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四边形分类：按四边、直角与平行边判别

四边形分类：按四边、直角与平行边判别
几何画板及绘图内容加载中，请稍等……
2012-11-02 13:13
多边形角向量数列几何作图
① 本范例演示如何根据四条边的长度、是否存在直角以及是否有平行边，将四边形分类为正方形、长方形、菱形、平行四边形、梯形、风筝形等。② 涉及的核心数学概念包括四边形的定义与性质、平行线判别、直角判别、对称性以及分类学的逻辑推理。③ 作图要点：使用 Point、Segment、Polygon 创建顶点与边；用 Circle、Semicircle、Intersect 构造辅助圆与交点；利用 Sequence、List 生成多组四边形；通过 Midpoint、Angle、Rotate、Vector、If 实现动态判别与标注。④ 适合初中几何分类教学、高中几何复习以及 GeoGebra 动态演示课件制作。

GeoGebra 命令及参数

  1. Point(Segment[(1, 2), (10, 2)]) → B
  2. Segment((0, 2), (round(10 * x(B)) / 10, 2)) → b
  3. Point(Segment[(1, 1), (10, 1)]) → A
  4. Segment((0, 1), (round(10 * x(A)) / 10, 1)) → a
  5. Semicircle((x(Q) - b, y(Q)), (x(Q) + b, y(Q))) → c_q
  6. Point(c_q) → R1
  7. Point(Segment[(1, 3), (10, 3)]) → C
  8. Segment((0, 3), (round(10 * x(C)) / 10, 3)) → c
  9. Circle(R, c) → c_r
  10. Point(Segment[(1, 4), (10, 4)]) → D
  11. Segment((0, 4), (round(10 * x(D)) / 10, 4)) → d
  12. Circle(P, d) → c_p
  13. Intersect(c_r, c_p, 2) → S1
  14. Sequence(Segment[(k, 0), (k, -1 * 0.5)], k, 0, 10) → list1
  15. Polygon((-0.3, 0.02662), (-0.3, -1.27338), (10.3, -1.27338), (10.3, 0.02662)) → poly1, poly1_1, poly1_2, poly1_3, poly1_4
  16. Sequence(Segment[(k + 0.5, 0), (k + 0.5, -1 * 0.375)], k, 0, 9) → list2
  17. Sequence(Sequence[Segment[(0.5 * (i - 1) + j / 10, 0), (0.5 * (i - 1) + j / 10, -1 * 0.25)], j, 1, 4], i, 1, 20) → list3
  18. Midpoint((0, 1), A) → A'
  19. Midpoint((0, 2), B) → B'
  20. Midpoint((0, 3), C) → C'
  21. Midpoint((0, 4), D) → D'
  22. Segment((0, 4), (0, 0)) → i
  23. Segment(P, Q) → j
  24. Segment(P, R) → k
  25. Intersect(c_p, k, 1) → E
  26. Intersect(c_r, k, 1) → F
  27. Segment((round(10 * x(C)) / 10, 3), (round(10 * x(C)) / 10, 0)) → g
  28. Segment((round(10 * x(B)) / 10, 2), (round(10 * x(B)) / 10, 0)) → f
  29. Segment((round(10 * x(A)) / 10, 1), (round(10 * x(A)) / 10, 0)) → e
  30. Segment((round(10 * x(D)) / 10, 4), (round(10 * x(D)) / 10, 0)) → h
  31. Angle(Rotate[P, (-β1)°, Q], Q, P) → β
  32. Angle(Rotate[Q, (-γ1)°, R], R, Q) → γ
  33. Angle(Q, P, Rotate[Q, α1°, P]) → α
  34. Polygon(P, Q, R1, S1) → poly2, t, a_1, b_1, c_1
  35. Angle(Q, P, Rotate[Q, 90*°, P]) → α_0
  36. Angle(R, Q, Rotate[R, 90*°, Q]) → β_0
  37. Angle(S, R, Rotate[S, 90*°, R]) → γ_0
  38. Angle(P, S, Rotate[P, 90*°, S]) → δ_0
  39. Segment(Midpoint[P, Q], Rotate[Intersect[Circle[Midpoint[P, Q], 0.2], Ray[Midpoint[P, Q], P]], 30*°, Midpoint[P, Q]]) → u_2
  40. Segment(Midpoint[P, Q], Rotate[Intersect[Circle[Midpoint[P, Q], 0.2], Ray[Midpoint[P, Q], P]], 330*°, Midpoint[P, Q]]) → u_1
  41. Segment(Midpoint[S, R], Rotate[Intersect[Circle[Midpoint[S, R], 0.2], Ray[Midpoint[S, R], S]], 330*°, Midpoint[S, R]]) → v_1
