Author: 藍邦偉　老師
Date: 23 3 月 2014

$O$

$β=%v$

$α=%v$

單位向量

座標軸

Start

x

y

z

$ 顯示文字說明$

$P$

$Q$

$R$

$Step=%v$

$A$

$B$

$C$

$D$

$M$

$θ$

$H$

$ 正\Delta PQR $

$  分別取三邊中點A,B,C $

$連\overline{AB},\overline{BC},\overline{CA} $

$ 做正四面體ABCD $

$ 過D作正\Delta ABC的高\overline{DH} $

$ 連\overline{AH},\overline{BH},\overline{CH} $

$ 是真的正四面體 $

$ \Delta AHD≅\Delta BHD≅\Delta CHD(RHS) $
$⇒\overline{HA}=\overline{HB}=\overline{HC} $

$H\;為正\Delta ABC\;的外心, \;$
$又正三角形的外心與重心重合$
$∴H\;為正\Delta ABC\;的重心$

$ 取\overline{AB}的中點M $

$ 連\overline{CM}與\overline{DM} $
$則θ為兩相鄰平面的兩面角$

$ cosθ=\frac{\overline{MH}}{\overline{DM}} =\frac{\frac{1}{3}\overline{CM}}{\overline{CM}}=\frac{1}{3}$

$ 正四面體 的性質$

O = (16.36756972977987, 1.059650522662307)
β = 0 rad (slider from 0 to 6.283185307179586, step 0.1*°)
Z_{unit}: O + (0, sin(β))
w = Vector(O, Z_{unit})
α = 0 rad (slider from 0 to 6.283185307179586, step 0.1*°)
X_{unit}: O + (cos(α), sin(α) * cos(β))
u = Vector(O, X_{unit})
Y_{unit}: O + (-1 * sin(α), cos(α) * cos(β))
v = Vector(O, Y_{unit})
L = 1 (slider from 1 to 10, step 1)
yy = Vector(O - L * v, O + L * v)
zz = Vector(O - L * w, O + L * w)
xx = Vector(O - L * u, O + L * u)
c = false
d = false
Q_2: 0.25 * Corner[2] + 0.75 * Corner[1]
Q_3: (x(Q_2), y(Corner[3]))
poly1, q_2, q_1, q_4, q_3 = Polygon(Corner[1], Q_2, Q_3, Corner[4])
P: O - u - 1 / sqrt(3) * v
Q: O + u - 1 / sqrt(3) * v
R: O + 2 / sqrt(3) * v
a_1 = Segment(P, Q)
a_2 = Segment(R, Q)
a_3 = Segment(R, P)
n = 0 (slider from 0 to 12, step 1)
A: O - 1 / sqrt(3) * v
B: O + 0.5 * u + sqrt(3) / 6 * v
C: O - 0.5 * u + sqrt(3) / 6 * v
t3 = 0 (slider from 0 to 1, step 0.01)
A_1: A + t3 * Vector[A, C]
B_1: B + t3 * Vector[B, A]
C_1: C + t3 * Vector[C, B]
b_a = Segment(A, A_1)
b_b = Segment(B, B_1)
b_c = Segment(C, C_1)
t4 = 0 rad (slider from 0 to 1.9106332362490186, step 0.1*°)
A': O + sqrt(3) / 2 * sin(t4) * w + (0.25 + 0.75 * cos(t4)) * u + ((-sqrt(3)) / 12 - sqrt(3) / 4 * cos(t4)) * v
B': O + sqrt(3) / 2 * sin(t4) * w + (-0.25 - 0.75 * cos(t4)) * u + ((-sqrt(3)) / 12 - sqrt(3) / 4 * cos(t4)) * v
C': O + sqrt(3) / 2 * sin(t4) * w + (sqrt(3) / 6 + sqrt(3) / 2 * cos(t4)) * v
c_1 = Segment(C', B)
c_2 = Segment(C', C)
c_3 = Segment(A', A)
c_4 = Segment(A', B)
c_5 = Segment(A, B')
c_6 = Segment(C, B')
D: O + sqrt(6) / 3 * w
d_1, d_2 = SDLine(D, C, B, A)
d_3, d_4 = SDLine(D, B, A, C)
d_5, d_6 = SDLine(D, A, C, B)
d_7, d_8 = SDLine(A, B, C, D)
d_9, d_{10} = SDLine(B, C, A, D)
d_{11}, d_{12} = SDLine(C, A, B, D)
t6 = 0 (slider from 0 to 1, step 0.01)
D_1: D + t6 * Vector[D, O]
e_1 = Segment(D, D_1)
t7 = 0 (slider from 0 to 1, step 0.01)
D_A: O + t7 * Vector[O, A]
D_B: O + t7 * Vector[O, B]
D_C: O + t7 * Vector[O, C]
f_1 = Segment(O, D_A)
f_2 = Segment(O, D_B)
f_3 = Segment(O, D_C)
Trig_1, z_a, z_d, z_g = Polygon(D, O, A)
Trig_2, z_b, z_e, z_h = Polygon(D, O, B)
Trig_3, z_c, z_f, z_i = Polygon(D, O, C)
g_1 = Segment(O, A)
g_2 = Segment(O, B)
g_3 = Segment(O, C)
M = Midpoint(A, B)
t11 = 0 (slider from 0 to 1, step 0.01)
E_1: M + t11 * Vector[M, D]
E_2: M + t11 * Vector[M, C]
h_1 = Segment(M, E_1)
h_2 = Segment(M, E_2)
E_4: 2 / 3 * C + 1 / 3 * M
E_3: 2 / 3 * M + 1 / 3 * C
c_0 = Circle(O, 0.25 * Segment[O, M])
F_1 = Intersect(c_0, e_1)
F_2, F_3 = Intersect(c_0, h_2)
F_4: F_1 + F_2 - O
i_0 = PolyLine(F_1, F_4, F_2)
h_3 = Segment(O, M)
Trig_4, z_3, z_1, z_2 = Polygon(D, M, O)
γ = Angle(O, M, D)
j_1 = Segment(P, Q)
j_2 = Segment(R, Q)
j_3 = Segment(R, P)
H: O
a = false
Trig_A = Poly_C3(D, C, A)
Trig_B = Poly_C3(D, A, B)
Trig_C = Poly_C3(D, B, C)
Trig_D = Poly_C3(C, B, A)

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)

Tool: Poly_B3
Input: A, B, C
Output: poly1'

α = Angle(C, B, A)
d: sqrt(sgn(sin(α)))
A' = Dilate(A, d, A)
B' = Dilate(B, d, A)
C' = Dilate(C, d, A)
poly1', c', a', b' = Polygon(A', B', C')

Tool: Poly_C3
Input: A, B, C
Output: poly1

poly1 = Poly_B3(A, B, C)