1�Ù õ�õª
§ 1 õ�õª
1. � f(x1, · · · , xn)´ê K þ� n�àgõª.
y²: XJ3ê K þ� n�õª g(x1, · · · , xn) h(x1, · · · , xn),
¦
f(x1, · · · , xn) = g(x1, · · · , xn)h(x1, · · · , xn),
K g(x1, · · · , xn) h(x1, · · · , xn)Ñ´àgõª.
y²: � deg f = m, deg g = k, deg h = l. -
g = gp + gp+1 + · · · + gk, h = hq + hq+1 · · ·+ hl,
Ù¥ gi, hj ©O i, j gàgõª,
gp, hq ´©)¥gê$�àgõ
ª, k + l = m, K
f = gphq +
m∑
t=p+q+1
(∑
i+j=t
gihj
)
.
Ïd� p + q < m f Ø´àgõª.
p+ q = k + l = m íÑ p = k,
q = l, Ïd g = gk, h = hl Ñ´àgõª.
2. � f(x, y) ∈ K[x, y]. y²: XJ f(x, x) = 0, K x− y | f(x, y).
y²: � f(x, y) =
n∑
k=0
ak(x)y
k, K
f(x, y) = f(x, y)− f(x, x) =
n∑
k=0
ak(x)(y
k − xk)
= (y − x)
n∑
k=1
ak(x)(y
k−1 + yk−2x+ · · · + yxk−2 + xk−1).
Ïd x− y | f(x, y).
· 160 · 1�Ù õ�õª
∗3. Oe�1�ª:
∣∣∣∣∣∣∣∣∣∣∣∣∣
1
x1 − a1
1
x1 − a2 · · ·
1
x1 − an
1
x2 − a1
1
x2 − a2 · · ·
1
x2 − an
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
xn − a1
1
xn − a2 · · ·
1
xn − an
∣∣∣∣∣∣∣∣∣∣∣∣∣
.
): r�1�ªP Dn(x1, · · · , xn, a1, · · · , an). K
Dn(x1, · · · , xn, a1, · · · , an) = G(x1, · · · , xn, a1, · · · , an)
F (x1, · · · , xn, a1, · · · , an) ,
Ù¥ G F Ñ´ x1, · · · , xn, a1, · · · , an �õª. ´
F (x1, · · · , xn, a1, · · · , an) =
∏
1≤i,j≤n
(xi − aj).
du
Dn(x1, · · · , xi, · · · , xi, · · · , xn, a1, · · · , an) = 0,
Dn(x1, · · · , xn, a1, · · · , ai, · · · , ai, · · · , an) = 0,
�
G(x1, · · · , xn, a1, · · · , an)
=
∏
1≤i xi aj �gê (Ñ´ n− 1 g), G1 = cn ´~ê. Ïd
Dn(x1, · · · , xn, a1, · · · , an) =
∏
1≤i n.
y²: � g(x) =
n∑
i=1
xk+1i f(x)
x− xi , K g(x) = 0½ deg g(x) < n.
xk+1f ′(x)− g(x) =
n∑
i=1
xk+1f(x)
x− xi −
n∑
i=1
xk+1i f(x)
x− xi =
(
xk+1 − xk+1i
x− xi
)
f(x)
=
n∑
i=1
k∑
j=0
(xk−jxjif(x) =
k∑
j=0
(
n∑
i=1
(xk−jxji
)
f(x)
=
(
k∑
j=0
xk−jsj
)
f(x)
= (s0x
k + s1x
k−1 + · · ·+ sk−1x+ sk)f(x).
=�¤y.
(2) '��ª
xk+1f ′(x) = (s0x
k + s1x
k−1 + · · ·+ sk−1x+ sk)f(x) + g(x)
ü> n gXê, du g(x)�gê < n½ g(x) = 0, ¤±
xk+1f ′(x)�ngXê=(s0x
k+ s1x
k−1 + · · ·+ sk−1x+ sk)f(x)�ngXê,
¤±� k ≤ n,
(n− k)(−1)kσk = sk − σ1sk−1 + σ2sk−2 + · · ·+ (−1)kσks0,
=
sk − σ1sk−1 + σ2sk−2 + · · ·+ (−1)k−1σk−1s1 + (−1)kkσk = 0.
