9901第二次北市公立模考
Author: 藍邦偉  老師

這題是華東大學"多功能題典"多面體的最後一題。

x

y

z

顯示xy平面

顯示單位向量

座標軸

四個內切球球心所形成的正四面體

正四面體容器

Key

相似三角形

$ 將大小相同且半徑為1的4顆球置入正四面體的容$
$器裡，\;這個正四面體高的最小值是? $

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 = (1.663294021242252, -0.8249133093343435)
I' = Translate(I, Vector[C, O])
J' = Translate(J, Vector[C, O])
K' = Translate(K, Vector[C, O])
L = 7 (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 + 1.9916 * u + 3.4495 * v
A_2: O + 1.9916 * u - 3.4495 * v
A_4: O + (4 + 2 * sqrt(6) / 3) * w
A_3: O - 3.9631 * u
m = Segment(A_1, A_2)
n_1 = Segment(A_2, A_3)
p = Segment(A_3, A_1)
q = Segment(A_1, A_4)
r = Segment(A_4, A_3)
s = Segment(A_2, A_4)
H = Midpoint(A_1, A_2)
M = Midpoint(A_2, A_3)
N = Midpoint(A_3, A_1)
O_1: O - 2 * w
T_1: O + 0.57735 * u - v + w
T_2: O + 0.57735 * u + v + w
T_3: O - 1.1547 * u + w
T_4: O + 2.63299 * w
t = Segment(T_1, T_2)
a_1 = Segment(T_2, T_3)
b_1 = Segment(T_3, T_1)
c_1 = Segment(T_4, T_1)
d_1 = Segment(T_4, T_3)
e_1 = Segment(T_4, T_2)
O_2: 1 / 3 * (T_1 + T_2 + T_3)
P = Midpoint(T_1, T_2)
Q = Midpoint(A_1, A_2)
多邊形1, p_1, t_4, o_2 = Polygon(T_4, O_2, P)
多邊形2, n, f_1, g_1 = Polygon(A_4, O_1, O)
o = false
z = true
l = true
h_1 = true
e1 = true
j_1 = OrthogonalLine(T_4, q)
R: O - 0.47141 * u + 0.8165 * v + 2.96633 * w
k_1 = Segment(R, T_4)
S = Midpoint(p)
l_1 = Segment(A_4, S)
m_1 = Segment(S, O)
i_1 = Segment(O, A_4)