3.\;兩直線的關係
\
\\;\;\;\;\;(1)\;\fgcolor{yellow}{平行}
\\;\;\;\;\;(2)\;\fgcolor{yellow}{垂直}

x

y

Forward

$θ+90°$

兩平行直線的斜率相等
\設兩非鉛直的相異直線L_1與L_2，\;其斜率分
\別為m_1與m_2，
\(1)\;若L_1//L_2，\;則：\;m_1=m_2
\(2)\;若m_1=m_2，\;則：\;L_1//L_2

(2)\;相互垂直兩直線的斜率關係
\\;\;\;\;\;\;設兩直線L_1與L_2均非鉛直線，\;且其斜
\\;\;\;\;\;\;率分別為m_1與m_2，\;若L_1與L_2互相垂
\\;\;\;\;\;\;直，\;則：\;\fgcolor{yellow}{m_1·m_2=-1}

A(x,y)

B(x+1,b)

C(x+1,c)

$%v$

$%v$

$θ$

已知a為實數，坐標平面上四點A(2,1),
\B(3,5),C(0,-1),D(2,a)，\;則：
\(1)\;若\overline{AB}//\overline{CD}，\;試求a值。
\(2)\;若\overline{AB}⊥\overline{CD}，\;試求a值。

若坐標平面上三直線\;L_1:x-2y+3=0,
\L_2:2x+3y=0,L_3:ax-y-1=0，\;若
\L_1,L_2,L_3不能圍成一個三角形，\;試求a值。

$a=%v$

A(2,-7)

B(5,2)

C(9,0)

G

H

M_1(7,1)

M_2(\frac{11}{2},-\frac{7}{2})

O

G

就k值討論\left\{ (k-3)x-2y=2k\3x+(2k+1)y=-k-2 \right.的關係。

A(2,1)

B(3,5)

C(0,-1)

D(2,7)

D

(2,5)