∗§ 3 (ª · 165 ·
� k > n,
0 = sk − σ1sk−1 + σ2sk−2 + · · ·+ (−1)nσnsk−n,
=�¤y.
∗8. ^Ð�é¡õªL« s2, s3, s4, s5.
): s2 = σ
2
1 − 2σ2,
s3 = σ
3
1 − 3σ1σ2 + 3σ3,
s4 = σ
4
1 − 4σ21σ2 + 2σ22 + 4σ1σ3 − 4σ4,
s5 = σ
5
1 − 5σ31σ2 + 5σ1σ22 + 5σ21σ3 − 5σ2σ3 − 5σ1σ4 + 5σ5.
∗§ 3 (ª
1. Oe�õª�(ª:
(1) f(x) = x3 − 3x2 + 2x+ 1, g(x) = 2x2 − x− 1;
(2) f(x) = 2x3 − 3x2 − x+ 2, g(x) = x4 − 2x3 − 3x+ 4;
): (1) Res(f, g) = (−1)2·3 Res(2x2−x−1, f) = (−1)6 ·23 ·f
(
−1
2
)
f(1)
= −7.
(2) f(x), g(x) kú� 1, ¤±(ª Res(f, g) = 0.
2. � λ�Û, e�õªkú�:
(1) f(x) = x3 − λx+ 2, g(x) = x2 + λx+ 2;
(2) f(x) = x3 + λx2 − 9, g(x) = x3 + λx− 3.
): (1) Res(f, g) = −4(λ+ 1)2(λ− 3), �� λ = −1½ 3kú�.
(2) Res(f, g) = 9(λ2 + 12)(λ2 + 2), �� λ = ±2√3i½ ±√2ikú�
.
3. ¦e���I§:
(1) x = t2 + t− 1, y = 2t2 + t− 1;
(2) x =
t− 1
t2 + 1
, y =
t2 + t− 1
t2 + 1
.
): (1) 4x2 − 4xy + y2 + 5x− 3y + 1 = 0.
(2) 5x2 − 6xy + 2y2 + 5x− 3y + 1 = 0.
4. � λÛ, e�õªk?
(1) f(x) = x3 − 3x+ λ; (2) f(x) = x4 − 4x3 + (2− λ)x2 +
2x− 2.
): (1) 2,−2;
(2) −1,−3
2
,
7
2
+
9
2
√
3i,
7
2
− 9
2
√
3i.
· 166 · 1�Ù õ�õª
5. ¦e�§|�):
(1)
5x
2 − 6xy + 5y2 = 16,
2x2 − xy + y2 − x− y = 4;
(2)
x
2 + y2 + 4x− 2y = −3,
x2 + 4xy − y2 + 10y = 9.
): (1) Resy(f, g) = 32(y
4 − y3 − 3y2 + y + 2),
x = 1y = −1
x = −1y = 1
x = 2y = 2
(2) Resx(f, g) = 4(5x
4 + 40x3 + 106x2 + 104x + 33),
x = −1y = 2
x = −3y = 0
x = −2 + 3
5
√
5
y = 1 +
1
5
√
5
x = −2− 3
5
√
5
y = 1− 1
5
√
5
6. ¦e��I��:I:
(1) � x2 + y2 − 3x− y = 0 V x2 + 2xy − y2 − 4y − 2 = 0;
(2) V 4x2 − 7xy + y2 + 13x − 2y − 3 = 0 V 9x2 − 14xy +
y2 + 28x − 4y − 5 = 0.
): (1) (1,−1),
(
3
2
+
1
2
√
2,
1
2
+
√
2
)
,
(
3
2
− 1
2
√
2,
1
2
−√2
)
;
(2) (0,−1), (1, 2), (2, 3), (−2, 1).
7. y²(ª�e�5: � f(x), g(x)©O´ n g m gõª. K
(1) Res(f, g) = (−1)mnRes(g, f);
(2) Res(af, bg) = ambnRes(f, g);
∗(3) Res((x− a)f, g) = g(a)Res(f, g).
y²: (1), (2) w,. 8y (3). �
f(x) = a0x
n + a1x
n−1 + · · ·+ an, g(x) = b0xm + b1xm−1 + · · ·+ bm,
K
(x− a)f(x) = a0xn+1 + (a1 − a0a)xn + · · ·+ (an − an−1a)x− ana.
