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动态几何演示：等边三角形与金字塔

动态几何演示：等边三角形与金字塔
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2018-02-09 00:57
测试改名0圆向量几何作图
①本范例动态演示了两个不同来源（Antonio Gutierrez 和 Alexander Bogomolny）的等边三角形与金字塔相关问题；②涉及核心数学概念包括等边三角形性质、单位圆、中点、向量、变换等；③作图要点：运用了 Rotate（旋转）、Dilate（缩放）、Polygon（多边形）、Midpoint（中点）、CircleArc（圆弧）、Vector（向量）等 GeoGebra 命令；④适合高中及以上数学学习者用于理解几何变换与空间关系。

GeoGebra 命令及参数

  1. Rotate(A, If[1 < e ≤ 2, ((-(e - 1)) * π / 2), -1.5707963267948966], B_2) → A'_1
  2. Dilate(A, If[0 < e ≤ 1, 1 - e, 0], B_2) → A'
  3. Rotate(A', If[1 < e ≤ 2, ((-(e - 1)) * π / 2), -1.5707963267948966], B_2) → A''
  4. Polygon(A, B_2, 4) → poly1, a, b, c, d, C, D
  5. Midpoint(A, C) → V
  6. CircleArc(V, C, D) → g_2
  7. Point(g_2) → B
  8. Segment(C, B) → j_1
  9. Segment(A, B) → n
  10. Rotate(A'', If[2 < e ≤ 3, ((-(e - 2)) * π / 2), -1.5707963267948966], C) → A'''
  11. Rotate(A'_1, If[2 < e ≤ 3, ((-(e - 2)) * π / 2), -1.5707963267948966], C) → R
  12. Segment(R, A''') → f''
  13. Intersect(n, f'') → Z
  14. Segment(B, Z) → l_1
  15. Segment(A, A') → f
  16. Segment(A'_1, A'') → f'
  17. Segment(A, B_2) → g
  18. Segment(B_2, C) → h
  19. Segment(C, D) → i
  20. Segment(D, A) → j
  21. Circle(B_2, (0.15 * f)) → p
  22. Circle(C, Radius[p]) → k
  23. Circle(A, (0.1 * j)) → q
  24. Circle(D, (0.1 * j)) → r
  25. Segment(A, C) → l
  26. Segment(D, B_2) → m
  27. Intersect(r, j, 1) → F
  28. Intersect(r, m, 1) → G
  29. Intersect(r, i, 1) → H
  30. Intersect(k, i, 1) → I
  31. Intersect(k, l, 1) → J
  32. Intersect(k, h, 1) → K
  33. Intersect(p, h, 1) → L
  34. Intersect(p, m, 1) → M
  35. Intersect(p, g, 1) → N
  36. Intersect(q, g, 1) → O
  37. Intersect(q, l, 1) → P
  38. Intersect(q, j, 1) → Q
  39. CircleSector(B_2, L, N) → s
  40. Rotate(R, If[3 < e ≤ 4, ((-(e - 3)) * π / 2), -1.5707963267948966], D) → R'
  41. Rotate(A''', If[3 < e ≤ 4, ((-(e - 3)) * π / 2), -1.5707963267948966], D) → S
  42. Segment(R', S) → f'''
  43. Dilate(B_2, If[4 < e ≤ 5, 5 - e, 0], C) → B'
  44. Vector(B_2, B') → u
  45. Translate(s, u) → s'
  46. Rotate(s', If[5 < e ≤ 6, ((e - 5) * π / 2), 1.5707963267948966], C) → s''
  47. Dilate(C, If[6 < e ≤ 7, 7 - e, 0], D) → C'
  48. Vector(C, C') → v
  49. Translate(s'', v) → s'''
  50. Rotate(s''', If[7 < e ≤ 8, ((e - 7) * π / 2), 1.5707963267948966], D) → t
  51. Dilate(D, If[8 < e ≤ 9, 9 - e, 0], A) → D'
  52. Vector(D, D') → w
  53. Translate(t, w) → t'
  54. Rotate(t', If[9 < e ≤ 10, ((e - 9) * π / 2), 1.5707963267948966], A) → t''
  55. Dilate(B_2, If[18 < e ≤ 19, 19 - e, 0], C) → B'_1
  56. Dilate(B_2, If[18 < e ≤ 19, 19 - e, 0], A) → B'_2
  57. Dilate(B_2, If[18 < e ≤ 19, 19 - e, 0], D) → B'_3
  58. Vector(B_2, B'_1) → v_1
  59. Vector(B_2, B'_3) → w_1
  60. Vector(B_2, B'_2) → i_1
  61. Circle(A, n - j_1) → h_2
  62. Dilate(B, If[10 < e ≤ 11, 11 - e, 0], C) → W'
  63. Dilate(B_2, If[13 < e ≤ 14, 14 - e, 0], B) → B'_4
  64. Vector(B_2, B'_4) → u_1
  65. Translate(s, u_1) → s'_1
  66. Line(B, f') → a_1
  67. Point(a_1) → U
  68. Angle(U, B, C) → α
  69. Rotate(B, If[14 < e ≤ 15, ((-(e - 14)) * π / 2), ((-π)) / 2], B) → W'_2