α = 0.8377580409572781 rad (slider from 0.1*° to 179.9*°, step 0.1*°)
B = (14.481819047619071, -6.482476190476204)
poly1, a_0, a_1, a_3, a_4 = Polygon(B, B + (18, 0), B + (18, 12), B + (0, 12))
C = PointIn(poly1)
O_1 = DynamicCoordinates(C, floor(x(C)), floor(y(C)))
W: (floor(x(B)), floor(y(B)))
list = Sequence(Sequence[W + (h, k), h, 1, 18], k, 1, 13)
D_C = Point(list)
L_3: y - y(D_C) = (tan(α) * (x - x(D_C)))
F = Intersect(L_3, y = y(O_1))
A = Corner(4)
s = 0 (slider from 0 to 132, step 1)
u = Vector((x(B), y(O_1)), (x(B) + 18, y(O_1)))
v = Vector((x(O_1), y(B)), (x(O_1), y(B) + 12))
D_A = Point(u)
L_1: y - y(D_A) = (tan(α) * (x - x(D_A)))
k = 3.5 (slider from -20 to 20, step 0.5)
L_{27}: y - y(D_A) = (0.5 * (x - x(D_A))) + k
D = PointIn(poly1)
E_5: O_1 + (16 / 3, ((-5)) / 3)
E_6: O_1 + (7, 1)
E_7: O_1 + (5.5, (-3.5))
E_8: O_1 + (5, (-3))
E_9 = Point(list)
E_a: O_1 + (2, ((-3)) / 2)
L_{17} = Line(E_9, E_a)
F_1: O_1 + (2, 0)
F_2: O_1 + (0, 4)
P_1 = Point(list)
P_2 = Point(list)
F_4: O_1 + ((-2), 6)
F_5: O_1 + (8, 4)
F_6: O_1 + (5, 0)
F_7: O_1 + (0, (-2))
P_3: O_1 + (2, 3)
β = Angle(F + (1, 0), F, F + (cos(α), sin(α)))
L_{26}: y - y(P_3) = (tan(β) * (x - x(P_3)))
P_4 = Intersect(L_{26}, y = y(O_1))
P_5 = Intersect(L_{26}, x = x(O_1))
D_B = Point(u)
L_2: y - y(D_B) = (tan(α) * (x - x(D_B)))
L_5: y - y(D_C) = ((-cot(α)) * (x - x(D_C)))
poly2, a_x, b, c, d = Polygon(W, W + (19, 0), W + (19, 13), W + (0, 13))
L_6: x = x(D_C) + 1
D_D = Intersect(L_5, L_6)
D_F = Intersect(L_3, L_6)
d_1 = Segment(D_D, D_F)
G = Intersect(L_5, y = y(O_1))
δ = Angle(D_A + (1, 0), D_A, D_A + (cos(α), sin(α)))
e = Angle(D_B + (1, 0), D_B, D_B + (cos(α), sin(α)))
text9: "" + (LaTeX[If[(s ≥ 9) ∧ (s ≤ 14), "證明：\;法一"]]) + "
\\" + (LaTeX[If[(s ≥ 10) ∧ (s ≤ 14), "設兩直線L_1與L_2交於點A(x,y)，\;"]]) + "" + (LaTeX[If[(s ≥ 11) ∧ (s ≤ 14), "且 \\B(x+1,b)在直線L_1上，\;C(x+1,c) \\在L_2上。\;"]]) + "" + (LaTeX[If[(s ≥ 12) ∧ (s ≤ 14), "m_1=\frac{b-y}{1}=b-y, \\m_2=\frac{c-y}{1}=c-y"]]) + "
\\" + (LaTeX[If[s ≟ 13 ∨ s ≟ 14, "∵\Delta ABC是直角三角形"]]) + "
\\" + (LaTeX[If[s ≟ 14, "∴\overline{AB}^2+\overline{AC}^2=\overline{BC}^2"]]) + ""