∗§ 3 (ª · 167 ·
Res((x− a)f, g) =∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣
a0 a1−a0a a2−a1a · · · an−an−1a −ana
a0 a1−a0a a2−a1a · · · an−an−1a −ana
. . .
. . .
. . . · · · . . . . . .
n
a0 · · · · · · · · · an−an−1a −ana
b0 b1 b2 · · · bm−1 bm
b0 b1 b2 · · · bm−1 bm
. . .
. . .
. . . · · · . . . . . .
m+1
b0 · · · · · · · · · bm−1 bm
∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣
g1�å, �¦ a \���, ��, �
Res((x− a)f, g) =∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣
a0 a1 a2 · · · an 0
a0 a1 a2 · · · an 0
. . .
. . .
. . . · · · . . . . . .
n
a0 · · · · · · · · · an 0
b0 b1+b0a b2+b1a+b0a
2 · · · · · · g(a) · · · g(a)am
b0 b1+b0a · · · · · · · · · · · · g(a)am−1
. . .
. . .
. . . · · · . . . . . . ...
m+1
b0 · · · · · · · · · · · · g(a)
∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣
l�1å, 1¦ (−a) \�c1, �1 n+ 11, 2U��Ðm,
�
Res((x− a)f, g) = g(a)Res(f, g).
∗8. � f(x) = a(x− x1) · · · (x− xn), g(x) = b(x− y1) · · · (x− ym).
y²: Res(f, g) = am
n∏
i=1
g(xi) = (−1)mnbn
m∏
j=1
f(yj) = a
mbn
n∏
i=1
m∏
j=1
(xi −
yj).
y²: Res(f, g)= amRes((x− x1) · · · (x− xn), g(x))
= amg(x1)Res((x− x2) · · · (x− xn), g(x))
= amg(x1)g(x2) · · · g(xn)
= ambn
n∏
i=1
m∏
j=1
(xi − yj) = (−1)mnbn
m∏
j=1
f(yj).
∗9. y²: Res(f(x), g1(x)g2(x)) = Res(f(x), g1(x))Res(f(x), g2(x)).
· 168 · 1�Ù õ�õª
y²: �
f(x) = a(x− x1) · · · (x− xn),
K
Res(f, g1g2) = a
deg g1g2
n∏
i=1
g1(xi)g2(xi)
= adeg g1
n∏
i=1
g1(xi)a
deg g2
n∏
i=1
g2(xi)
= Res(f, g1)Res(f, g2).
∗10. � f Äõª, y²: é?¿õª h, Res(f, g) = Res(f, g +
hf).
y²: � f(x) = (x− x1)(x− x2) · · · (x− xn), K
Res(f, g + hf) =
n∏
i=1
(g(xi) + h(xi)f(xi))
=
n∏
i=1
g(xi) = Res(f, g).
∗11. |^SK 7 10y²�(ª5Oe�õª�(ª:
(1) f(x) = xn + x+ 1, g(x) = x2 − 3x+ 2;
(2) f(x) = xn + 1, g(x) = (x− 1)n;
(3) f(x) = a0x
n + a1x
n−1 + · · ·+ an−1x+ an,
g(x) = a0x
n−1 + a1x
n−2 + · · ·+ an−2x+ an−1.
(4) f(x) =
xn − 1
x− 1 , g(x) =
xm − 1
x− 1 ;
): (1) Res(f, g) = (−1)2n(1 + 1 + 1)(2n + 2 + 1) = 3(2n + 3).
(2) Res(f, g) = (−1)n · 2n.
(3) du f(x) = xg(x) + an, ¤±
Res(f, g) = (−1)n(n−1) Res(g, f) = (−1)n(n−1) Res(g, an) = an−1n .
(4) (a) X (m,n) = d > 1, K
xn − 1
x− 1
xm − 1
x− 1 kú�, Ïd Res(f, g)
= 0.
