < ( + )( + 1) > 0

48 1 3 D , 0, + 8, = k 1 k k A A 4 8 (0, 0), (4, 0), (3, ), (0, 4) = k 1 (3, ) k 1 A A k 6 k 4 5 (0, 0) k = 0 (4, 0) k = 4 (3, ) k = 5 (0, 4) k = 4 A 0 < k < 5 + = 3 = 5 = 0 = 0 0

49 , 0, + 3 5,

50 = 0 P(1, ) A(, 6) B( 1, 3) C(3, ) 8 (1) () 4 + 9

51 A C 8 (1) () = 1 + =

52 A 1 P A( 1, ) B(4, 3) P 3 A(1, 5) B(6, 3) C(, ) ABC (1, 3) = = 0 (1) + 3 = 0 () + 3 = 0

53 ( a) = 5 a a = 1 5 C(, 1) = 0 6 A(, 1) Q(a, b) P (1) P a b () Q = 0 P

54 (1) () (, 4) ( 1, 1) (5, 1) B 8 3 a A(1, ) B(3, a) C(a, 0)

55 = 0 + = = a = 0 A(4, ) l (1) a () C (3) l (4) l

56 = a + a + a + 3 (1) P (, ) a () a P 1 A B 1 P Q P Q A A 3 B B P 8 Q 1 1 A B A B 1

57 A B AB C 9 11 () (1) a 1 (, 0) P(, 0) ( + 1) + (0 ) = ( 4) + (0 3) = 4 = 7 3 (1) = 0 () = < a < ( ) + ( 1) = 5 C 6 (1) (4 a, b) () + 11 = 0 (1) PQ (, 1) () a + b + 1 = 0 a = 4 b = 7 (1) () a = 3, 9 9 [ ( ) ( 6 (0, 0) 5 5, 18 3 ) ] 5 5, (1) a = () (, 1) (3) (4) = + 10 (3) CA l 11 (1) = a = a + 3 () = A 3 B 1 A B = k

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