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平面上，\;到一定點的距離是固定
\長的點所成的圖形稱為圓。\;
\其中該定點是圓的對稱中心，\;
\稱為圓心；\;此固定長稱為半徑。

圓的標準式
\已知Q(h,k)為圓C的圓心，\;r為
\半徑(r>0)，\;則圓C的方程式為
\(x-h)^2+(y-k)^2=r^2

圓的直徑式
\已知A(x_1,y_1)與B(x_2,y_2)為直徑
\上兩端點，\;則以\overline{AB}為直徑之圓
\方程式為：\;
\(x-x_1)(x-x_2)
\\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;+(y-y_1)(y-y_2)=0

圓的一般式
\x^2+y^2+dx+ey+f=0,其中\;d,e,f∈ℝ

求圓2x^2+2y^2-6x+4y+5=0
\之圓心Q與半徑r。

設\;x^2+y^2+2(k-1)x-2ky+(3k^2-2)=0...★\;試求下列k值或
\範圍：
\(1)\;★表一點；
\(2)\;★表一圓；
\(3)\;★表一圓，且此圓半徑最大。

A(0,2)

Q(t,2)

A(4,2)

B(5,1)

Q(t+3,t)

$α=%v$

Q(h,k)

P(x,y)

A(x_1,y_1)

B(x_2,y_2)

P(x,y)

Q(t,t+1)

A(0,-1)

B(-2,1)

試求通過A(0,0),B(0,4),C(3,3)
\三點的圓方程式。

A(0,0)

C(3,3)

B(0,4)

已知坐標平面上兩點A(-9,0)與
\B(-1,0)，\;試求滿足\frac{\overline{PA}}{\overline{PB}}=3之點
\的軌跡方程式。(Apollonius圓)

A(-9,0)

B(-1,0)

P(x,y)

自圓 C:(x-1)^2+(y-2)^2=3外
\一點 P(4,1)，\;對圓C作兩切線，\;切
\點為A,B ，\;試求 \Delta PAB之外接圓
\方程式。

P(4,1)