(b) X (m,n) = 1, Ø� n > m, K n = mq + r, 0 ≤ r < m. w,
(m, r) = 1. K
xn − 1
x− 1 =
xmqxr − 1
x− 1 =
(xmq − 1)xr + xr − 1
x− 1 ,
∗§ 4 Ç�{ · 169 ·
l
Res
(
xn − 1
x− 1 ,
xm − 1
x− 1
)
= (−1)m−1)(n+r) Res
(
xr − 1
x− 1 ,
xm − 1
x− 1
)
.
·y² (m− 1)(n + r)½´óê.
X m − 1´óê, K(ؤá. y� m− 1´Ûê, K móê, l
n
´Ûê, r´Ûê, u´ n+ r´óê. l
Res
(
xn − 1
x− 1 ,
xm − 1
x− 1
)
= Res
(
xr − 1
x− 1 ,
xm − 1
x− 1
)
.
2^ rØm, âÎ=Ø{��n, d (m, r) = 1 �
Res
(
xr − 1
x− 1 ,
xm − 1
x− 1
)
= · · · = Res
(
xr
′ − 1
x− 1 , 1
)
= 1.
=� (m,n) = 1 Res
(
xn − 1
x− 1 ,
xm − 1
x− 1
)
= 1.
∗12. � f(x) = a0x
n + a1x
n−1 + · · ·+ an−1x+ an ∈ K[x],
y²: f(x)��Oª
D(f) = (−1)n(n−1)2 a−10 Res(f, f ′).
y²:
D(f) = a2n−20
∏
1≤iread ‘d:/mapleuser/wsolve2‘:
>P1:=-12*x2^2+7*x1*x2-2;
>P2:=-2*x3+x1^2;
>P3:=-x3^2+x1*x2+2;
>PS1:={P1,P2,P3};
>ord:=[x3,x2,x1];
>B1:=basset(PS1,ord);
B1 := [−2x3 + x21,−12x22 + 7x1x2 − 2]
>R1:=remseta(PS1,B1,ord);
R1 := {8 + 4x1x2 − x41}
>PS2:={op(B1)} union R1;
>B2:=basset(PS2,ord);
B2 := [−2x3 + x21, 8 + 4x1x2 − x41]
>R2:=remseta(PS2,B2,ord);
R2 := {64x21 + 192− 48x41 − 7x61 + 3x81}
>PS3:={op(B2)} union R2;
>B3:=basset(PS3,ord);
B3 := [−2x3 + x21, 8 + 4x1x2 − x41, 64x21 + 192 − 48x41 − 7x61 + 3x81]
>R3:=remseta(PS3,B3,ord);
R3 := {}
>J:=Initial(B3[1],ord)*Initial(B3[2],ord)*Initial(B3[3],
ord);
J := x1
>solveas(B3,ord,{x1});
ù�)� 8 |):
∗§ 4 Ç�{ · 171 ·
x1 = 2,
x2 = 1,
x3 = 2;
x1 = −2,
x2 = −1,
x3 = 2;
x1 =
√
3i,
x2 = −
√
3
12
i,
x3 = −3
2
;
x1 = −
√
3i,
x2 =
√
3
12
i,
x3 = −3
2
;
x1 =
√
−6 + 6√13
3
i,
x2 =
2(2 +
√
13)
3
√
−6 + 6√13
i,
x3 =
1−√13
3
;
x1 = −
√
−6 + 6√13
3
i,
x2 = − 2(2 +
√
13)
3
√
−6 + 6√13
i,
x3 =
1−√13
3
;
x1 =
√
6 + 6
√
13
3
i,
x2 =
2(−2 +√13)
3
√
6 + 6
√
13
i,
x3 =
1 +
√
13
3
;
x1 = −
√
6 + 6
√
13
3
i,
x2 = −2(−2 +
√
13)
3
√
6 + 6
√
13
i,
x3 =
1 +
√
13
3
.
2. )õª§|:
2x23 − x21 − x22 = 0,
x1x3 − 2x3 + x1x2 = 0,
x21 − x22 = 0.
): ùp�Ñ©Ú{�L§, ¿�ÑÜ©$(J.
