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五边形内角与外角定理的变换几何探究 by Mrs. Cole

五边形内角与外角定理的变换几何探究 by Mrs. Cole
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2019-01-02 11:45
多边形角变换几何作图滑块动画
本范例通过动态几何Applet，利用平移、旋转、镜像等变换工具，直观演示五边形内角和与外角和的定理。核心数学概念包括多边形内角和公式（(n-2)×180°）、外角和恒为360°，以及变换几何中的保角性质。作图要点：使用GeoGebra的Segment、Midpoint、Ray、Circle、Intersect等命令构造基本图形，再通过Vector、Translate、Mirror、Dilate、Rotate等变换命令将角或边进行移动、旋转，从而将分散的角集中到一点或一条直线上，便于学生观察角度之和。适合初中几何教学或高中复习，用于帮助学生从变换角度理解多边形角度定理，培养几何直观与推理能力。

GeoGebra 命令及参数

  1. Segment(D, E) → i_3
  2. Midpoint(i_3) → M_3
  3. Ray(A, B) → a
  4. Ray(B, C) → b
  5. Ray(C, D) → c
  6. Ray(D, E) → d
  7. Ray(E, A) → e
  8. Circle(A, 1.5) → f
  9. Circle(B, 1.5) → g
  10. Circle(C, 1.5) → h
  11. Circle(D, 1.5) → k
  12. Circle(E, 1.5) → p
  13. Segment(C, A) → i
  14. Segment(C, E) → j
  15. Intersect(f, a, 1) → F
  16. Intersect(f, i, 1) → G
  17. Intersect(f, e, 1) → H
  18. Intersect(f, e, 2) → I
  19. Intersect(p, e) → J,
  20. Intersect(p, j) → K,
  21. Intersect(p, d, 1) → L
  22. Intersect(p, d, 2) → M
  23. Intersect(k, d, 1) → N
  24. Intersect(k, c, 2) → O
  25. Intersect(h, c, 1) → P
  26. Intersect(h, j, 1) → Q
  27. Intersect(h, i, 1) → R
  28. Intersect(h, b, 1) → S
  29. Intersect(g, a, 2) → T
  30. Intersect(g, b, 1) → U
  31. Intersect(g, a, 1) → V
  32. CircleSector(A, F, G) → q
  33. CircleSector(A, G, H) → r
  34. CircleSector(B, U, V) → s
  35. CircleSector(C, P, Q) → t
  36. CircleSector(C, Q, R) → c_1
  37. CircleSector(C, R, S) → d_1
  38. Intersect(k, c, 1) → W
  39. CircleSector(D, O, N) → e_1
  40. CircleSector(D, N, W) → f_1
  41. CircleSector(E, M, J) → g_1
  42. CircleSector(E, J, K) → h_1
  43. CircleSector(E, K, L) → k_1
  44. CircleSector(A, I, F) → p_1
  45. CircleSector(B, T, U) → q_1
  46. Intersect(h, b, 2) → Z
  47. CircleSector(C, Z, P) → r_1
  48. Vector(A, B) → u
  49. Vector(C, B) → v
  50. Vector(E, D) → w
  51. Vector(C, D) → u_1
  52. Vector(C, E) → v_1
  53. Vector(A, E) → w_1
  54. Translate(d_1, v) → s_1
  55. Mirror(s_1, B) → t_1
  56. Dilate(A, If[0 < l ≤ 1, 1 - l, 0], B) → A'
  57. Dilate(C, If[0 < l ≤ 1, 1 - l, 0], B) → C'
  58. Dilate(A, If[1 < l ≤ 2, 2 - l, 0], E) → A'_1
  59. Dilate(C, If[1 < l ≤ 2, 2 - l, 0], E) → C'_1
  60. Dilate(C, If[2 < l ≤ 3, 3 - l, 0], D) → C'_2
  61. Vector(A, B) → a_1
  62. Vector(C, C') → b_1
  63. Vector(A, A'_1) → i_1
  64. Rotate(E, If[2 < l ≤ 3, (((-(l - 2)) * 180))°, -180°], M_3) → E'
  65. Vector(E, E') → j_1
  66. Vector(C, C'_2) → l_1
  67. Vector(C, C'_1) → m_1
  68. Translate(d_1, b_1) → c_2
  69. Translate(r, i_1) → r'_1
  70. Translate(r, w_1) → r'_2
  71. Translate(k_1, w) → d_2
  72. Translate(k_1, j_1) → e_2
  73. Translate(c_1, v_1) → f_2