γ = Angle(G + (1, 0), G, G + (cos(α + 90°), sin(α + 90°)))
L_7: (3 * (x - x(O_1))) + (2 * (y - y(O_1))) = 1
text39: "" + (LaTeX[If[(s ≥ 39) ∧ (s ≤ 43), "解"]]) + "
\\" + (LaTeX[If[(s ≥ 40) ∧ (s ≤ 43), "(1)\;2y=-3x+1⇒y=-\frac{3}{2}x+\frac{1}{2}"]]) + "
\\" + (LaTeX[If[(s ≥ 41) ∧ (s ≤ 43), "\;\;\;\;\;\;∴直線的斜率為-\frac{3}{2}"]]) + "
\\" + (LaTeX[If[(s ≥ 42) ∧ (s ≤ 43), "\;\;\;\;\;\;故所求之直線方程式為y=-\frac{3}{2}x+6"]]) + "
\\" + (LaTeX[If[s ≟ 43, "\;\;\;\;\;\;⇒3x+2y-12=0"]]) + "
\\" + (LaTeX[If[s ≥ 45, "(2)\;3y=x-7⇒y=\frac{1}{3}x-\frac{7}{3}"]]) + "
\\" + (LaTeX[If[s ≥ 46, "\;\;\;\;\;\;∴直線的斜率為\frac{1}{3}"]]) + "
\\" + (LaTeX[If[s ≥ 47, "\;\;\;\;\;\;故與其垂直的直線斜率為-3"]]) + "
\\" + (LaTeX[If[s ≥ 48, "\\\;\;\;\;\;\;故所求之直線方程式為\\\;\;\;\;\;\;y-5=-3(x-2)"]]) + "" + (LaTeX[If[s ≥ 49, "\\\;\;\;\;\;\;⇒3x+y-11=0"]]) + ""
L_8: (3 * (x - x(O_1))) + (2 * (y - y(O_1))) = 12
L_9: x - x(O_1) - (3 * (y - y(O_1))) = 7
L_{10}: y - y(O_1) = ((-3) * (x - x(O_1))) + 11
L_{11}: (2 * (y - y(O_1))) = x - x(O_1) + 3
L_{12}: (3 * (y - y(O_1))) = ((-2) * (x - x(O_1)))
a = -1.4 (slider from -10 to 10, step 0.1)
L_{13}: (a * (x - x(O_1))) - (y - y(O_1)) = 1
text51: "" + (LaTeX[If[(s ≥ 51) ∧ (s ≤ 54), "L_1:y=\frac{1}{2}x+\frac{3}{2}⇒m_1=\frac{1}{2}"]]) + "
\\" + (LaTeX[If[(s ≥ 52) ∧ (s ≤ 54), "L_2:y=-\frac{2}{3}x⇒m_2=-\frac{2}{3}"]]) + "
\\" + (LaTeX[If[(s ≥ 53) ∧ (s ≤ 54), "L_3:y=ax-1⇒m_3=a"]]) + "
\\" + (LaTeX[If[(s ≥ 54) ∧ (s ≤ 57), "⇒m_1=\frac{1}{2},m_2=-\frac{2}{3},m_3=a"]]) + "
\\" + (LaTeX[If[s ≥ 56, "L_1,L_2,L_3不能圍成三角形的情形有三種："]]) + "
\\" + (LaTeX[If[s ≥ 57, "(1)\;L_1//L_3：\;m_1=m_3⇒a=\frac{1}{2}"]]) + "
\\" + (LaTeX[If[s ≥ 58, "(2)\;L_2//L_3：\;m_2=m_3⇒a=-\frac{2}{3}"]]) + "
\\" + (LaTeX[If[s ≥ 59, "(3)\;L_1,L_2,L_3交於一點：\;"]]) + "
\\" + (LaTeX[If[s ≥ 60, "\left\{ x-2y+3=0\\2x+3y=0 \right.\ "]]) + "" + (LaTeX[If[s ≥ 61, "⇒x=-\frac{9}{7},y=\frac{6}{7}"]]) + "
\\" + (LaTeX[If[s ≥ 62, "⇒代入a(-\frac{9}{7})=\frac{6}{7}+1"]]) + "" + (LaTeX[If[s ≟ 63, "⇒a=-\frac{13}{9}"]]) + ""
E_2: O_1 + (2, (-7))
E_3: O_1 + (5, 2)
E_4: O_1 + (9, 0)
d_a = Segment(E_2, E_3)
