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等面积动态演示

等面积动态演示
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2017-02-19 23:01
圆多边形变换
所属专题 Perimeter & Area
该范例演示了等面积的概念，通过动态变换展示了不同平面图形面积相等的现象。核心数学概念包括面积计算和图形变换。使用了圆、多边形、向量等GeoGebra对象进行构造。适合初中及以上学生学习面积计算和图形变换。

GeoGebra 命令及参数

  1. Circle(A, b) → d
  2. Point(d) → B
  3. Circle(B, c) → e
  4. Point(e) → C
  5. Polygon(A, B, 4) → poly1, f, g, h, i, D, E
  6. Polygon(C, B, 4) → poly2, j, k, l, m, F, G
  7. Dilate(B, If[0 < a ≤ 1, 1 - a, 0], A) → B'
  8. Dilate(B, If[0 < a ≤ 1, 1 - a, 0], F) → B'_1
  9. Segment(B, B') → n
  10. Segment(B, B'_1) → p
  11. Rotate(B, If[1 < a ≤ 2, ((-(a - 1)) * π / 2), ((-π)) / 2], F) → B'_2
  12. Rotate(B'_1, If[1 < a ≤ 2, ((-(a - 1)) * π / 2), ((-π)) / 2], F) → L
  13. Segment(B'_2, L) → p'
  14. Rotate(B, If[1 < a ≤ 2, ((a - 1) * π / 2), π / 2], A) → B'_3
  15. Rotate(B', If[1 < a ≤ 2, ((a - 1) * π / 2), π / 2], A) → B''
  16. Segment(B'_3, B'') → n'
  17. Rotate(B'_2, If[2 < a ≤ 3, ((-(a - 2)) * π / 2), ((-π)) / 2], G) → M
  18. Rotate(L, If[2 < a ≤ 3, ((-(a - 2)) * π / 2), ((-π)) / 2], G) → L'
  19. Segment(M, L') → p''
  20. Rotate(B'_3, If[2 < a ≤ 3, ((a - 2) * π / 2), π / 2], E) → N
  21. Rotate(B'', If[2 < a ≤ 3, ((a - 2) * π / 2), π / 2], E) → B'''
  22. Segment(N, B''') → n''
  23. Rotate(N, If[3 < a ≤ 4, ((a - 3) * π / 2), π / 2], D) → N'
  24. Rotate(B''', If[3 < a ≤ 4, ((a - 3) * π / 2), π / 2], D) → O
  25. Segment(N', O) → n'''
  26. Rotate(M, If[3 < a ≤ 4, ((-(a - 3)) * π / 2), ((-π)) / 2], C) → M'
  27. Rotate(L', If[3 < a ≤ 4, ((-(a - 3)) * π / 2), ((-π)) / 2], C) → L''
  28. Segment(M', L'') → p'''
  29. Segment(F, G) → q
  30. Segment(F, B) → r
  31. Segment(A, B) → s
  32. Min(r, s) → o
  33. Circle(F, (0.15 * o)) → t
  34. Intersect(l, q) → P
  35. Intersect(t, q) → Q,
  36. Intersect(t, r) → R,
  37. CircleSector(F, Q, R) → c_1
  38. Dilate(F, If[4 < a ≤ 5, 5 - a, 0], C) → F'
  39. Vector(F, F') → u
  40. Translate(c_1, u) → d_1
  41. Rotate(d_1, If[5 < a ≤ 6, ((a - 5) * π), π], C) → e_1
  42. Dilate(F, If[6 < a ≤ 7, 7 - a, 0], A) → F'_1
  43. Dilate(C, If[6 < a ≤ 7, 7 - a, 0], E) → C'
  44. Vector(F, F'_1) → v
  45. Vector(C, C') → w
  46. Translate(e_1, w) → f_1
  47. Translate(c_1, v) → g_1
  48. Line(A, p) → h_1
  49. Line(E, p''') → i_1
  50. Point(h_1) → S
  51. Angle(S, A, E) → α
  52. Rotate(g_1, If[7 < a ≤ 8, ((-(a - 7)) * ((2 * π) - α)), (-((2 * π) - α))], A) → k_1
  53. Line(E, h_1) → j_1
  54. Point(j_1) → T
  55. Angle(D, E, T) → β
  56. Rotate(f_1, If[7 < a ≤ 8, ((-(a - 7)) * β), (-β)], E) → p_1
