求正四面體的內接球半徑
Author: 藍邦偉  老師

運用四個小的正三角錐的體積和等於正四面體的體積

x

y

z

顯示xy平面

顯示單位向量

顯示座標軸

d = 0 (slider from 0 to 3, step 1)
O = (0.9313735648045363, -1.7465867663976733)
A: (x((Corner[3.0] + 39.0 * Corner[4.0]) / 40.0), y((Corner[1.0] + 9.0 * Corner[4.0]) / 10.0))
B: (x((4.0 * Corner[3.0] + 16.0 * Corner[4.0]) / 20.0), y((Corner[1.0] + 9.0 * Corner[4.0]) / 10.0))
D: (3.0 * B + A) / 4.0
C: (3.0 * A + B) / 4.0
a = Segment(A, B)
E = Point(a)
E' = Rotate(Translate[E, Vector[C, D]], 90.0*°, D)
c = Circle(C, E)
c' = Translate(c, Vector[Vector[C, D]])
G = Point(c')
c'' = Rotate(c', 270.0*°, C)
F = Point(c'')
A' = Rotate(A, 270.0*°, C)
B' = Rotate(B, 270.0*°, C)
a' = Segment(A', B')
D' = Rotate(D, 270.0*°, C)
b = Segment(D', F)
E'' = Rotate(E', 270.0*°, C)
α = Angle(E'', D', F)
d_1 = Segment(D, G)
β = Angle(G, D, E')
K: C + (0.0, b * cos(β))
K' = Translate(K, Vector[Vector[C, O]])
w = Vector(O, K')
J: C + (b * cos(α + 90.0°), b * sin(α + 90.0°) * sin(β))
J' = Translate(J, Vector[Vector[C, O]])
v = Vector(O, J')
I: C + (b * cos(α), b * sin(α) * sin(β))
I' = Translate(I, Vector[Vector[C, O]])
u = Vector(O, I')
e = Segment(C, K)
f = Segment(C, I)
g = Segment(C, J)
L = 9 (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'', -1.0, O)
J''' = Dilate(J'', -1.0, O)
K''' = Dilate(K'', -1.0, 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
l = true
list1 = Sequence(CurveCartesian[x(O + t * u + s * v), y(O + t * u + s * v), t, -(L), L], s, -(L), L, 1.0)
list2 = Sequence(CurveCartesian[x(O + t * u + s * v), y(O + t * u + s * v), s, -(L), L], t, -(L), L, 1.0)
A_2: O + 2.0 * sqrt(2.0) * u + 2.0 * sqrt(6.0) * v - d * w
A_4: O + 8.0 * w - 1.0 / 2.0 * d * u + d * sqrt(3.0) / 2.0 * v
A_3: O - 4.0 * sqrt(2.0) * u - d * w
A_1: O + 2.0 * sqrt(2.0) * u - 2.0 * sqrt(6.0) * v - d * w
m = Segment(A_1, A_2)
n = Segment(A_2, A_3)
p = Segment(A_3, A_1)
H = Midpoint(A_1, A_2)
M = Midpoint(A_2, A_3)
N = Midpoint(A_3, A_1)
A_{G1}: O + 3.0 / 2.0 * sqrt(2.0) * w - d * w
t = Segment(A_1, A_{G1})
a_1 = Segment(A_{G1}, A_2)
b_1 = Segment(A_{G1}, A_3)
A_{G2}: O + 3.0 / 2.0 * sqrt(2.0) * w + d * u
A_5: O + 2.0 * sqrt(2.0) * u - 2.0 * sqrt(6.0) * v + d * u
A_6: O + 2.0 * sqrt(2.0) * u + 2.0 * sqrt(6.0) * v + d * u
A_7: O + 8.0 * w + d * u
q = Segment(A_7, A_5)
r = Segment(A_5, A_6)
s = Segment(A_6, A_7)
c_1 = Segment(A_7, A_{G2})
e_1 = Segment(A_{G2}, A_5)
f_1 = Segment(A_{G2}, A_6)
A_{G3}: O + 3.0 / 2.0 * sqrt(2.0) * w - 1.0 / 2.0 * d * u + d * sqrt(3.0) / 2.0 * v
A_8: O + 2.0 * sqrt(2.0) * u + 2.0 * sqrt(6.0) * v - 1.0 / 2.0 * d * u + d * sqrt(3.0) / 2.0 * v
A_9: O - 4.0 * sqrt(2.0) * u - 1.0 / 2.0 * d * u + d * sqrt(3.0) / 2.0 * v
g_1 = Segment(A_4, A_9)
h_1 = Segment(A_9, A_8)
i_1 = Segment(A_8, A_4)
j_1 = Segment(A_4, A_{G3})
k_1 = Segment(A_{G3}, A_9)
l_1 = Segment(A_{G3}, A_8)
A_{G4}: O + 3.0 / 2.0 * sqrt(2.0) * w - 1.0 / 2.0 * d * u - d * sqrt(3.0) / 2.0 * v
A_{10}: O + 2.0 * sqrt(2.0) * u - 2.0 * sqrt(6.0) * v + d * u - 1.0 / 2.0 * d * u - d * sqrt(3.0) / 2.0 * v
A_{11}: O - 4.0 * sqrt(2.0) * u - 1.0 / 2.0 * d * u - d * sqrt(3.0) / 2.0 * v
A_{12}: O + 8.0 * w + d * u - 1.0 / 2.0 * d * u - d * sqrt(3.0) / 2.0 * v
m_1 = Segment(A_{12}, A_{11})
n_1 = Segment(A_{11}, A_{10})
p_1 = Segment(A_{10}, A_{12})
q_1 = Segment(A_{G4}, A_{11})
r_1 = Segment(A_{G4}, A_{12})
s_1 = Segment(A_{G4}, A_{10})
多邊形1, a_3, a_2, a_4 = Polygon(A_1, A_2, A_3)
多邊形2, a_9, a_5, a_8 = Polygon(A_4, A_8, A_9)
多邊形3, a_6, a_7, a_{10} = Polygon(A_7, A_5, A_6)
多邊形4, a_{11}, a_{12}, a_{13} = Polygon(A_{12}, A_{10}, A_{11})
O_{G1}: 1.0 / 3.0 * A_1 + 1.0 / 3.0 * A_2 + 1.0 / 3.0 * A_3
O_{G2}: 1.0 / 3.0 * A_{10} + 1.0 / 3.0 * A_{12} + 1.0 / 3.0 * A_{11}
O_{G3}: 1.0 / 3.0 * A_4 + 1.0 / 3.0 * A_8 + 1.0 / 3.0 * A_9
O_{G4}: 1.0 / 3.0 * A_5 + 1.0 / 3.0 * A_6 + 1.0 / 3.0 * A_7
t_1 = Segment(A_{G1}, O_{G1})
b_2 = Segment(A_{G3}, O_{G3})
c_2 = Segment(O_{G2}, A_{G4})
d_2 = Segment(O_{G4}, A_{G2})
e_2 = Circle(A_{G1}, O_{G1})