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Animation 42: Composite Geometric Transformations with Equilateral Triangles, Hy

Animation 42: Composite Geometric Transformations with Equilateral Triangles, Hy
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2016-06-18 07:56
TriangleIntegralTransformationGeometric ConstructionSlider Animation
This GeoGebra example demonstrates a dynamic animation combining polygons, circles, circular sectors, vector translations, and rotations, featuring equilateral triangles, scalene triangles, a hyperbola, and integral-related elements. The core mathematical concepts include properties of equilateral triangles, graphs of quadratic functions and hyperbolas, and the application of integrals in area calculation. The construction uses GeoGebra commands such as Polygon, Dilate, Segment, Circle, Intersect, CircleSector, Rotate, Vector, Translate, CircleArc, and Midpoint, creating multiple points, segments, arcs, and transformed objects (e.g., A', B', C') with dynamic changes via sliders or animation. Suitable for high school and above students to explore the integration of geometry, conic sections, and calculus, and useful for teachers creating dynamic lesson demonstrations.

GeoGebra Commands & Parameters

  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. Segment(B, B') → f
  6. Dilate(C, If[2 < d ≤ 3, 3 - d, 0], A) → C'
  7. Segment(C, C') → g
  8. Segment(A, B) → h
  9. Segment(A, C) → i
  10. Segment(B, C) → j
  11. Circle(A, (0.15 * h)) → k
  12. Intersect(k, i, 1) → D
  13. Intersect(k, h, 1) → E
  14. CircleSector(A, E, D) → p
  15. Rotate(A, If[3.5 < d ≤ 4.5, ((d - 3.5) * π / 3), 1.0471975511965976], A) → A'_1
  16. Rotate(A', If[3.5 < d ≤ 4.5, ((d - 3.5) * π / 3), 1.0471975511965976], A) → A''
  17. Segment(A'_1, A'') → e'
  18. Rotate(A, If[4.5 < d ≤ 5.5, ((-(d - 4.5)) * π / 3), -1.0471975511965976], B) → A'_2
  19. Rotate(A', If[4.5 < d ≤ 5.5, ((-(d - 4.5)) * π / 3), -1.0471975511965976], B) → A''_1
  20. Segment(A'_2, A''_1) → e'_1
  21. Dilate(A, If[6 < d ≤ 7, 7 - d, 0], B) → A'_3
  22. Vector(A, A'_3) → u
  23. Translate(p, u) → p'
  24. CircleArc(A, E, D) → q
  25. Rotate(p', If[7 < d ≤ 8, (((d - 7) * 2) * π / 3), 2.0943951023931953], B) → p''
  26. Circle(B, (0.15 * h)) → r
  27. Intersect(f, e'_1) → G
  28. Intersect(f, e'_1) → H
  29. Intersect(f, e'_1) → I
  30. Intersect(f, e'_1) → J
  31. Intersect(r, j, 1) → F
  32. Intersect(r, h, 1) → K
  33. CircleArc(B, F, K) → s
  34. Dilate(B, If[8 < d ≤ 9, 9 - d, 0], C) → B'_1
  35. Vector(B, B'_1) → v
  36. Translate(p'', v) → p'''
  37. Rotate(p''', If[9 < d ≤ 10, (((d - 9) * 2) * π / 3), 2.0943951023931953], C) → t
  38. Circle(C, (0.15 * h)) → c_1
  39. Intersect(g, e') → L
  40. Intersect(c_1, i) → M,
  41. Intersect(c_1, j) → N,
  42. Midpoint(h) → O
  43. Dilate(C, If[10 < d ≤ 11, 11 - d, 0], O) → C'_1
  44. Segment(C, C'_1) → l
  45. Circle(O, (0.15 * h)) → d_1
  46. Segment(C, O) → m
  47. Intersect(d_1, m, 1) → P
  48. Intersect(d_1, h) → Q, R
  49. Polygon(A, O, C) → poly2, c_2, a_1, o
  50. Intersect(c_1, m) → S,
  51. CircleSector(C, S, N) → e_1
  52. CircleSector(O, R, P) → f_1
  53. Segment(O, B) → n
  54. Rotate(f_1, If[11.25 < d ≤ 12.25, ((d - 11.25) * π / 2), 1.5708], O) → g_1
  55. Rotate(e_1, If[12.25 < d ≤ 13.25, ((-(d - 12.25)) * π / 6), -0.5236], C) → h_1
  56. CircleArc(C, S, N) → k_1
  57. CircleArc(O, R, P) → p_1
  58. Rotate(O, If[13.25 < d ≤ 14.25, ((-(d - 13.25)) * π), -3.141592653589793], O) → O'
  59. Rotate(B, If[13.25 < d ≤ 14.25, ((d - 13.25) * π), 3.14159], O) → B'_2
  60. Segment(O', B'_2) → n'
  61. Segment(O, B) → b_1
  62. Dilate(C, If[14.25 < d ≤ 15.25, 15.25 - d, 0], A) → C'_2
  63. Vector(C, C'_2) → w
  64. Translate(h_1, w) → q_1
  65. Rotate(q_1, If[15.25 < d ≤ 16.25, (((-(d - 15.25)) * 4) * π / 3), -4.1887902047863905], A) → r_1
  66. Dilate(C, If[16.25 < d ≤ 17.25, 17.25 - d, 0], A) → C'_3
  67. Vector(C, C'_3) → u_1
  68. Translate(h_1, u_1) → s_1
  69. Rotate(s_1, If[17.25 < d ≤ 18.25, (((-(d - 17.25)) * 7) * π / 6), -3.665191429188092], A) → t_1
  70. Midpoint(g) → T
  71. Rotate(O', If[18.25 < d ≤ 19.25, ((d - 18.25) * π / 3), 1.0471975511965976], A) → O''
  72. Rotate(B'_2, If[18.25 < d ≤ 19.25, ((d - 18.25) * π / 3), 1.0471975511965976], A) → U
  73. Segment(O'', U) → n''
  74. Dilate(A, If[19.25 < d ≤ 20.25, 20.25 - d, 0], T) → A'_4
  75. Vector(A, A'_4) → v_1
  76. Translate(O', v_1) → O''_1
  77. Translate(B'_2, v_1) → V
  78. Segment(O''_1, V) → n''_1
  79. Rotate(O''_1, If[20.25 < d ≤ 21.25, ((d - 20.25) * π / 3), 1.0471975511965976], T) → W
  80. Rotate(V, If[20.25 < d ≤ 21.25, ((d - 20.25) * π / 3), 1.0471975511965976], T) → V'
  81. Segment(W, V') → i_1