A

B

Q

Q

s = 0 (slider from 0 to 107, step 1)
B = (15.28, -8.3)
poly_0, b, a, c, d = Polygon(B, B + (18, 0), B + (18, 12), B + (0, 12))
C = PointIn(poly_0)
O = DynamicCoordinates(C, floor(x(C)), floor(y(C)))
u = Vector((x(B), y(O)), (x(B) + 18, y(O)))
v = Vector((x(O), y(B)), (x(O), y(B) + 12))
max = 107
A: Corner[4] - (0, 1.4)
D: A + (2.7, (-14))
E: D + (8, 0)
text12: "例
\\試求圓心為" + (LaTeX[If[s  <  15, "O(0,0)", "Q(1,-1)"]]) + "，\;且半徑
\\為3的圓方程式。"
Title: "" + (LaTeX[If[s ≤ 4, "圓方程式"]]) + "
\\" + (LaTeX[If[(s ≥ 1) ∧ (s ≤ 5), "(1)\;圓的定義與標準式"]]) + "
\\" + (LaTeX[If[(s ≥ 2) ∧ (s ≤ 4) ∨ s ≟ 18, "(2)\;圓的一般式與直徑式"]]) + "
\\" + (LaTeX[If[(s ≥ 3) ∧ (s ≤ 4), "(3)\;圓與點的關係"]]) + "
\\" + (LaTeX[If[s ≟ 4, "(4)\;圓與直線的關係"]]) + ""
A_{70}: O + (0, 2)
C_{69}: (x - x(O) - 4)^(2) + (y - y(O) - 2)^(2) = 16
C_{76}: (x - x(O) + 4)^(2) + (y - y(O) - 2)^(2) = 16
Q_{71}: O + (4, 2)
e: y = y(O)
F_{70} = Intersect(C_{69}, e, 1)
F_{71} = Intersect(C_{69}, e, 2)
F = Midpoint(F_{70}, F_{71})
f_{70} = Segment(Q_{71}, F_{70})
f_{71} = Segment(Q_{71}, F)
h_{69} = Segment(F_{70}, F_{71})
text71: "∵與y軸相切於A(0,2)
\\" + (LaTeX[If[s ≥ 72, "⇒設圓心Q(t,2),半徑長為|t|"]]) + "
\\" + (LaTeX[If[s ≥ 73, "⇒t^2=2^2+(2\sqrt{3})^2=16"]]) + "
\\" + (LaTeX[If[s ≥ 74, "⇒t=±4"]]) + "" + (LaTeX[If[s ≥ 75, "⇒Q(±4,2),r=4"]]) + "
\\" + (LaTeX[If[s ≥ 76, "(x-4)^2+(y-3)^2=16,\\或(x+4)^2+(y-3)^2=16"]]) + ""
text61: "先求\overline{AB}的中垂線x-y+1=0
\\" + (LaTeX[If[s ≥ 62, "又圓心Q在其上，\;"]]) + "" + (LaTeX[If[s ≥ 63, "故設Q(t,t+1)，"]]) + "" + (LaTeX[If[s ≥ 64, "\\又\overline{QA}^2=r^2"]]) + "" + (LaTeX[If[s ≥ 65, "⇒t^2+(t+2)^2=10"]]) + "
\\" + (LaTeX[If[s ≥ 66, "⇒2(t^2+2t-3)=0"]]) + "" + (LaTeX[If[s ≥ 67, "⇒t=1或3"]]) + "
\\" + (LaTeX[If[s ≥ 68, "⇒Q(1,2)或(-3,-2),"]]) + "" + (LaTeX[If[s ≥ 69, " \\⇒(x-1)^2+(y-2)^2=10\\或(x+3)^2+(y+2)^2=10"]]) + ""
text60: "" + (LaTeX[If[s ≟ 60, "求滿足下列條件之圓方程式"]]) + "
\\" + (LaTeX[If[(s ≥ 60) ∧ (s ≤ 69), "(1)\;通過A(0,-1)與B(-2,1)且半 \\\;\;\;\;\;\;徑長為\sqrt{10}"]]) + "
\\" + (LaTeX[If[(s ≥ 70) ∧ (s ≤ 76), "(2)\; 與y軸相切於A(0,2)點，且與 \\\;\;\;\;\;\;x軸所截之線段長為4\sqrt{3}"]]) + "
\\" + (LaTeX[If[s ≥ 77, "(3)\;通過A(4,2),B(5,1)兩點且與 \\\;\;\;\;\;\;x軸相切"]]) + ""
A_{79}: O + (4, 2)
B_{79}: O + (5, 1)