d_b = Segment(E_4, E_3)
d_c = Segment(E_4, E_2)
f = OrthogonalLine(E_3, d_c)
g = OrthogonalLine(E_2, d_b)
H = Intersect(f, g)
I = Intersect(d_c, f)
J = Intersect(g, d_b)
h = Segment(J, E_2)
i = Segment(E_3, I)
p = Circle(E_2, E_3, E_4)
p_1 = Segment(E_8, E_2)
p_2 = Segment(E_8, E_3)
p_3 = Segment(E_8, E_4)
text64: "" + (LaTeX[If[s  <  102, "已知坐標平面上三點A(2,-7),B(5,2), \\C(9,0)，\;試求\Delta ABC之："]]) + "" + (LaTeX[If[(s ≥ 64) ∧ (s ≤ 67), "(1)\;重心坐標G；"]]) + "" + (LaTeX[If[(s ≥ 68) ∧ (s ≤ 78), "(2)\;垂心座標H；\;"]]) + "" + (LaTeX[If[(s ≥ 79) ∧ (s ≤ 100), "(3)\;外心坐標O；\;"]]) + "
\\" + (LaTeX[If[s ≥ 101, "(4)請問G,H,O三點是否共線？\\\;\;\;\;\;\;若共線，\;試求\frac{\overline{OG}}{\overline{GH}}。"]]) + ""
L_{14} = Line(E_5, E_8)
text110: "(3)兩直線方程式聯立方程組解
\\\;\;\;\;\;的幾何意義與代數關係
\\\;\;\;\;\;坐標平面上兩直線\left\{ L_1:a_1x+b_1y=c_1\\L_2:a_2x+b_2y=c_2 \right.
\\" + (LaTeX[If[s ≥ 111, "\;\;\;\;\;(1)\;兩直線交於一點"]]) + "
\\" + (LaTeX[If[s ≥ 112, "\;\;\;\;\;\;\;\;\;\;⇔恰一解"]]) + "" + (LaTeX[If[s ≥ 113, "⇔\frac{a_1}{a_2}≠\frac{b_1}{b_2}"]]) + "
\\" + (LaTeX[If[s ≥ 114, "\;\;\;\;\;(2)\;兩直線重合"]]) + "
\\" + (LaTeX[If[s ≥ 115, "\;\;\;\;\;\;\;\;\;\;⇔無限多解"]]) + "" + (LaTeX[If[s ≥ 116, "⇔\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}"]]) + "
\\" + (LaTeX[If[s ≥ 117, "\;\;\;\;\;(3)\;兩直線平行"]]) + "
\\" + (LaTeX[If[s ≥ 118, "\;\;\;\;\;\;\;\;\;\;⇔無解"]]) + "" + (LaTeX[If[s ≥ 119, "⇔\frac{a_1}{a_2}=\frac{b_1}{b_2}≠\frac{c_1}{c_2}"]]) + "
 "
text121: "" + (LaTeX[If[(s ≥ 121) ∧ (s ≤ 124), "\frac{k-3}{3}=\frac{-2}{2k+1}"]]) + "" + (LaTeX[If[(s ≥ 122) ∧ (s ≤ 12), "⇒2k^2-5k+3=0"]]) + "
\\" + (LaTeX[If[(s ≥ 123) ∧ (s ≤ 124), "⇒(k-1)(2k-3)=0"]]) + "" + (LaTeX[If[s ≥ 124, "\\⇒k=1,\frac{3}{2}"]]) + "
\\" + (LaTeX[If[s ≥ 126, "(1)\;當k=1時，\;\frac{k-3}{3}=\frac{-2}{3}=\frac{2k}{-k-2}"]]) + "
\\" + (LaTeX[If[s ≥ 127, "\;\;\;\;\;\;故L_1與L_2重合"]]) + "" + (LaTeX[If[s ≥ 128, "⇔方程組有無限多解"]]) + "
\\" + (LaTeX[If[s ≥ 129, "(2)\;當k=\frac{3}{2}時，\;\frac{k-3}{3}=\frac{-1}{2}≠\frac{-6}{7}=\frac{2k}{-k-2}"]]) + "
\\" + (LaTeX[If[s ≥ 130, "\;\;\;\;\;\;故L_1與L_2平行"]]) + "" + (LaTeX[If[s ≥ 131, "⇔方程組無解"]]) + "
