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G_8.02 Special right triangles

G_8.02 Special right triangles
几何画板及绘图内容加载中，请稍等……
2022-07-20 21:29
测试改名0圆几何作图
本范例演示了特殊直角三角形的几何构造与性质。涉及的核心数学概念包括直角三角形、勾股定理、三角函数以及几何作图方法。作图要点包括使用GeoGebra命令如Polygon、Dilate、Circle、Intersect等来构造三角形、圆和相关点。适合初中或高中数学学习，用于理解直角三角形的基本性质和作图技巧。

GeoGebra 命令及参数

  1. Polygon(A, B, 3) → poly1, a, b, c, C
  2. Dilate(A, If[0 < d ≤ 1, 1 - d, 0], B) → A'
  3. Segment(A, A') → e
  4. Dilate(B, If[1 < d ≤ 2, 2 - d, 0], C) → B'
  5. Dilate(C, If[2 < d ≤ 3, 3 - d, 0], A) → C'
  6. Segment(A, B) → h
  7. Segment(A, C) → i
  8. Segment(B, C) → j
  9. Circle(A, (0.15 * h)) → k
  10. Intersect(k, i, 1) → D
  11. Intersect(k, h, 1) → E
  12. CircleSector(A, E, D) → p
  13. Rotate(A, If[3.5 < d ≤ 4.5, ((d - 3.5) * pi / 3), 1.0471975511965976], A) → A'_1
  14. Rotate(A', If[3.5 < d ≤ 4.5, ((d - 3.5) * pi / 3), 1.0471975511965976], A) → A''
  15. Segment(A'_1, A'') → e'
  16. Rotate(A, If[4.5 < d ≤ 5.5, ((-(d - 4.5)) * pi / 3), -1.0471975511965976], B) → A'_2
  17. Rotate(A', If[4.5 < d ≤ 5.5, ((-(d - 4.5)) * pi / 3), -1.0471975511965976], B) → A''_1
  18. Segment(A'_2, A''_1) → e'_1
  19. Dilate(A, If[6 < d ≤ 7, 7 - d, 0], B) → A'_3
  20. Vector(A, A'_3) → u
  21. Translate(p, u) → p'
  22. Rotate(p', If[7 < d ≤ 8, (((d - 7) * 2) * pi / 3), 2.0943951023931953], B) → p''
  23. Circle(B, (0.15 * h)) → r
  24. Intersect(r, j, 1) → F
  25. Intersect(r, h, 1) → K
  26. Dilate(B, If[8 < d ≤ 9, 9 - d, 0], C) → B'_1
  27. Vector(B, B'_1) → v
  28. Translate(p'', v) → p'''
  29. Rotate(p''', If[9 < d ≤ 10, (((d - 9) * 2) * pi / 3), 2.0943951023931953], C) → t
  30. Circle(C, (0.15 * h)) → c_1
  31. Intersect(c_1, i) → M,
  32. Intersect(c_1, j) → N,
  33. Midpoint(h) → O
  34. Dilate(C, If[10 < d ≤ 11, 11 - d, 0], O) → C'_1
  35. Segment(C, C'_1) → l
  36. Circle(O, (0.15 * h)) → d_1
  37. Intersect(d_1, h) → V_{1}, V_{2}
  38. Segment(C, O) → m
  39. Intersect(d_1, m, 1) → P
  40. Polygon(A, O, C) → poly2, c_2, a_1, o
  41. Intersect(c_1, m) → S,
  42. CircleSector(C, S, N) → e_1
  43. CircleSector(O, V_{2}, P) → f_1
  44. Segment(O, B) → n
  45. Rotate(e_1, If[13 < d ≤ 14, ((-(d - 13)) * pi / 6), -0.5236], C) → h_1
  46. Rotate(O, If[13.25 < d ≤ 14.25, ((-(d - 13.25)) * pi), -3.141592653589793], O) → O'
  47. Rotate(B, If[14 < d ≤ 15, ((d - 14) * pi), 3.14159], O) → B'_2
  48. Segment(O', B'_2) → n'
  49. Segment(O, B) → b_1
  50. Dilate(C, If[15 < d ≤ 16, 16 - d, 0], A) → C'_2
  51. Vector(C, C'_2) → w
  52. Translate(h_1, w) → q_1
  53. Dilate(C, If[17 < d ≤ 18, 18 - d, 0], A) → C'_3
  54. Vector(C, C'_3) → u_1
  55. Translate(h_1, u_1) → s_1
  56. Rotate(O', If[19 < d ≤ 20, ((d - 19) * pi / 3), 1.0472], A) → O''
