88年大學聯考一題
Author: 藍邦偉  老師

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x

y

z

顯示xy平面

顯示單位向量

顯示座標軸

相似三角形

O = (5.466131555997508, -0.653903058945323)
A: (x((Corner[3] + 39 * Corner[4]) / 40), y((Corner[1] + 9 * Corner[4]) / 10))
B: (x((4 * Corner[3] + 16 * Corner[4]) / 20), y((Corner[1] + 9 * Corner[4]) / 10))
D: (3 * B + A) / 4
C: (3 * A + B) / 4
a = Segment(A, B)
E = Point(a)
E' = Rotate(Translate[E, Vector[C, D]], 90*°, D)
c = Circle(C, E)
c' = Translate(c, Vector[C, D])
G = Point(c')
c'' = Rotate(c', 270*°, C)
F = Point(c'')
A' = Rotate(A, 270*°, C)
B' = Rotate(B, 270*°, C)
a' = Segment(A', B')
D' = Rotate(D, 270*°, C)
b = Segment(D', F)
E'' = Rotate(E', 270*°, C)
α = Angle(E'', D', F)
d_1 = Segment(D, G)
β = Angle(G, D, E')
K: C + (0, b * cos(β))
K' = Translate(K, Vector[C, O])
w = Vector(O, K')
J: C + (b * cos(α + 90°), b * sin(α + 90°) * sin(β))
J' = Translate(J, Vector[C, O])
v = Vector(O, J')
I: C + (b * cos(α), b * sin(α) * sin(β))
I' = Translate(I, Vector[C, O])
u = Vector(O, I')
e = Segment(C, K)
f = Segment(C, I)
g = Segment(C, J)
L = 10 (slider from 1 to 10, step 1)
I'' = Dilate(I', L, O)
J'' = Dilate(J', L, O)
K'' = Dilate(K', L, O)
I''' = Dilate(I'', -1, O)
J''' = Dilate(J'', -1, O)
K''' = Dilate(K'', -1, O)
u' = Vector(I''', I'')
v' = Vector(J''', J'')
w' = Vector(K''', K'')
h = Segment(D', E'')
i = Segment(D, E')
軌跡1 = Locus(I, F)
L_{axis}: L
j = true
k = false
list1 = Sequence(CurveCartesian[x(O + t * u + s * v), y(O + t * u + s * v), t, -L, L], s, -L, L, 1)
list2 = Sequence(CurveCartesian[x(O + t * u + s * v), y(O + t * u + s * v), s, -L, L], t, -L, L, 1)
C_1: O + 4 * cos(0°) * u + 4 * sin(0°) * v + 4 / sqrt(3) * w
C_5: O + 4 * sqrt(3) * w
C_3: O + 4 * cos(120°) * u + 4 * sin(120°) * v + 4 / sqrt(3) * w
C_7: O + 4 * cos(240°) * u + 4 * sin(240°) * v + 4 / sqrt(3) * w
C_2: O + 4 * cos(60°) * u + 4 * sin(60°) * v + 8 / sqrt(3) * w
C_6: O + 4 * cos(180°) * u + 4 * sin(180°) * v + 8 / sqrt(3) * w
C_4: O + 4 * cos(300°) * u + 4 * sin(300°) * v + 8 / sqrt(3) * w
n = 1 (slider from 0 to 1, step 1)
l = true
多邊形1, b_2, o, d_2 = Polygon(O, C_3, C_5)
z = true
d, m = SDLine(C_2, C_1, C_5, C_3)
p, q = SDLine(C_5, C_2, C_4, C_6)
r, s = SDLine(C_2, C_3, C_1, C_5)
t, a_1 = SDLine(C_3, O, C_2, C_6)
b_1, c_1 = SDLine(C_5, C_4, C_6, C_2)
e_1, f_1 = SDLine(C_1, O, C_7, C_3)
g_1, h_1 = SDLine(C_6, C_3, C_5, C_7)
i_1, j_1 = SDLine(C_5, C_6, C_2, C_4)
k_1, l_1 = SDLine(C_6, C_7, C_3, C_5)
m_1, n_1 = SDLine(C_4, C_1, C_7, C_5)
p_1, q_1 = SDLine(C_4, C_7, C_5, C_1)
r_1, s_1 = SDLine(C_7, O, C_6, C_4)
A_1: O + 4 * cos(120°) * u + 4 * sin(120°) * v
poly1, c_3, o_1, a_2 = Polygon(O, A_1, C_3)

Tool: SDLine
Input: A, B, C, D
Output: g, h

b: sgn(sin(Angle[D, B, A]))
a: sgn(sin(Angle[A, B, C]))
c: sgn(a + b + 0.5)
d: sqrt(c)
e: sqrt(-c)
E = Midpoint(A, B)
A' = Dilate(A, d, E)
B' = Dilate(B, d, E)
g = Segment(A', B')
A'_1 = Dilate(A, e, E)
B'_1 = Dilate(B, e, E)
h = Segment(A'_1, B'_1)