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GoGeometry Action 134: Dynamic Circumcircle and Polygon Sequence

GoGeometry Action 134: Dynamic Circumcircle and Polygon Sequence
几何画板及绘图内容加载中，请稍等……
2018-03-27 12:33
圆多边形数列变换几何作图
① This example dynamically illustrates a geometry problem posted by Antonio Gutierrez, involving a circumcircle, polygons, and a sequence of points. ② Core mathematical concepts include circumcircle, polygons, point sequences, distance ratios, sequences and series, and transformations such as rotation and dilation. ③ Construction highlights: from the summary, GeoGebra commands such as Circle, Point, Line, OrthogonalLine, Intersect, Ray, Dilate, Angle, Polygon, Segment, Distance, Rotate, and CircleArc are used to create multiple points (A, B, C, D, F, F', B_1, C_1, etc.), circles (c, c_1), lines (l, e, f, g, h), and polygons; distances like distanceB1C1 and distanceC1A1 are computed, and Dilate and Rotate enable dynamic transformations. ④ Suitable for high school and above, or geometry enthusiasts, to explore proportional relationships, sequence patterns, and dynamic geometry software applications.

GeoGebra 命令及参数

  1. Circle(O, B_2) → c
  2. Point(c) → A
  3. Point(c) → B
  4. Line(A, B) → l
  5. Point(c) → C
  6. Circle(C, 3) → e
  7. Line(O, C) → f
  8. OrthogonalLine(C, f) → g
  9. Intersect(e, g, 2) → D
  10. Ray(C, D) → h
  11. Intersect(l, h) → F
  12. Dilate(F, If[1 < a ≤ 2, 2 - a, 0], C) → F'
  13. Circle(F, C) → p
  14. Point(p) → P
  15. Angle(C, F, P) → β
  16. Circle(C, O) → d
  17. Polygon(C, B, A) → t1, g_1, c_1, f_1
  18. Segment(A, B) → i
  19. Segment(B, C) → j
  20. Segment(C, A) → k
  21. Line(A, P) → m
  22. Line(B, P) → n
  23. Line(C, P) → q
  24. Intersect(c, n, 2) → B_1
  25. Intersect(c, m, 2) → A_1
  26. Intersect(c, q) → E, C_1
  27. Distance(B_1, C_1) → distanceB1C1
  28. Distance(C_1, A_1) → distanceC1A1
  29. Dilate(B, If[0 < a ≤ 1, 1 - a, 0], F) → B'
  30. Segment(B, B') → r
  31. Dilate(O, If[1 < a ≤ 2, 2 - a, 0], C) → O'
  32. Segment(O, O') → s
  33. Segment(F, F') → t
  34. Angle(F, C, O) → α
  35. Rotate(C, If[2 < a ≤ 4, ((a - 2) * π), (2 * π)], F) → C'
  36. Rotate(F, If[2 < a ≤ 4, ((a - 2) * π), (2 * π)], F) → F'_1
  37. Rotate(C, If[2 < a ≤ 4, ((a - 2) * β / 2), β], F) → C'_1
  38. Segment(F, C'_1) → b
  39. CircleArc(F, C, C'_1) → d_1
  40. Rotate(P, If[4 < a ≤ 5, ((a - 4) * ((2 * π) - β)), (2 * π) - β], F) → P'
  41. Segment(F, P') → h_1
  42. CircleArc(F, P, P') → e_1
  43. Dilate(A, If[5 < a ≤ 6, 6 - a, 0], P) → A'
  44. Segment(A, A') → i_1
  45. Dilate(P, If[6 < a ≤ 7, 7 - a, 0], A_1) → P'_1
  46. Segment(P, P'_1) → j_1
  47. Dilate(B, If[7 < a ≤ 8, 8 - a, 0], P) → B'_1
  48. Segment(B, B'_1) → k_1
  49. Dilate(P, If[8 < a ≤ 9, 9 - a, 0], B_1) → P'_2
  50. Segment(P, P'_2) → l_1
  51. Dilate(C, If[9 < a ≤ 10, 10 - a, 0], P) → C'_2
  52. Segment(C, C'_2) → m_1
  53. Dilate(P, If[10 < a ≤ 11, 11 - a, 0], C_1) → P'_3
  54. Segment(P, P'_3) → n_1
  55. Angle(B_1, C_1, A_1) → γ
  56. Dilate(B_1, If[12 < a ≤ 13, 13 - a, 0], C_1) → G
  57. Segment(B_1, G) → p_1
  58. Dilate(A_1, If[13 < a ≤ 14, 14 - a, 0], C_1) → H
  59. Segment(A_1, H) → q_1
  60. Rotate(A_1, If[15 < a ≤ 16, ((-(a - 15)) * γ), (-γ)], C_1) → I
  61. Rotate(H, If[15 < a ≤ 16, ((-(a - 15)) * γ), (-γ)], C_1) → H'
  62. Segment(I, H') → r_1

AI 绘图提示语参考

绘制一个圆，在其上取三个点构成三角形。从圆心向三角形的一个顶点作垂线，与以该顶点为圆心、半径为3的圆相交，得到一点，并作射线与三角形对边相交。根据参数变化，对交点进行缩放或旋转，生成动态点。连接三角形各顶点与圆上一点，与圆相交得到新点，并测量这些点之间的距离。随后，根据参数范围，对多个点进行缩放或旋转，并绘制相应的线段与圆弧，展示几何序列与比例关系。
以上提示语为根据原始绘图命令整理而来，仅供参考。