  42. Segment(Midpoint[S, R], Rotate[Intersect[Circle[Midpoint[S, R], 0.2], Ray[Midpoint[S, R], S]], 30*°, Midpoint[S, R]]) → v_2
  43. Segment(Midpoint[Q, R], Rotate[Intersect[Circle[Midpoint[Q, R], 0.2], Ray[Midpoint[Q, R], Q]], 330*°, Midpoint[Q, R]]) → w_1
  44. Segment(Midpoint[Q, R], Rotate[Intersect[Circle[Midpoint[Q, R], 0.2], Ray[Midpoint[Q, R], Q]], 30*°, Midpoint[Q, R]]) → w_2
  45. Segment(Translate[Midpoint[Q, R], Vector[Q, R] / b * 0.2], Translate[Rotate[Intersect[Circle[Midpoint[Q, R], 0.2], Ray[Midpoint[Q, R], Q]], 330*°, Midpoint[Q, R]], Vector[Q, R] / b * 0.2]) → w_3
  46. Segment(Translate[Midpoint[Q, R], Vector[Q, R] / b * 0.2], Translate[Rotate[Intersect[Circle[Midpoint[Q, R], 0.2], Ray[Midpoint[Q, R], Q]], 30*°, Midpoint[Q, R]], Vector[Q, R] / b * 0.2]) → w_4
  47. Segment(Midpoint[P, S], Rotate[Intersect[Circle[Midpoint[P, S], 0.2], Ray[Midpoint[P, S], P]], 330*°, Midpoint[P, S]]) → z_1
  48. Segment(Midpoint[P, S], Rotate[Intersect[Circle[Midpoint[P, S], 0.2], Ray[Midpoint[P, S], P]], 30*°, Midpoint[P, S]]) → z_2
  49. Angle(P, S, Rotate[P, δ1°, S]) → δ
  50. Midpoint(Midpoint[P, R], Midpoint[Q, S]) → G
  51. Circle(H, 1) → r
  52. Point(r) → I
  53. Rotate(P, ε1°, G) → P'
  54. Rotate(Q, ε1°, G) → Q'
  55. Rotate(R, ε1°, G) → R'
  56. Rotate(S, ε1°, G) → S'
  57. Polygon(P', Q', R', S') → poly2', p', q', r'_1, s'_1
  58. Angle(Q', P', Rotate[Q', α1°, P']) → α'
  59. Angle(Rotate[P', (-β1)°, Q'], Q', P') → β'
  60. Rotate(S_0, ε1°, G) → S'_1
  61. Angle(Rotate[Q', (-γ1)°, R'], R', Q') → γ'
  62. Angle(P', S', Rotate[P', δ1°, S']) → δ'
  63. Angle(Q', P', Rotate[Q', 90*°, P']) → α'_0
  64. Angle(R', Q', Rotate[R', 90*°, Q']) → β'_0
  65. Angle(S', R', Rotate[S', 90*°, R']) → γ'_0
  66. Angle(P', S', Rotate[P', 90*°, S']) → δ'_0
  67. Segment(Midpoint[P', Q'], Rotate[Intersect[Circle[Midpoint[P', Q'], 0.2], Ray[Midpoint[P', Q'], P']], 330*°, Midpoint[P', Q']]) → u'_1
  68. Segment(Midpoint[P', Q'], Rotate[Intersect[Circle[Midpoint[P', Q'], 0.2], Ray[Midpoint[P', Q'], P']], 30*°, Midpoint[P', Q']]) → u'_2
  69. Segment(Midpoint[S', R'], Rotate[Intersect[Circle[Midpoint[S', R'], 0.2], Ray[Midpoint[S', R'], S']], 330*°, Midpoint[S', R']]) → v'_1
  70. Segment(Midpoint[S', R'], Rotate[Intersect[Circle[Midpoint[S', R'], 0.2], Ray[Midpoint[S', R'], S']], 30*°, Midpoint[S', R']]) → v'_2
  71. Segment(Midpoint[Q', R'], Rotate[Intersect[Circle[Midpoint[Q', R'], 0.2], Ray[Midpoint[Q', R'], Q']], 330*°, Midpoint[Q', R']]) → w'_1
  72. Segment(Midpoint[Q', R'], Rotate[Intersect[Circle[Midpoint[Q', R'], 0.2], Ray[Midpoint[Q', R'], Q']], 30*°, Midpoint[Q', R']]) → w'_2