\\" + (LaTeX[If[s ≥ 132, "(3)\;當k≠1且k≠\frac{3}{2}時，\;L_1與L_2相交於一點"]]) + "" + (LaTeX[If[s ≥ 133, "⇔方程組恰一解"]]) + ""
L_{15}: ((k - 3) * (x - x(O_1))) - (2 * (y - y(O_1))) - (2 * k) = 0
L_{16}: (3 * (x - x(O_1))) + (((2 * k) + 1) * (y - y(O_1))) = (-k) - 2
L_{18} = Line(F_2, E_9)
L_{20} = Line(P_1, P_2)
E_A: O_1 + (2, 1)
E_B: O_1 + (3, 5)
E_C: O_1 + (0, -1)
L_{19} = Line(E_A, E_B)
E_D: O_1 + (2, 7)
L_{21} = Line(E_C, E_D)
L_{22} = Line(E_a, E_C)
L_{23} = LineBisector(d_b)
L_{24} = LineBisector(d_a)
L_{25} = LineBisector(d_c)
E_E = Intersect(L_{15}, L_{16})
F_8: O_1 + (2, 5)
F_9: O_1 + (0, -1)
text65: "" + (LaTeX[If[(s ≥ 65) ∧ (s ≤ 67), "(1)\;G(\frac{x_1+x_2+x_3}{3},\frac{y_1+y_2+y_3}{3})"]]) + "
\\" + (LaTeX[If[(s ≥ 66) ∧ (s ≤ 67), "\;\;\;\;\;\;⇒G(\frac{2+5+9}{3},\frac{-7+2+0}{3})"]]) + "
\\" + (LaTeX[If[s ≟ 67, "\;\;\;\;\;\;⇒G(\frac{16}{3},-\frac{5}{3})"]]) + "
\\" + (LaTeX[If[(s ≥ 69) ∧ (s ≤ 75), "(2)\;垂心為三高之交點"]]) + "
\\" + (LaTeX[If[(s ≥ 70) ∧ (s ≤ 75), "\;\;\;\;\;\;∵\overline{AH}⊥\overline{BC},\overline{BH}⊥\overline{AC}"]]) + "
\\" + (LaTeX[If[(s ≥ 71) ∧ (s ≤ 75), "\;\;\;\;\;\;∴m_{\overline{AH}}×m_{\overline{BC}}=-1,m_{\overline{BH}}×m_{\overline{AC}}=-1"]]) + "
\\" + (LaTeX[If[(s ≥ 72) ∧ (s ≤ 75), "\;\;\;\;\;\;m_{\overline{BC}}=\frac{2-0}{5-9}=-\frac{1}{2},⇒m_{\overline{AH}}=2"]]) + "
\\" + (LaTeX[If[(s ≥ 73) ∧ (s ≤ 75), "\;\;\;\;\;\;m_{\overline{AC}}=\frac{-7-0}{2-9}=1⇒m_{\overline{BH}}=-1"]]) + "
\\" + (LaTeX[If[s ≟ 74 ∨ s ≟ 75, "\;\;\;\;\;\;\overleftrightarrow{AH}:y+7=2(x-2)⇒2x-y=11"]]) + "
\\" + (LaTeX[If[s ≟ 75 ∧ (s ≤ 78), "\;\;\;\;\;\;\overleftrightarrow{BH}:y-2=-(x-5)⇒x+y=7"]]) + "
\\" + (LaTeX[If[(s ≥ 76) ∧ (s ≤ 78), "\;\;\;\;\;\;H:\left\{2x-y=11\\x+y=7  \right.\ "]]) + "
\\" + (LaTeX[If[s ≟ 77 ∨ s ≟ 78, "\;\;\;\;\;\;⇒x=6,y=1"]]) + "\\" + (LaTeX[If[s ≟ 78, "\;\;\;\;\;\;⇒H(6,1)"]]) + ""
text15: "\\∴\overline{AB}^2+\overline{AC}^2=\overline{BC}^2
\\" + (LaTeX[If[s ≥ 16, "⇒1^2+(b-y)^2+1^2+(c-y)^2\\\;\;\;\;\;\;=(b-c)^2"]]) + "
\\" + (LaTeX[If[s ≥ 17, "⇒2-2by-2cy+2y^2=-2bc"]]) + "
\\" + (LaTeX[If[s ≥ 18, "⇒bc-by-cy+y^2=-1"]]) + "