L_{79}: x - x(O) - (y - y(O)) = 3
C_{85}: (x - x(O) - 8)^(2) + (y - y(O) - 5)^(2) = 25
C_{84}: (x - x(O) - 4)^(2) + (y - y(O) - 1)^(2) = 1
text78: "先求\overline{AB}的中垂線x-y=3
\\" + (LaTeX[If[s ≥ 79, "又圓心Q在其上，\;設Q(t+3,t)，\;\\半徑為|t|\;"]]) + "" + (LaTeX[If[s ≥ 80, "∵\overline{QA}^2=r^2"]]) + "
\\" + (LaTeX[If[s ≥ 81, "⇒(t-1)^2+(t+2)^2=t^2"]]) + "
\\" + (LaTeX[If[s ≥ 82, "⇒t^2-6t+5=0"]]) + "" + (LaTeX[If[s ≥ 83, "⇒t=1,5"]]) + "
\\" + (LaTeX[If[s ≥ 84, "⇒Q(4,1)或Q(8,5)"]]) + "
\\" + (LaTeX[If[s ≥ 85, "(x-4)^2+(y-2)^2=1或(x-8)^2+(y-5)^2=25"]]) + ""
G_{79} = Point(L_{79})
Q_{84}: O + (4, 1)
Q_{85}: O + (8, 5)
t = 3.3 (slider from 0.1 to 5, step 0.1)
c_{79} = Circle(G_{79}, t)
G = (25.3, -0.5)
H = DynamicCoordinates(G, floor(x(G)), floor(y(G)))
c_6 = Circle(H, 3)
α = 0.5864306286700948 rad (slider from 0*° to 6.283185307179586, step 0.1*°)
I: H + (3, 0)
I_{6} = Rotate(I, α, H)
f_{6} = Segment(H, I_{6})
A_{12}: O
c_{12} = Circle(A_{12}, 3)
A_{15}: O + (1, -1)
c_{15} = Circle(A_{15}, 3)
c_{40} = Circle(O + (1.5, -1), 0.8660254037844386)
J = (20.62, -0.74)
K = (28.06, -3.54)
c_{29} = Circle(Midpoint[J, K], (0.5 * Segment[J, K]))
A_{31} = Point(c_{29})
g_{32} = Segment(A_{31}, J)
f_{32} = Segment(K, A_{31})
β = Angle(K, A_{31}, J)
f_{29} = Segment(K, J)
PolyA, f, g, h, i = Polygon(Corner[1], Corner[2], Corner[3], Corner[4])
A_{60}: O + (0, -1)
B_{60}: O + ((-2), 1)
L_{71}: x - x(O) - (y - y(O)) + 1 = 0
Q_{62} = Point(L_{71})
j_{64} = Segment(Q_{62}, A_{60})
A_{67}: O + (1, 2)
A_{68}: O - (3, 2)
c_{69} = Circle(A_{68}, A_{60})
c_{68} = Circle(A_{67}, B_{60})
B_{86}: O + (0, 4)
B_{85}: O + (3, 3)
B_{87}: O
c_{90} = Circle(B_{86}, B_{85}, B_{87})
text92: "設P(x,y)
\\" + (LaTeX[If[s ≥ 93, "∵\overline{PA}=3\overline{PB}"]]) + "" + (LaTeX[If[s ≥ 94, "⇒\overline{PA}^2=9\overline{PB}^2"]]) + "
\\" + (LaTeX[If[s ≥ 95, "∴(x+9)^2+y^2=9[(x+1)^2+y^2] "]]) + "
\\" + (LaTeX[If[s ≥ 96, "⇒8x^2+8y^2=72"]]) + "" + (LaTeX[If[s ≥ 97, "⇒x^2+y^2=9"]]) + ""
A_{91}: O + ((-9), 0)
A_1: O + (-1, 0)
c_{97}: (x - x(O))^(2) + (y - y(O))^(2) = 9
A_{92}: O - ((3 * cos(α)), (3 * sin(α)))
j_{92} = Segment(A_{92}, A_1)
k = Segment(A_{92}, A_{91})