>read ‘d:/mapleuser/wsolve2‘:
>P1:=2*x3^2-x1^2-x2^2;
>P2:=x1*x3-2*x3+x1*x2;
>P3:=x1^2-x2^2;
>PS1:={P1,P2,P3};
>ord:=[x3,x2,x1];
>B1:=basset(PS1,ord);
B1 := [x1x3 − 2x3 + x1x2, x21 − x22]
>R1:=remseta(PS1,B1,ord);
R1 := {x21(x1 − 1)}
· 172 · 1�Ù õ�õª
>PS2:={op(B1)} union R1;
>B2:=basset(PS2,ord);
B2 := [x1x3 − 2x3 + x1x2, x21 − x22, x21(x1 − 1)]
>R2:=remseta(PS2,B2,ord);
R2 := {}
>J:=Initial(B2[1],ord)*Initial(B2[2],ord)*Initial(B2[3],
ord);
J := x1 − 2
>solveas(B2,ord,{x1-2});
ù�)� 4 |):
x1 = 0,
x2 = 0,
x3 = 0,
x1 = 1,
x2 = −1,
x3 = −1,
x1 = 1,
x2 = 1,
x3 = 1.
∗§ 5 AÛ½n�Åìy²
1. y²: !/�é�pR.
A(0, 0) B(u1, 0)
C(x2, x3)D(u2, x1)
1 1K
y²: âb�^±��e�õª§:
P1
def
= u22 + x
2
1 − u21 = 0, (|AD| = |AB|)
P2
def
= (x2 − u2)2 + (x3 − x1)2 − u21 = 0, (|DC| = |AB|)
P3
def
= (x2 − u1)2 + x23 − u21 = 0. (|BC| = |AB|)
ù�½nb�±8(¤õª| P = {P1, P2, P3}.
½n(Ø´ AC ⊥ BD, ±8(õª§
G
def
= (u2 − u1)x2 + x1x3 = 0.
∗§ 5 AÛ½n�Åìy² · 173 ·
�Cþ�S x1, x2, x3, ¦�üA�� Ci = {Ci1, Ci2, Ci3}©O:
C11 = x
2
1 − u21 + u22,
C12 = x2,
C13 = x3.
±9
C21 = x
2
1 − u21 + u22,
C22 = −x2 + u1 + u2,
C33 = x1x3 − u21 + u22.
l C1 �Ñ x2 = x3 = 0, w,´O. O Rem(G,C2) = 0, ½n¤á.
òz^ J2 = x1, lAÛ¿Âw, ù´Ø±�.
2. y²: ��F/.��.
A(0, 0) B(u1, 0)
C(x1, x2)D(u2, u3)
1 2K
y²: âb�^±��e�õª§:
P1
def
= x2 − u3 = 0, (AB//CD)
P2
def
= u1 − x1 − u2 = 0. (−−→AD −−→CB 3 AB þ�ÝK�)
ù�½nb�±8(¤õª| P = {P1, P2}.
½n(Ø´ ∠BAD = ∠CBA, ±8(õª§
G
def
= −u21u3(x1 − u1)− u21u2x2 = 0.
¦�A�� C = {C1, C2}Ò´P gC,
J = 1, vkòz^. O
Rem(G,C ) = 0, ½n¤á.
XJr½n�1�^U¤
P2
def
= u22 + u
2
3 − (x1 − u1)2 − x22 = 0, (|AD| = |BC|)
O�¬��üA��, A��Óc, ,A��´
C1 = −x1 + u1 + u2,
· 174 · 1�Ù õ�õª
C2 = x2 − u3.
G'uùA���{ª�u u21u2u3, Ò´`(ØØé. lAÛ¿Â5w,
C1 = 0 �u u1 = x1 − u2, = ABCD´²1o>/ (eã). ù´ØÎÜ
K¿�O, Ïd ∠BAD 6= ∠CBA.
A(0, 0) B(u1, 0)
C(x1, x2)D(u2, u3)
1 2K�,«/
3. y²: n�/�ü^¥��:©º:é>¥:¤ 2 : 1.
A(0, 0) B(2u1, 0)
C(2u2, 2u3)
D(u1+u2, u3)E(u2, u3)
G(x1,x2)
1 3K
y²: âb�^±��e�õª§:
P1
def
= u3x1 − (u1 + u2)x2 = 0, (AGD �)
P2
def
= (u2 − 2u1)x2 − (x1 − 2u1)u3 = 0, (BGE �)
ù�½nb�±8(¤õª| P = {P1, P2}.