\\" + (LaTeX[If[s ≥ 19, "⇒(b-y)(c-y)=-1"]]) + "
\\" + (LaTeX[If[s ≟ 20, "⇒m_1·m_2=-1"]]) + ""
text21: "證明：\;法二
\\" + (LaTeX[If[s ≥ 22, "設L_1與L_2的斜角分別為θ與90+θ"]]) + "
\\" + (LaTeX[If[s ≥ 23, "故m_1=tanθ,"]]) + "
\\" + (LaTeX[If[s ≥ 24, "m_2=tan(90°+θ)"]]) + "" + (LaTeX[If[s ≥ 25, "=\frac{sin(90°+θ)}{cos(90°+θ)}"]]) + "
\\" + (LaTeX[If[s ≥ 26, "=\frac{cosθ}{-sinθ}"]]) + "" + (LaTeX[If[s ≥ 27, "=-\frac{1}{tanθ}"]]) + "" + (LaTeX[If[s ≥ 28, "=-\frac{1}{m_1}"]]) + "
\\" + (LaTeX[If[s ≥ 29, "∴m_1·m_2=-1"]]) + ""
text31: "\\" + (LaTeX[If[s ≥ 32, "m_{\overline{AB}}=\frac{5-1}{3-2}=4,"]]) + "" + (LaTeX[If[s ≥ 33, "\\m_{\overline{CD}}=\frac{-1-a}{0-2}=\frac{a+1}{2}"]]) + "
\\" + (LaTeX[If[s ≥ 34, "(1)\;∵\overline{AB}//\overline{CD}⇒m_{\overline{AB}}=m_{\overline{CD}}"]]) + "
\\" + (LaTeX[If[s ≥ 35, "∴\frac{a+1}{2}=4⇒a=7"]]) + "
\\" + (LaTeX[If[s ≥ 36, "(2)\;∵\overline{AB}⊥\overline{CD}⇒m_{\overline{AB}}×m_{\overline{CD}}=-1"]]) + "
\\" + (LaTeX[If[s ≥ 37, "∴4×\frac{a+1}{2}=-1⇒a=-\frac{3}{2}"]]) + ""
text38: "範例
\\" + (LaTeX[If[(s ≥ 38) ∧ (s  <  44), "(1)\;求y截距為6，\;且與直線3x+2y=1 \\\;\;\;\;\;\;平行的直線方程式。"]]) + "
\\" + (LaTeX[If[s ≥ 44, "(2)\;求過點(2,5)，\;且與直線x-3y=7 \\\;\;\;\;\;\;垂直的直線方程式。"]]) + ""
text80: "(3)\;外心O是" + (LaTeX[If[(s ≥ 80) ∧ (s ≤ 91), "三中垂線的交點", "外接圓的圓心"]]) + "
\\" + (LaTeX[If[(s ≥ 81) ∧ (s ≤ 88), "\;\;\;\;\;\;取\overline{BC}的中點M_1(7,1)"]]) + "
\\" + (LaTeX[If[(s ≥ 82) ∧ (s ≤ 88), "\;\;\;\;\;\;取\overline{AC}的中點M_2(\frac{11}{2},-\frac{7}{2})"]]) + "
\\" + (LaTeX[If[(s ≥ 83) ∧ (s ≤ 88), "\;\;\;\;\;\;m_{\overline{BC}}=\frac{2-0}{5-9}=-\frac{1}{2},⇒m_{\overline{OM_1}}=2"]]) + "
\\" + (LaTeX[If[(s ≥ 84) ∧ (s ≤ 88), "\;\;\;\;\;\;m_{\overline{AC}}=\frac{-7-0}{2-9}=1⇒m_{\overline{OM_2}}=-1"]]) + "
\\" + (LaTeX[If[(s ≥ 85) ∧ (s ≤ 88), "\;\;\;\;\;\;\overleftrightarrow{OM_1}:y-1=2(x-7)"]]) + "" + (LaTeX[If[(s ≥ 86) ∧ (s ≤ 88), "⇒2x-y=13"]]) + "
\\" + (LaTeX[If[s ≟ 87 ∨ s ≟ 88, "\;\;\;\;\;\;\overleftrightarrow{OM_2}:y+\frac{7}{2}=-(x-\frac{11}{2})"]]) + "" + (LaTeX[If[s ≟ 88, "⇒x+y=2"]]) + "