A_{98}: O + (4, 1)
c_{98}: (x - x(O) - 1)^(2) + (y - y(O) - 2)^(2) = 3
L_{98}, L_{99} = Tangent(A_{98}, c_{98})
A_{100} = Intersect(c_{98}, L_{98}, 1)
A_{101} = Intersect(c_{98}, L_{99}, 1)
A_{99}: O + (1, 2)
j_{99} = Segment(A_{99}, A_{100})
j_{98} = Segment(A_{99}, A_{101})
c_{102} = Circle(A_{98}, A_{100}, A_{101})
text32: "" + (LaTeX[If[(s ≥ 99) ∧ (s ≤ 104), "∵切線垂直半徑"]]) + "
\\" + (LaTeX[If[(s ≥ 100) ∧ (s ≤ 104), "∴∠PAQ=∠PBQ=90°"]]) + "
\\" + (LaTeX[If[(s ≥ 101) ∧ (s ≤ 104), "⇒∠PAQ+∠PBQ=180°"]]) + "
\\" + (LaTeX[If[(s ≥ 102) ∧ (s ≤ 104), "⇒P,A,Q,B四點共圓"]]) + "
\\" + (LaTeX[If[(s ≥ 103) ∧ (s ≤ 104), "又半圓的圓周角是直角"]]) + "
\\" + (LaTeX[If[s ≟ 104 ∨ s ≟ 105, "所求即以\overline{PQ}為直徑之圓"]]) + "
\\" + (LaTeX[If[s ≥ 106, "⇒(x-4)(x-1)\\\;\;\;\;\;\;\;\;\;\;+(y-1)(y-2)=0"]]) + "
\\" + (LaTeX[If[s ≟ 107, "⇒x^2+y^2-5x-3y+6=0"]]) + ""
list0: {"f=0", "16+4e+f=0", "18+3d+3e+f=0"}
lista: {"4e=-16", "d+e=-6"}
text8: "\\證明：
\\" + (LaTeX[If[s ≥ 9, "設動點P(x,y)為圓上一點"]]) + "
\\" + (LaTeX[If[s ≥ 10, "則：\;\overline{PQ}=r，"]]) + "" + (LaTeX[If[s ≥ 11, "\;故 \\\overline{PQ}^2=(x-h)^2+(y-k)^2=r^2"]]) + ""
text13: "" + (LaTeX[If[s ≟ 13 ∨ s ≟ 14, "\sqrt{(x-0)^2+(y-0)^2}=3"]]) + "
\\" + (LaTeX[If[s ≟ 14, "⇒x^2+y^2=9"]]) + "
\\" + (LaTeX[If[s ≥ 16, "\sqrt{(x-1)^2+(y+1)^2}=3"]]) + "
\\" + (LaTeX[If[s ≥ 17, "⇒(x-1)^2+(y+1)^2=9"]]) + ""
text25: "通常稱：\;d^2+e^2-4f\;為圓的判別式。通常以D表示。\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;
\\" + (LaTeX[If[s ≥ 26, "(1)\;當D>0時，\;圖形為一圓，\;其圓心為(-\frac{d}{2},-\frac{e}{2})， \\\;\;\;\;\;\;半徑為\frac{1}{2}\sqrt{d^2+e^2-4f}。"]]) + "
\\" + (LaTeX[If[s ≥ 27, "(2)\;當D=0時，\;圖形為一點，\;其坐標為(-\frac{d}{2},-\frac{e}{2})。"]]) + "
\\" + (LaTeX[If[s ≥ 28, "(3)\;當D<0時，\;沒有圖形。"]]) + "
\\\;
\\\;
\\\;
\\\;
\\\;
\\\;
\\\;
\\\;
\\\;"
text42: "" + (LaTeX[If[s ≤ 55, "考慮判別式D"]]) + "
\\ " + (LaTeX[If[(s ≥ 42) ∧ (s ≤ 55), "D=4(k-1)^2+(-2k)^2-4(3k^2-2)"]]) + "" + (LaTeX[If[(s ≥ 44) ∧ (s ≤ 55), "=-4(k^2+2k-3)"]]) + "
\\" + (LaTeX[If[(s ≥ 45) ∧ (s ≤ 55), "(1)\;表一點，\;"]]) + "" + (LaTeX[If[(s ≥ 46) ∧ (s ≤ 55), "D=0"]]) + "" + (LaTeX[If[(s ≥ 47) ∧ (s ≤ 55), "⇒k^2+2k-3=0"]]) + "
\\" + (LaTeX[If[(s ≥ 48) ∧ (s ≤ 55), "\;\;\;\;\;⇒(k+3)(k-1)=0"]]) + "" + (LaTeX[If[(s ≥ 49) ∧ (s ≤ 55), "⇒k=1,-3"]]) + "
