正立方體的8個頂點，可以決定幾個平面？
Author: 藍邦偉  老師

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正立方體的8個頂點，可以決定幾個平面?

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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))
a = Segment(A, B)
C: (3 * A + B) / 4
D: (3 * B + A) / 4
E = Point(a)
c = Circle(C, E)
c' = Translate(c, Vector[C, D])
A' = Rotate(A, 270*°, C)
B' = Rotate(B, 270*°, C)
a' = Segment(A', B')
c'' = Rotate(c', 270*°, C)
D' = Rotate(D, 270*°, C)
E' = Rotate(Translate[E, Vector[C, D]], 90*°, D)
E'' = Rotate(E', 270*°, C)
F = Point(c'')
G = Point(c')
α = Angle(E'', D', F)
b = Segment(D', F)
d = Segment(D, G)
β = Angle(G, D, E')
I: C + (b * cos(α), b * sin(α) * sin(β))
J: C + (b * cos(α + 90°), b * sin(α + 90°) * sin(β))
K: C + (0, b * cos(β))
e = Segment(C, K)
f = Segment(C, I)
g = Segment(C, J)
O = (-0.17051405149215051, -1.499865806280506)
I' = Translate(I, Vector[C, O])
J' = Translate(J, Vector[C, O])
K' = Translate(K, Vector[C, O])
L = 6 (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 = false
w = Vector(O, K')
u = Vector(O, I')
v = Vector(O, J')
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)
A_1: O + 4 * u + 4 * v + 4 * w
A_2: O + 4 * u + 4 * v - 4 * w
A_3: O + 4 * u - 4 * v + 4 * w
A_4: O - 4 * u + 4 * v + 4 * w
A_6: O + 4 * u - 4 * v - 4 * w
A_7: O - 4 * u + 4 * v - 4 * w
A_8: O - 4 * u - 4 * v - 4 * w
A_5: O - 4 * u - 4 * v + 4 * w
l = false
n = 0 (slider from 0 to 20, step 1)
多邊形1, a_2, a_3, a_7, a_4 = Polygon(A_1, A_2, A_7, A_4)
多邊形2, a_5, a_6, a_8, a_9 = Polygon(A_3, A_6, A_8, A_5)
多邊形3, a_{10}, a_{11}, a_{12}, a_{13} = Polygon(A_1, A_3, A_6, A_2)
多邊形4, a_{14}, a_{15}, a_{16}, a_{17} = Polygon(A_5, A_8, A_7, A_4)
多邊形5, a_{18}, a_{19}, a_{20}, a_{21} = Polygon(A_2, A_7, A_8, A_6)
多邊形6, a_{22}, a_{23}, a_{24}, a_{25} = Polygon(A_1, A_4, A_5, A_3)
多邊形7, a_{26}, a_{27}, a_{28}, a_{29} = Polygon(A_3, A_2, A_7, A_5)
多邊形8, a_{30}, a_{31}, a_{32}, a_{33} = Polygon(A_1, A_6, A_8, A_4)
多邊形9, a_{34}, a_{35}, a_{36}, a_{37} = Polygon(A_1, A_5, A_8, A_2)
多邊形10, a_{38}, a_{39}, a_{40}, a_{41} = Polygon(A_3, A_6, A_7, A_4)
多邊形11, a_{42}, a_{43}, a_{44}, a_{45} = Polygon(A_1, A_3, A_8, A_7)
多邊形12, a_{46}, a_{47}, a_{48}, a_{49} = Polygon(A_5, A_4, A_2, A_6)
多邊形13, a_{50}, a_{51}, a_{52} = Polygon(A_3, A_2, A_8)
多邊形14, a_{53}, a_{54}, a_{55} = Polygon(A_1, A_7, A_6)
多邊形15, a_{56}, a_{57}, a_{58} = Polygon(A_4, A_2, A_8)
多邊形16, a_{59}, a_{60}, a_{61} = Polygon(A_7, A_6, A_5)
多邊形17, a_{62}, a_{63}, a_{64} = Polygon(A_1, A_5, A_6)
多邊形18, a_{65}, a_{66}, a_{67} = Polygon(A_3, A_4, A_2)
多邊形19, a_{68}, a_{69}, a_{70} = Polygon(A_1, A_5, A_7)
多邊形20, a_{71}, a_{72}, a_{73} = Polygon(A_3, A_4, A_8)
m, p = SDLine(A_3, A_1, A_6, A_5)
q, r = SDLine(A_1, A_4, A_2, A_3)
s, t = SDLine(A_1, A_2, A_3, A_4)
a_1, b_1 = SDLine(A_3, A_6, A_5, A_1)
c_1, d_1 = SDLine(A_4, A_7, A_1, A_5)
e_1, f_1 = SDLine(A_5, A_8, A_4, A_3)
g_1, h_1 = SDLine(A_4, A_5, A_7, A_1)
i_1, j_1 = SDLine(A_5, A_3, A_8, A_4)
k_1, l_1 = SDLine(A_8, A_6, A_7, A_5)
m_1, n_1 = SDLine(A_7, A_8, A_6, A_5)
p_1, q_1 = SDLine(A_2, A_6, A_1, A_7)
r_1, s_1 = SDLine(A_2, A_7, A_8, A_4)

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)