  74. Translate(c_1, m_1) → g_2
  75. Translate(t, u_1) → t'
  76. Translate(t, l_1) → t'_1
  77. Translate(q, a_1) → r'
  78. Translate(d_1, b_1) → h_2
  79. Vector(A, A') → n_1
  80. Line(E, C) → a_2
  81. Line(E, D) → b_2
  82. Segment(E, C) → i_2
  83. Vector(D, C) → u_2
  84. Vector(E, C) → v_2
  85. Vector(A, C) → w_2
  86. Vector(B, C) → l_2
  87. Dilate(B, If[0 < o < 1, 1 - o, 0], C) → B'
  88. Dilate(A, If[1 < o < 2, 2 - o, 0], C) → A'_2
  89. Dilate(E, If[2 < o < 3, 3 - o, 0], C) → E'_1
  90. Dilate(D, If[3 < o < 4, 4 - o, 0], C) → D'
  91. Vector(B, B') → m_2
  92. Vector(A, A'_2) → n_2
  93. Vector(E, E'_1) → s_2
  94. Vector(D, D') → t_2
  95. Translate(q_1, m_2) → c_3
  96. Translate(q_1, l_2) → d_3
  97. Translate(p_1, w_2) → e_3
  98. Translate(g_1, v_2) → f_3
  99. Translate(e_1, u_2) → g_3
  100. If(1 < o < 2, Translate[p_1, n_2]) → h_3
  101. If(2 < o < 3, Translate[g_1, s_2]) → k_3
  102. If(3 < o < 4, Translate[e_1, t_2]) → p_3
  103. Point(a_2) → A_1
  104. Point(a_2) → B_1
  105. Ray(C, B_1) → a_3
  106. Ray(E, A_1) → b_3
  107. Polygon(A, B, C, D, E) → poly1, a_4, b_4, c_4, d_4, e_4
  108. Polygon(A, B, C) → poly2, c_5, a_5, b_5
  109. Polygon(A, C, E) → poly3, e_5, a_6, c_6
  110. Polygon(E, C, D) → poly4, d_5, e_6, c_7
  111. Segment(E, A) → j_3
  112. Segment(A, B) → l_3
  113. Segment(B, C) → m_3
  114. Segment(C, D) → n_3
  115. Midpoint(j_3) → M_4
  116. Midpoint(l_3) → M_5
  117. Midpoint(m_3) → M_1
  118. Midpoint(n_3) → M_2
  119. Rotate(d_1, If[0 < l ≤ 1, (((-l) * 180))°, -180°], M_1) → k_2
  120. Rotate(C, If[0 < l ≤ 1, (((-l) * 180))°, -180°], M_1) → C'_3
  121. Segment(M_1, C'_3) → q_3
  122. Rotate(q, If[0 < l ≤ 1, ((l * 180))°, 180°], M_5) → q'
  123. Rotate(A, If[0 < l ≤ 1, ((l * 180))°, 180°], M_5) → A'_3
  124. Rotate(R, 180°, M_1) → R'
  125. Line(B, R') → s_3
  126. Segment(M_5, A'_3) → r_3
  127. Rotate(r, If[1 < l ≤ 2, (((-(l - 1)) * 180))°, -180°], M_4) → r'_3
  128. Rotate(A, If[1 < l ≤ 2, (((-(l - 1)) * 180))°, -180°], M_4) → A'_4
  129. Segment(M_4, A'_4) → p_2
  130. Rotate(t, If[2 < l ≤ 3, (((l - 2) * 180))°, 180°], M_2) → t'_2
  131. Rotate(C, If[2 < l ≤ 3, (((l - 2) * 180))°, 180°], M_2) → C'_4
  132. Segment(M_2, C'_4) → q_2
  133. Vector(E, D) → j_2
  134. Translate(L, j_2) → L'
  135. Ray(D, L') → r_2
  136. CircleArc(C, R, S) → t_3
  137. CircleArc(C, Q, R) → f_4
  138. CircleArc(C, P, Q) → g_4
  139. CircleArc(A, F, G) → h_4
  140. CircleArc(A, G, H) → k_4
  141. CircleArc(E, K, L) → p_4
  142. Segment(M_3, E') → i_4

AI 绘图提示语参考

绘制一个五边形ABCDE，并使用变换几何工具展示其内角和外角的性质。首先画出五边形的边，然后以每个顶点为中心画圆，接着通过交点找到内角和外角的位置，最后利用平移、旋转和反射等变换将所有内角或外角集中到一个顶点或一条线上，直观展示内角和为540度，外角和为360度。
以上提示语为根据原始绘图命令整理而来，仅供参考。