  73. Segment(Translate[Midpoint[Q', R'], Vector[Q', R'] / b * 0.2], Translate[Rotate[Intersect[Circle[Midpoint[Q', R'], 0.2], Ray[Midpoint[Q', R'], Q']], 330*°, Midpoint[Q', R']], Vector[Q', R'] / b * 0.2]) → w'_3
  74. Segment(Translate[Midpoint[Q', R'], Vector[Q', R'] / b * 0.2], Translate[Rotate[Intersect[Circle[Midpoint[Q', R'], 0.2], Ray[Midpoint[Q', R'], Q']], 30*°, Midpoint[Q', R']], Vector[Q', R'] / b * 0.2]) → w'_4
  75. Segment(Midpoint[P', S'], Rotate[Intersect[Circle[Midpoint[P', S'], 0.2], Ray[Midpoint[P', S'], P']], 330*°, Midpoint[P', S']]) → z'_1
  76. Segment(Midpoint[P', S'], Rotate[Intersect[Circle[Midpoint[P', S'], 0.2], Ray[Midpoint[P', S'], P']], 30*°, Midpoint[P', S']]) → z'_2
  77. Vector(H, I) → u
  78. Angle((x(H) + 1, y(H)), H, I) → ε
  79. Segment((x(H) + 1, y(H)), H) → s
  80. Midpoint(P, Q) → J
  81. Midpoint(Q, R) → K
  82. Midpoint(R, S) → L
  83. Midpoint(S, P) → M
  84. Rotate(J, ε1°, G) → J'
  85. Rotate(K, ε1°, G) → K'
  86. Rotate(L, ε1°, G) → L'
  87. Rotate(M, ε1°, G) → M'
  88. Segment(Translate[Midpoint[P, S], Vector[P, S] / d * 0.2], Translate[Rotate[Intersect[Circle[Midpoint[P, S], 0.2], Ray[Midpoint[P, S], P]], 330*°, Midpoint[P, S]], Vector[P, S] / d * 0.2]) → z_3
  89. Segment(Translate[Midpoint[P, S], Vector[P, S] / d * 0.2], Translate[Rotate[Intersect[Circle[Midpoint[P, S], 0.2], Ray[Midpoint[P, S], P]], 30*°, Midpoint[P, S]], Vector[P, S] / d * 0.2]) → z_4
  90. Segment(Translate[Midpoint[P', S'], Vector[P', S'] / d * 0.2], Translate[Rotate[Intersect[Circle[Midpoint[P', S'], 0.2], Ray[Midpoint[P', S'], P']], 330*°, Midpoint[P', S']], Vector[P', S'] / d * 0.2]) → z'_3
  91. Segment(Translate[Midpoint[P', S'], Vector[P', S'] / d * 0.2], Translate[Rotate[Intersect[Circle[Midpoint[P', S'], 0.2], Ray[Midpoint[P', S'], P']], 30*°, Midpoint[P', S']], Vector[P', S'] / d * 0.2]) → z'_4
  92. Midpoint(P, E) → N
  93. Midpoint(F, R) → O
  94. If(d ≟ c ∧ (c ≠ b) ∧ (c ≠ a), 1, If[d ≟ b ∧ (b ≠ a), 0, If[d ≟ a, 1, 0]]) → d1
  95. If(d ≟ c ∧ (c ≠ b) ∧ (c ≠ a), 0, If[d ≟ b ∧ (b ≠ a), 0, If[d ≟ a, 0, 0.8]]) → d2
  96. If(d ≟ c ∧ (c ≠ b) ∧ (c ≠ a), 1, If[d ≟ b ∧ (b ≠ a), 1, If[d ≟ a, 0, 0.8]]) → d3
  97. Segment(R1, S1) → n
  98. Segment(P, S1) → m
  99. Segment(Q, R1) → l
  100. Point(c_r) → T
  101. Point(c_p) → U
  102. Segment(R, T) → d_1
  103. Segment(P, U) → e_1
  104. Midpoint(R, T) → V
  105. Midpoint(P, U) → W
  106. Segment(F, R) → q
  107. Segment(P, E) → p

AI 绘图提示语参考

绘制一个用于四边形分类的教学示意图。底部有一条水平刻度线，上方有四条平行且长度可调的水平线段，分别代表四边形的四条边。在刻度线上方，通过辅助圆和交点构造一个动态四边形，其顶点可沿水平线移动。四边形内部显示边长、角度和垂直标记，并标注平行、垂直或等长关系。同时，在底部刻度线上生成多个小刻度标记，用于测量和比较边长。整体图形强调通过边长、角度和平行性来区分正方形、矩形、菱形、平行四边形、梯形和风筝形等四边形类型。
以上提示语为根据原始绘图命令整理而来，仅供参考。