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Dynamic Triangle Phenomena Illustration

Dynamic Triangle Phenomena Illustration
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2017-05-24 20:22
TriangleCircleOptimizationGeometric Construction
This example dynamically illustrates some cool phenomena within a triangle! ① It visually demonstrates geometric relationships and patterns in the triangle through dynamic manipulation. ② Core mathematical concepts include triangle properties, rational numbers, exponential functions, equilateral triangles, and optimization problems. ③ Construction highlights the use of GeoGebra commands/objects such as Rotate, Angle, Segment, Ray, Point, Intersect, Min, Circle, CircleArc, Mirror, Line, OrthogonalLine, Midpoint, Polygon, CircleSector, and Dilate. ④ Suitable for high school and above students to explore and understand complex geometric problems and function properties.

GeoGebra Commands & Parameters

  1. Rotate(A, (a * π / 180), B) → A'
  2. Angle(A, B, A') → α
  3. Segment(B, A) → f
  4. Ray(B, A') → g
  5. Point(g) → C
  6. Segment(B, C) → h
  7. Segment(A, C) → i
  8. Rotate(A, α / 2, B) → A'_1
  9. Angle(A, B, A'_1) → β
  10. Ray(B, A'_1) → j
  11. Intersect(j, i) → D
  12. Segment(B, D) → k
  13. Min({f, h, k}) → b
  14. Circle(B, (0.25 * b)) → c
  15. Intersect(c, f, 1) → E
  16. Intersect(j, k) → F
  17. Intersect(c, h, 1) → G
  18. Intersect(c, j) → H,
  19. CircleArc(B, E, H) → d
  20. Point(d) → I
  21. Mirror(I, j) → I'
  22. Line(B, I') → l
  23. Line(B, I) → m
  24. Ray(B, I') → n
  25. Ray(B, I) → p
  26. Point(p) → J
  27. Line(A, B) → q
  28. OrthogonalLine(J, q) → r
  29. Intersect(r, q) → K
  30. Point(n) → L
  31. Line(B, C) → s
  32. OrthogonalLine(L, s) → t
  33. Intersect(s, t) → M
  34. OrthogonalLine(L, q) → e
  35. Intersect(q, e) → N
  36. OrthogonalLine(J, s) → f_1
  37. Intersect(s, f_1) → O
  38. Midpoint(J, L) → P
  39. Circle(P, K) → c_1
  40. Polygon(A, C, B) → poly1, b_1, a_1, c_2
  41. CircleSector(B, E, I) → d_1
  42. Angle(I, B, G) → γ
  43. Rotate(d_1, If[0 < o ≤ 1, (o * γ), γ], B) → e_1
  44. Angle(L, M, C) → δ
  45. Angle(K, N, L) → ε
  46. Angle(A, K, J) → ζ
  47. Angle(J, O, C) → η
  48. Dilate(L, If[2 < o ≤ 3, 3 - o, 0], M) → L'
  49. Dilate(L, If[2 < o ≤ 3, 3 - o, 0], N) → L'_1
  50. Segment(L, L') → g_1
  51. Segment(L, L'_1) → h_1
  52. Dilate(J, If[3 < o ≤ 4, 4 - o, 0], O) → J'
  53. Dilate(J, If[3 < o ≤ 4, 4 - o, 0], K) → J'_1
  54. Segment(J, J') → i_1
  55. Segment(J, J'_1) → j_1
  56. Dilate(J, If[4 < o ≤ 5, 5 - o, 0], P) → J'_2
  57. Segment(J, J'_2) → k_1
  58. Dilate(P, If[5 < o ≤ 6, 6 - o, 0], L) → P'
  59. Segment(P, P') → l_1
  60. Rotate(J, If[6 < o ≤ 7, ((o - 6) * π), π], P) → J'_3
  61. Rotate(J'_2, If[6 < o ≤ 7, ((o - 6) * π), π], P) → Q
  62. Segment(J'_3, Q) → m_1
  63. Dilate(P, If[7 < o ≤ 8, 8 - o, 0], N) → P'_1
  64. Segment(P, P'_1) → n_1
  65. Circle(P, P'_1) → p_1