  57. Dilate(E, If[8 < a ≤ 9, 9 - a, 0], G) → E'
  58. Dilate(F, If[8 < a ≤ 9, 9 - a, 0], D) → F'_2
  59. Vector(E, E') → u_1
  60. Vector(F, F'_2) → v_1
  61. Translate(p_1, u_1) → q_1
  62. Translate(c_1, v_1) → r_1
  63. Line(G, n'') → l_1
  64. Point(l_1) → U
  65. Line(G, n') → m_1
  66. Point(m_1) → V
  67. Angle(C, G, V) → γ
  68. Rotate(q_1, If[8 < a ≤ 9, ((-(a - 8)) * γ), (-γ)], E') → s_1
  69. Line(D, p) → n_1
  70. Point(n_1) → W
  71. Angle(W, D, B) → δ
  72. Rotate(r_1, If[8 < a ≤ 9, ((a - 8) * δ), δ], F'_2) → t_1
  73. Midpoint(A, B) → Z
  74. Midpoint(B, C) → A_1
  75. Midpoint(Z, B) → B_1
  76. Midpoint(A_1, B) → C_1
  77. Dilate(C, If[9 < a ≤ 10, 10 - a, 0], C_1) → C'_1
  78. Dilate(A, If[9 < a ≤ 10, 10 - a, 0], B_1) → A'
  79. Vector(A, A') → w_1
  80. Vector(C, C'_1) → a_1
  81. Translate(e_1, a_1) → c_2
  82. Translate(k_1, w_1) → d_2
  83. Rotate(d_2, If[10 < a ≤ 11, ((a - 10) * π / 2), π / 2], B_1) → e_2
  84. Rotate(c_2, If[10 < a ≤ 11, ((a - 10) * π / 2), π / 2], C_1) → f_2
  85. Dilate(C_1, If[11 < a ≤ 12, 12 - a, 0], B) → D_1
  86. Dilate(B_1, If[11 < a ≤ 12, 12 - a, 0], B) → E_1
  87. Vector(B_1, E_1) → b_1
  88. Vector(C_1, D_1) → u_2
  89. Translate(e_2, b_1) → g_2
  90. Translate(f_2, u_2) → h_2
  91. Segment(D, C) → a_3
  92. Segment(A, F) → b_3
  93. Line(A, B) → f_4
  94. Polygon(A, F, B) → poly4, b_4, a_4, f_5
  95. Polygon(B, C, D) → poly5, d_4, b_5, c_4
  96. OrthogonalLine(F, f_4) → h_4
  97. Line(D, B) → g_4
  98. OrthogonalLine(C, g_4) → i_4
  99. Intersect(f_4, h_4) → R_1
  100. Intersect(g_4, i_4) → S_1
  101. Dilate(B, If[13 < a ≤ 14, 14 - a, 0], R_1) → B'_4
  102. Dilate(F, If[13 < a ≤ 14, 14 - a, 0], R_1) → F'_3
  103. Angle(B, R_1, F) → ε
  104. Segment(B, B'_4) → i_2
  105. Segment(F, F'_3) → j_2
  106. Dilate(A, If[12 < a ≤ 13, 13 - a, 0], F) → A'_1
  107. Dilate(D, If[12 < a ≤ 13, 13 - a, 0], C) → D'
  108. Segment(D, D') → k_2
  109. Segment(A, A'_1) → l_2
  110. Dilate(C, If[14 < a ≤ 15, 15 - a, 0], S_1) → C'_2
  111. Dilate(D, If[14 < a ≤ 15, 15 - a, 0], S_1) → D'_1
  112. Segment(D, D'_1) → m_2
  113. Segment(C, C'_2) → n_2
  114. Angle(D, S_1, C) → ζ
  115. Dilate(F'_3, If[15 < a ≤ 16, 16 - a, 0], C) → H
  116. Vector(F'_3, H) → v_2
  117. Translate(F, v_2) → F'_4
  118. Translate(F'_3, v_2) → I
  119. Segment(F'_4, I) → p_2
  120. Line(C, j_2) → q_2
  121. Point(q_2) → J
  122. Angle(J, C, C'_2) → η
  123. Rotate(F'_4, If[16 < a ≤ 17, ((a - 16) * η), η], C) → K
  124. Rotate(I, If[16 < a ≤ 17, ((a - 16) * η), η], C) → I'
  125. Segment(K, I') → r_2