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黄金三角形

黄金三角形
几何画板及绘图内容加载中，请稍等……
2018-02-04 08:38
测试改名0圆多边形角不等式
所属专题 Polyeder, whose coordinates are golden numbers
①本范例演示黄金三角形的构造与性质；②涉及核心数学概念包括放物线、不等式、面积、定积分、指数函数；③作图要点：使用Segment、Circle、Intersect、Polygon、Rotate、Angle等GeoGebra命令构造图形；④适合高中及以上数学学习者和爱好者用于探索几何变换与函数图像。

GeoGebra 命令及参数

  1. Segment((0.5, 0), (-0.5, 0)) → a'
  2. Circle((-0.5, 0), φ) → c_{0}
  3. Circle((0.5, 0), φ) → c_{1}
  4. Intersect(c_{0}, c_{1}) → A, A', ,
  5. Segment(A, (-0.5, 0)) → c'
  6. Segment(A, (0.5, 0)) → b'
  7. Polygon(A, Intersect[a', c'], Intersect[a', b']) → T_{0}, c, a, b
  8. Segment(D, Midpoint[a]) → k_{1}
  9. Segment(Intersect[a', b'], Midpoint[c']) → l_{1}
  10. Intersect(c, a) → B
  11. Intersect(l_{1}, k_{1}) → O
  12. Intersect(a, b) → C
  13. Rotate(Dilate[T_{0}, (2 / (1 + sqrt(5)))^(R), O], ((R * (72 + 36) / 180) * π), O) → T_{v}
  14. Rotate(Dilate[a, (2 / (1 + sqrt(5)))^(R), O], ((R * (72 + 36) / 180) * π), O) → T_{a}
  15. Rotate(Dilate[b, (2 / (1 + sqrt(5)))^(R), O], ((R * (72 + 36) / 180) * π), O) → T_{b}
  16. Rotate(Dilate[c, (2 / (1 + sqrt(5)))^(R), O], ((R * (72 + 36) / 180) * π), O) → T_{c}
  17. Rotate(Dilate[T_{0}, (2 / (1 + sqrt(5)))^(1), O], ((72 + 36) / 180 * π), O) → T_{1}
  18. Rotate(Dilate[T_{0}, (2 / (1 + sqrt(5)))^(2), O], ((2 * (72 + 36) / 180) * π), O) → T_{2}
  19. Rotate(Dilate[T_{0}, (2 / (1 + sqrt(5)))^(3), O], ((3 * (72 + 36) / 180) * π), O) → T_{3}
  20. Rotate(Dilate[T_{0}, (2 / (1 + sqrt(5)))^(4), O], ((4 * (72 + 36) / 180) * π), O) → T_{4}
  21. Rotate(Dilate[T_{0}, (2 / (1 + sqrt(5)))^(5), O], ((5 * (72 + 36) / 180) * π), O) → T_{5}
  22. Rotate(Dilate[T_{0}, (2 / (1 + sqrt(5)))^(6), O], ((6 * (72 + 36) / 180) * π), O) → T_{6}
  23. Rotate(Dilate[T_{0}, (2 / (1 + sqrt(5)))^(7), O], ((7 * (72 + 36) / 180) * π), O) → T_{7}
  24. Rotate(Dilate[T_{0}, (2 / (1 + sqrt(5)))^(8), O], ((8 * (72 + 36) / 180) * π), O) → T_{8}
  25. Rotate(Dilate[T_{0}, (2 / (1 + sqrt(5)))^(9), O], ((9 * (72 + 36) / 180) * π), O) → T_{9}
  26. Rotate(Dilate[T_{0}, (2 / (1 + sqrt(5)))^(10), O], ((10 * (72 + 36) / 180) * π), O) → T_{10}
  27. Rotate(Dilate[T_{0}, (2 / (1 + sqrt(5)))^(-1), O], (((-(72 + 36))) / 180 * π), O) → T_{-1}
  28. Rotate(Dilate[T_{0}, (2 / (1 + sqrt(5)))^(-2), O], ((-2 * (72 + 36) / 180) * π), O) → T_{-2}
  29. Rotate(Dilate[T_{0}, (2 / (1 + sqrt(5)))^(-3), O], ((-3 * (72 + 36) / 180) * π), O) → T_{-3}
  30. Angle((-0.5, 0), A, (0.5, 0)) → α
  31. Angle(Intersect[a', b'], Intersect[a', c'], A) → β
  32. Segment(B, D) → m
  33. Angle(a, m) → α'
  34. Angle(A, Intersect[a', b'], Intersect[a', c']) → β'
  35. Rotate(Dilate[T_{0}, (2 / (1 + sqrt(5)))^(-4), O], ((-4 * (72 + 36) / 180) * π), O) → T_{-4}
  36. Rotate(Dilate[T_{0}, (2 / (1 + sqrt(5)))^(-5), O], ((-5 * (72 + 36) / 180) * π), O) → T_{-5}
  37. Rotate(Dilate[T_{0}, (2 / (1 + sqrt(5)))^(-6), O], ((-6 * (72 + 36) / 180) * π), O) → T_{-6}
  38. Segment(A, Midpoint[c']) → l_{11}
  39. Segment(Midpoint[c'], B) → l_{12}
  40. Segment(B, Midpoint[a']) → k_{11}
  41. Segment(Midpoint[a'], C) → k_{12}