\\" + (LaTeX[If[(s ≥ 89) ∧ (s ≤ 91), "\;\;\;\;\;\;O:\left\{ 2x-y=13\\x+y=2 \right.\ "]]) + "" + (LaTeX[If[(s ≥ 90) ∧ (s ≤ 91), "\\\;\;\;\;\;\;⇒x=5,y=-3"]]) + "" + (LaTeX[If[s ≟ 91, "⇒O(5,-3) "]]) + "
\\" + (LaTeX[If[(s ≥ 92) ∧ (s ≤ 97), "設\Delta ABC外心為O(x,y)"]]) + "" + (LaTeX[If[(s ≥ 93) ∧ (s ≤ 97), "\\⇒\overline{OA}=\overline{OB}=\overline{OC}"]]) + "" + (LaTeX[If[(s ≥ 94) ∧ (s ≤ 97), "\\∴\overline{OA}^2=\overline{OB}^2,\overline{OA}^2=\overline{OC}^2"]]) + "
\\" + (LaTeX[If[(s ≥ 95) ∧ (s ≤ 97), "\left\{ (x-2)^2+(y+7)^2=(x-5)^2+(y-2)^2\\(x-2)^2+(y+7)^2=(x-9)^2+y^2 \right.\ "]]) + "
\\" + (LaTeX[If[(s ≥ 96) ∧ (s ≤ 97), "⇒\left\{ 6x+18y=-24\\14x+14y=28 \right."]]) + "" + (LaTeX[If[s ≟ 97, "⇒\left\{ x+3y=-4\\x+y=2 \right."]]) + "
\\" + (LaTeX[If[s ≥ 98, "\;\;\;\;\;\;⇒\left\{ x+3y=-4\\x+y=2 \right."]]) + "" + (LaTeX[If[s ≥ 99, "\\\;\;\;\;\;\;⇒x=5,y=-3"]]) + "" + (LaTeX[If[s ≟ 100, "⇒O(5,-3)"]]) + "
 "
text102: "(4)∵\Delta ABC的外心O(5,-3),
\\\;\;\;\;\;\;垂心H(6,1),\;重心G(\frac{16}{3},-\frac{5}{3})
\\" + (LaTeX[If[(s ≥ 103) ∧ (s ≤ 105), "\;\;\;\;\;\;m_{\overline{OH}}=\frac{-3-1}{5-6}=4"]]) + "
\\" + (LaTeX[If[(s ≥ 104) ∧ (s ≤ 105), "\;\;\;\;\;\;m_{\overline{GH}}=\frac{-\frac{5}{3}-1}{\frac{16}{3}-6}=4"]]) + "
\\" + (LaTeX[If[s ≟ 105, "\;\;\;\;\;\;故O,G,H三點共線"]]) + "
\\" + (LaTeX[If[s ≥ 106, "\;\;\;\;\;\;\overline{OG}^2=(5-\frac{16}{3})^2+(-3+\frac{5}{3})^2=\frac{17}{9}"]]) + "
\\" + (LaTeX[If[s ≥ 107, "\;\;\;\;\;\;\overline{GH}^2=(6-\frac{16}{3})^2+(1+\frac{5}{3})^2=\frac{68}{9}"]]) + "
\\" + (LaTeX[If[s ≥ 108, "\;\;\;\;\;\;∴\overline{OG}=\frac{\sqrt{17}}{3},\overline{GH}=\frac{2\sqrt{17}}{3}"]]) + "" + (LaTeX[If[s ≥ 109, "\\\;\;\;\;\;\;⇒\frac{\overline{OG}}{\overline{GH}}=\frac{1}{2}"]]) + ""
text1: "(1)\;平行直線的斜率關係
\\" + (LaTeX[If[s ≥ 2, "\;\;\;\;\;\;如果兩直線互相平行，\;"]]) + "" + (LaTeX[If[s ≥ 3, "由於同位角相等"]]) + "
\\" + (LaTeX[If[s ≥ 4, "\;\;\;\;\;\;，\;則兩直線斜角相等，\;可推得兩直線的 \\\;\;\;\;\;\;斜率相同。"]]) + "
\\" + (LaTeX[If[s ≥ 5, "\;\;\;\;\;\;直觀看來，\;若兩條斜率存在的平行直 \\\;\;\;\;\;\;線，\;其傾斜程度一樣，\;斜率也就相等。"]]) + "
\\" + (LaTeX[If[s ≥ 6, "\;\;\;\;\;\;反之，\;若兩相異直線的斜率相等，\;則可 \\\;\;\;\;\;\;知兩直線有相同的傾斜程度，\;則此兩 \\\;\;\;\;\;\;直線互相平行。"]]) + ""