  57. Rotate(B'_2, If[18.25 < d ≤ 19.25, ((d - 18.25) * pi / 3), 1.0471975511965976], A) → U
  58. Segment(O'', U) → n''
  59. Rotate(O'', If[19.25 < d ≤ 20.25, ((d - 19.25) * pi), pi], O'') → O'''
  60. Rotate(U, If[20 < d ≤ 21, ((d - 20) * pi), pi], O'') → U'
  61. Segment(O''', U') → n'''
  62. Rotate(f_1, If[12 < d ≤ 13, ((d - 12) * pi / 2), 1.5708], O) → g_1
  63. Rotate(q_1, If[16 < d ≤ 17, (((-(d - 16)) * 4) * pi / 3), -4.18879], A) → r_1
  64. Rotate(s_1, If[18 < d ≤ 19, (((-(d - 18)) * 7) * pi / 6), -3.66519], A) → t_1
  65. Rotate(E, pi / 6, A) → E'
  66. Ray(A, E') → f
  67. Circle(A, (0.2 * h)) → q
  68. Intersect(q, f, 1) → G
  69. Segment(A, G) → g
  70. Sequence(Rotate[(1, 0), i°, (0, 0)], i, 15, 360, 15) → list1
  71. Segment(H, I) → s
  72. Circle(H, I) → k_1
  73. OrthogonalLine(H, s) → i_1
  74. Intersect(k_1, i_1, 2) → J
  75. Intersect(k_1, i_1, 1) → L
  76. Dilate(H, If[0 < j_1 ≤ 1, 1 - j_1, 0], I) → H'
  77. Segment(H, H') → l_1
  78. Rotate(H, If[1 < j_1 ≤ 2, ((j_1 - 1) * pi / 2), pi / 2], H) → H'_{1}
  79. Rotate(H', If[1 < j_1 ≤ 2, ((j_1 - 1) * pi / 2), pi / 2], H) → H''
  80. Segment(H'_{1}, H'') → m_1
  81. Angle(I, H, H'') → α
  82. Dilate(J, If[2 < j_1 ≤ 3, 3 - j_1, 0], I) → J'
  83. Segment(J, J') → n_1
  84. Polygon(H, I, J) → t1, j_2, h_2, i_2
  85. Segment(H, I) → p_1
  86. Segment(I, J) → f_2
  87. Segment(J, H) → g_2
  88. Angle(I, H, H'') → β
  89. Rotate(f_1, If[12 < d ≤ 13, ((d - 12) * pi / 2), 1.5708], O) → d_2
  90. CircleSector(O, V_{2}, P) → e_2
  91. Rotate(e_1, If[13 < d ≤ 14, ((-(d - 13)) * pi / 6), -0.5236], C) → k_2
  92. CircleSector(A, E, D) → p_2
  93. Translate(p, u) → p'_{1}
  94. Rotate(p', If[7 < d ≤ 8, (((d - 7) * 2) * pi / 3), 2.0943951023931953], B) → p''_{1}
  95. Translate(p'', v) → p'''_{1}
  96. Rotate(p''', If[9 < d ≤ 10, (((d - 9) * 2) * pi / 3), 2.0943951023931953], C) → q_2
  97. CircleSector(C, S, N) → r_2
  98. Translate(h_1, w) → s_2
  99. Translate(h_1, w) → t_2
  100. Rotate(q_1, If[16 < d ≤ 17, (((-(d - 16)) * 4) * pi / 3), -4.18879], A) → c_3
  101. Translate(h_1, u_1) → d_3
  102. Rotate(s_1, If[18 < d ≤ 19, (((-(d - 18)) * 7) * pi / 6), -3.66519], A) → e_3
  103. Intersect(d_1, m, 1) → T
  104. Rotate(C, If[12 < d ≤ 13, ((d - 12) * pi / 2), 1.5708], O) → T'
  105. Rotate(B, If[12 < d ≤ 13, ((d - 12) * pi / 2), 1.5708], O) → Q
  106. Angle(T', O, Q) → γ
  107. Angle(C, O, B) → δ
  108. Angle(T', O, Q) → ε
  109. Angle(C, O, B) → ζ
  110. Segment(A, C) → l_2
  111. Segment(A, O) → m_2

AI 绘图提示语参考

在同一平面构造一个以两条直角边为基础的“特殊直角三角形”示意图：先给定三角形的三个顶点，画出三角形边与以其中顶点为圆心、由另一边长度按比例确定半径的圆。圆与两条边的交点用于生成多个扇形区域；再把围绕顶点的旋转与移动得到的对应点连线，形成围绕三角形的对称/重复结构。最后用圆弧扇形标出从相关交点夹角对应的区域，并通过中点与以中点为圆心的辅助圆确定额外交点，再连接若干关键线段以体现几何关系。
以上提示语为根据原始绘图命令整理而来，仅供参考。