  70. Rotate(W', If[14 < e ≤ 15, ((-(e - 14)) * π / 2), ((-π)) / 2], B) → W''
  71. Line(A, B_2) → h_1
  72. Angle(B, A, D) → β
  73. Dilate(D, If[19 < e ≤ 20, 20 - e, 0], B) → D'_1
  74. Mirror(C, D) → C'_1
  75. Rotate(C, ((-π)) / 3, B_2) → C'_2
  76. Segment(B_2, C'_2) → f_1
  77. Segment(C, C'_2) → g_1
  78. Intersect(k, g_1, 1) → E
  79. Intersect(k, h) → T,
  80. CircleSector(C, T, E) → d_1
  81. Rotate(D, 15°, C) → D'_2
  82. Angle(D, C, D'_2) → γ
  83. Ray(C, D'_2) → k_1
  84. Rotate(C, (-(15°)), D) → C'_3
  85. Angle(C'_3, D, C) → δ
  86. Ray(D, C'_3) → m_1
  87. Intersect(k_1, m_1) → W
  88. Rotate(B_2, If[10 < e ≤ 11, (((-π)) / 3 * (e - 10)), ((-π)) / 3], B_2) → A_1
  89. Rotate(C, If[10 < e ≤ 11, (((-π)) / 3 * (e - 10)), ((-π)) / 3], B_2) → C'_4
  90. Segment(A_1, C'_4) → h'
  91. Rotate(B_2, If[11 < e ≤ 12, (π / 3 * (e - 11)), π / 3], C) → B_1
  92. Rotate(C, If[11 < e ≤ 12, (π / 3 * (e - 11)), π / 3], C) → C'_5
  93. Segment(B_1, C'_5) → h'_1
  94. Rotate(E, If[12 < e ≤ 13, (((-π)) / 6 * (e - 12)), ((-π)) / 6], C) → E'
  95. CircleSector(C, E', E) → c_1
  96. Polygon(C, W, D) → poly2, d_2, c_2, w_2
  97. Polygon(A, W, D) → poly3, d_3, a_2, w_3
  98. Polygon(B_2, W, C) → poly4, c_3, b_2, w_4
  99. Polygon(A, B_2, W) → poly5, w_5, a_3, b_1
  100. Polygon(B_2, C'_2, C) → poly6, c_4, b_3, c'_2
  101. Midpoint(h') → C_1
  102. Segment(C, C_1) → n_1
  103. Intersect(k, n_1) → D_1,
  104. Dilate(C, If[12 < e ≤ 13, 13 - e, 0], C_1) → C'_6
  105. Segment(C, C'_6) → p_1
  106. Rotate(T, 15°, C) → T'
  107. Angle(T, C, T') → ε
  108. Ray(C, T') → q_1
  109. Intersect(h', q_1) → E_1
  110. Rotate(c_1, If[13.5 < e ≤ 14.5, (((-π)) / 6 * (e - 13.5)), ((-π)) / 6], C) → e_1
  111. Dilate(C, If[16 < e ≤ 17, 17 - e, 0], E_1) → C'_7
  112. Segment(C, C'_7) → r_1
  113. Rotate(D_1, If[16 < e ≤ 17, (((-π)) / 12 * (e - 16)), ((-π)) / 12], C) → F_1
  114. CircleSector(C, F_1, D_1) → s_1
  115. Rotate(s_1, If[17.5 < e ≤ 18.5, (((-π)) / 12 * (e - 17.5)), ((-π)) / 12], C) → t_1
  116. Rotate(t_1, If[19.5 < e ≤ 20.5, (((-π)) / 2 * (e - 19.5)), ((-π)) / 2], C) → e_2
  117. Dilate(C, If[21 < e ≤ 22, 22 - e, 0], D) → C'_8
  118. Vector(C, C'_8) → u_2
  119. Translate(e_2, u_2) → k_2
  120. Rotate(k_2, If[22 < e ≤ 23, ((π - π / 12) * (e - 22)), π - π / 12], D) → p_2
  121. Dilate(D, If[23 < e ≤ 24, 24 - e, 0], W) → D'_3
  122. Dilate(C, If[23 < e ≤ 24, 24 - e, 0], W) → C'_9
  123. Segment(C, C'_9) → f_2
  124. Segment(D, D'_3) → i_2
  125. Dilate(B_2, If[25 < e ≤ 26, 26 - e, 0], W) → G_1
  126. Dilate(A, If[25 < e ≤ 26, 26 - e, 0], W) → A'_2
  127. Segment(A, A'_2) → j_2
  128. Segment(B_2, G_1) → l_2
  129. Rotate(B_2, If[26.75 < e ≤ 27.75, (π / 2 * (e - 26.75)), π / 2], B_2) → H_1
  130. Rotate(C'_2, If[26.75 < e ≤ 27.75, (π / 2 * (e - 26.75)), π / 2], B_2) → I_1
  131. Rotate(C, If[26.75 < e ≤ 27.75, (π / 2 * (e - 26.75)), π / 2], B_2) → C'_{10}
  132. Polygon(H_1, I_1, C'_{10}) → poly6', c'_{10}, h_3, i_3
  133. Rotate(C, ((-π)) / 3, B_2) → C'_{11}
  134. Dilate(D, If[30 < e ≤ 31, 31 - e, 0], W) → D'_4
  135. Segment(DD, D'_4) → m_2
  136. Dilate(W, If[31 < e ≤ 32, 32 - e, 0], C'_{11}) → W'_1
  137. Segment(W, W'_1) → n_2