½n(Ø´
−→
AG =
2
3
−−→
AD, ±8(õª§
G1
def
= 3x1 − 2(u1 + u2) = 0,
G2
def
= 3x2 − 2u3 = 0.
�Cþ�S x1, x2, ¦�A�� C = {C1, C2}:
C1 = 3x1 − 2u1 − 2u2,
C2 = −3x2 + 2u3.
J = 1, vkòz^. w, G1, G2 ÑU� C ئ.
4. y²: �n�/�>þ�p´�>þüã�'~¥.
∗§ 5 AÛ½n�Åìy² · 175 ·
A(0, 0) B(u1, 0)
C(0, u2)
D(x1, x2)
1 4K
y²: âb�^±��e�õª§:
P1
def
= u1x1 − u2x2 = 0, (AD ⊥ BC)
P2
def
= (x2 − u2)u1 + u2x1 = 0. (BDC �)
ù�½nb�±8(¤õª| P = {P1, P2}.
½n(Ø´ |AD|2 = |CD||DB|, ±8(õª§
G
def
= (x21 + x
2
2)
2 − (x21 + (x2 − u2)2)((u1 − x1)2 + x22) = 0.
�Cþ�S x1, x2, ¦�A�� C = {C1, C2}:
C1 = (u
2
1 + u
2
2)x1 − u1u22,
C2 = −(u21 − u22)x2 + u21u2.
O Rem(G,C ) = 0, ½n¤á.
òz^´ u2±9 J = (u
2
1 +u
2
2)
2,
lAÛ¿Âw, ù
/Ñ´Ø#N�.
5. Xã, � △ABC ¥ ∠A´�, M1,M2,M3 ©O´ AB,AC,BC >
�¥:. AH ⊥ BC ¿
H ´Rv. y² M1,M2,M3,H o:��.
b
A(0, 0)
B(u1, 0)
C(0, u2)
M1(x1, 0)
M2(0, x2) M3(x3, x4)
H(x5, x6)
O(x7, x8)
1 5K
· 176 · 1�Ù õ�õª
y²: âb�^±��e�õª§:
P1
def
= u1 − 2x1 = 0, (M1 ´ AB �¥:)
P2
def
= u2 − 2x2 = 0, (M2 ´ AC �¥:)
P3
def
= u1 − 2x3 = 0, (M3 ´ BC �¥:)
P4
def
= u2 − 2x4 = 0, (M3 ´ BC �¥:)
P5
def
= u1x5 − u2x6 = 0, (AH ⊥ BC)
P6
def
= u1(x6 − u2) + u2x5 = 0. (BHC �)
P7
def
= (x1 − x7)2 − x27 = 0, (|OM1| = |OA|)
P8
def
= (x2 − x8)2 − x28 = 0. (|OM2| = |OA|)
ù�½nb�±8(¤õª| P = {P1, P2, · · · , P8}.
½n(Ø´ |OH| = |OM3| = |OA|, ±8(õª§
G1
def
= (x5 − x7)2 + (x6 − x8)2 − x27 − x28 = 0,
G2
def
= (x3 − x7)2 + (x4 − x8)2 − x27 − x28 = 0.
�Cþ�S x1, x2, · · · , x8, ¦�A�� C = {C1, C2, · · · , C8}:
C1 = −2x1 + u1,
C2 = −2x2 + u2,
C3 = −2x3 + u1,
C4 = −2x4 + u2,
C5 = (u
2
1 + u
2
2)x5 − u1u22,
C6 = −(u21 + u22)x6 + u21u2,
C7 = −4x7 + u1,
C8 = −4x8 + u2.
O Rem(G1,C ) = 0, Rem(G2,C ) = 0, ½n¤á.
òz^´ u2
±9 J = (u21 + u
2
2)
2, lAÛ¿Âw, ù
/Ñ´Ø#N�.
6. Xã, A,B,C n:3^þ, A′, B′, C ′ n:3,^þ.
P,Q,R´§ë��:. y²: P,Q,Rn:�.
∗§ 5 AÛ½n�Åìy² · 177 ·
b
b
b
A(u1, 0) B(u2, 0) C(u3, 0)
A′(0, u4)
B′(0, u5)
C′(0, u6)
P (x1, x2)
R(x5, x6)
Q
(x3, x4)
1 6K
y²: Ïù´�¯K, Ïdïá�IXXþ㤫.
