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Golden Triangle

Golden Triangle
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2018-02-04 08:38
TriangleCirclePolygonAngleInequality
In collections Polyeder, whose coordinates are golden numbers
This example demonstrates the construction and properties of a Golden Triangle, involving concepts like parabola, inequality, area, definite integral, and exponential functions. It uses commands like Segment, Circle, Intersect, Polygon, Rotate, and Angle in GeoGebra for geometric construction, suitable for high school and above mathematics learners exploring geometric transformations and function graphs.

GeoGebra Commands & Parameters

  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}