\\" + (LaTeX[If[(s ≥ 50) ∧ (s ≤ 55), "(2)\;表一圓，\;"]]) + "" + (LaTeX[If[(s ≥ 51) ∧ (s ≤ 55), "D>0"]]) + "" + (LaTeX[If[(s ≥ 52) ∧ (s ≤ 55), "⇒-4(k^2+2k-3)>0"]]) + "" + (LaTeX[If[(s ≥ 53) ∧ (s ≤ 55), "\\\;\;\;\;\;\;⇒k^2+2k-3<0"]]) + "" + (LaTeX[If[(s ≥ 54) ∧ (s ≤ 55), "⇒(k+3)(k-1)<0"]]) + "" + (LaTeX[If[s ≟ 55, "\\\;\;\;\;\;\;⇒-3<k<1"]]) + "
\\" + (LaTeX[If[s ≥ 56, "(3)\;考慮半徑\\\;\;\;\;\;\;r=\frac{1}{2}\sqrt{D}"]]) + "" + (LaTeX[If[s ≥ 57, "=\sqrt{-k^2-2k+3}"]]) + "" + (LaTeX[If[s ≥ 58, "=\sqrt{-(k+1)^2+4}"]]) + "
\\" + (LaTeX[If[s ≥ 59, "\;\;\;\;\;故當k=-1時，\;半徑有最大值2"]]) + ""
text20: "
\\" + (LaTeX[If[s ≥ 20, "若將一般式配方可得標準式：\;(x^2+dx)+(y^2+ey)=-f"]]) + "
\\" + (LaTeX[If[s ≥ 21, "⇒(x^2+dx+\frac{d^2}{4})+(y^2+ey+\frac{e^2}{4})=-f+\frac{d^2}{4}+\frac{e^2}{4}"]]) + "
\\" + (LaTeX[If[s ≥ 22, "⇒(x+\frac{d}{2})^2+(y+\frac{e}{2})^2=\frac{d^2+e^2-4f}{4}＝\left(  \frac{1}{2}\sqrt{d^2+e^2-4f} \right) ^2"]]) + "
\\" + (LaTeX[If[s ≥ 23, "故圓心為Q(-\frac{d}{2},-\frac{e}{2}),半徑為 \frac{1}{2}\sqrt{d^2+e^2-4f},"]]) + "" + (LaTeX[If[s ≥ 24, "\\其中d^2+e^2-4f>0"]]) + ""
text30: "證明：
\\" + (LaTeX[If[s ≥ 31, "取P(x,y)為圓上異於A,B的一\\點，\;"]]) + "" + (LaTeX[If[s ≥ 31, "故∠APB=90°"]]) + "" + (LaTeX[If[s ≥ 32, "\\∴m_{\overline{PA}}·m_{\overline{PB}}=-1"]]) + "" + (LaTeX[If[s ≥ 33, "\\⇒\frac{y-y_1}{x-x_1}·\frac{y-y_2}{x-x_2}=-1"]]) + "
\\" + (LaTeX[If[s ≥ 34, "⇒(y-y_1)(y-y_2)\\\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;=-(x-x_1)(x-x_2)"]]) + "
\\" + (LaTeX[If[s ≥ 35, "⇒(x-x_1)(x-x_2)\\\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;+(y-y_1)(y-y_2)=0"]]) + ""
text37: "⇒x^2+y^2-3x+2y+\frac{5}{2}=0
\\" + (LaTeX[If[s ≥ 38, "⇒(x^2-3x)+(y^2+2y)=-\frac{5}{2}"]]) + "
\\" + (LaTeX[If[s ≥ 39, "⇒( x-\frac{3}{2})^2 +( y+1)^2 =\frac{3}{4}"]]) + "
\\" + (LaTeX[If[s ≥ 40, "⇒Q(\frac{3}{2},-1),r=\frac{\sqrt{3}}{2}"]]) + ""
j = Segment(B_{60}, A_{60})
l_{78} = Segment(A_{79}, B_{79})
text87: "設所求之圓方程式為
\\x^2+y^2+dx+ey+f=0
\\" + (LaTeX[If[s ≥ 88, TableText[{list0}, "v{"]]]) + "
\\" + (LaTeX[If[s ≥ 89, TableText[{lista}, "v{"]]]) + "" + (LaTeX[If[s ≥ 89, "\\⇒e=-4 ,d=-2"]]) + "" + (LaTeX[If[s ≟ 90, "⇒x^2+y^2-2x-4y=0"]]) + ""