âb�^±��e�õª§:
P1
def
= x1(−u4)− u2(x2 − u4) = 0, (A′PB �)
P2
def
= x1(−u5)− u1(x2 − u5) = 0, (B′PA�)
P3
def
= x3(−u4)− u3(x4 − u4) = 0, (A′QC �)
P4
def
= x3(−u6)− u1(x4 − u6) = 0, (C ′QA�)
P5
def
= x5(−u5)− u3(x6 − u5) = 0, (B′RC �)
P6
def
= x5(−u6)− u2(x6 − u6) = 0. (C ′RB �)
ù�½nb�±8(¤õª| P = {P1, P2, · · · , P6}.
½n(Ø´ PQR�, ±8(õª§
G
def
= (x3 − x1)(x6 − x2)− (x5 − x1)(x4 − x2) = 0,
�Cþ�S x1, x2, · · · , x6, ¦�A�� C = {C1, C2, · · · , C6}:
C1 = (u1u4 − u2u5)x1 − u1u2(u4 − u5),
C2 = (u1u4 − u2u5)x2 − (u1 − u2)u4u5,
C3 = (u1u4 − u3u6)x3 − u1u3(u4 − u6),
C4 = (u1u4 − u3u6)x4 − (u1 − u3)u4u6,
C5 = −(u2u5 − u3u6)x5 + u2u3(u5 − u6),
C6 = −(u2u5 − u3u6)x6 + (u2 − u3)u5u6.
O Rem(G,C ) = 0, ½n¤á.
òz^´ u2, u3 ±9 J =
(u1u4 − u2u5)2(u1u4 − u3u6)2(u2u5 − u3u6)2.
éòz^�Aۿ©Û:
(1) e u2 = 0½ u3 = 0, §�Aۿ´ B ½ C A
′B′C ′ �, l
P,Q,RØ(½, ¯KÿÂ.
· 178 · 1�Ù õ�õª
(2) u1u4 − u2u5 = 0 =⇒ A′B//B′A, u1u4 − u3u6 = 0 =⇒ A′C//C ′A,
u2u5− u3u6 = 0 =⇒ B′C//C ′B, ?Û«/ÑyѬ¦ P,Q,R¥�
:Ã{(½, ¯KÿÂ.
7. y²: �%��uA�±��ü�.
A(u1, x1) B(u2, x2)
C(u3, x3)
O(0, 0)
r
1 7K
y²: âb�^±��e�õª§:
P1
def
= u21 + x
2
1 − r2 = 0, (|OA| = r)
P2
def
= u22 + x
2
2 − r2 = 0, (|OB| = r)
P3
def
= u23 + x
2
3 − r2 = 0, (|OC| = r)
ù�½nb�±8(¤õª| P = {P1, P2, P3}.
½n(Ø´ ∠AOB = 2∠ACB, =
tan∠AOB = tan(2∠ACB) =
2 tan∠ACB
1− tan2 ∠ACB . (∗)
du
∠AOB =
kOB − kOA
1 + kOBkOA
=
u1x2 − u2x1
u1u2 + x1x2
,
∠ACB =
kCB − kCA
1 + kCBkCA
=
(u1 − u3)(x2 − x3)− (u2 − u3)(x1 − x3)
(u1 − u3)(u2 − u3) + (x1 − x3)(x2 − x3) =
α
β
.
\ (∗)ª�
u1x2 − u2x1
u1u2 + x1x2
=
2α
β
1− α2
β2
=
2αβ
β2 − α2 .
Ïd·K�(ر8(õª§
G
def
= (u1x2 − u2x1)(((u1 − u3)(u2 − u3) + (x1 − x3)(x2 − x3))2
− ((u1 − u3)(x2 − x3)− (u2 − u3)(x1 − x3))2)− 2(u1u2 + x1x2)
× ((u1 − u3)(x2 − x3)− (u2 − u3)(x1 − x3))((u1 − u3)(u2 − u3)
+ (x1 − x3)(x2 − x3)) = 0.
�Cþ�S x1, x2, x3, ¦�A�� C Ò´�5�õª| P. O
Rem(G,C ) = 0, ½n¤á.
vkòz^.
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