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Dynamic and Modifiable Geometric Problem Illustration

Dynamic and Modifiable Geometric Problem Illustration
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2018-02-04 11:16
CircleVectorGeometric Construction
This example dynamically demonstrates a geometric problem from Antonio Gutierrez, involving elements like 2D vectors, prisms, and circumcircles. Key math concepts include 2D vectors, circumcircles, definite integrals, and equations. Construction highlights the use of GeoGebra commands/objects such as Circle, Point, Line, OrthogonalLine, Intersect, Ray, Angle, Radius, Dilate, Segment, Rotate, Polygon, Midpoint, CircleSector, Vector, and Translate. Suitable for high school and above, helping learners understand vector operations, geometric transformations, and integral applications.

GeoGebra Commands & Parameters

  1. Circle(A, B) → c
  2. Point(c) → C
  3. Circle(C, A) → d
  4. Line(A, C) → f
  5. OrthogonalLine(C, f) → g
  6. Intersect(d, g, 2) → D
  7. Intersect(d, g, 1) → E
  8. Line(D, A) → h
  9. Intersect(c, h, 1) → F
  10. Intersect(c, d, 2) → G
  11. Ray(F, G) → i
  12. Angle(A, F, G) → α
  13. Radius(c) → b
  14. Dilate(A, If[0 < e ≤ 1, 1 - e, 0], C) → A'
  15. Segment(A, A') → j
  16. Rotate(A, If[1 < e ≤ 2, ((e - 1) * π / 3), π / 3], C) → A'_1
  17. Rotate(A', If[1 < e ≤ 2, ((e - 1) * π / 3), π / 3], C) → A''
  18. Segment(A'_1, A'') → j'
  19. Rotate(A'_1, If[2 < e ≤ 3, ((e - 2) * π / 3), π / 3], G) → I
  20. Rotate(A'', If[2 < e ≤ 3, ((e - 2) * π / 3), π / 3], G) → A'''
  21. Segment(I, A''') → j''
  22. Polygon(A, G, C) → t1, c_1, a_1, g_1
  23. Rotate(G, If[3 < e ≤ 4, ((e - 2) * π / 3), π / 3], G) → G'
  24. Midpoint(C, G) → J
  25. Dilate(A, If[4 < e ≤ 5, 5 - e, 0], J) → A'_2
  26. Segment(A, A'_2) → l
  27. Circle(A, (0.3 * b)) → p
  28. Line(A, C) → m
  29. Intersect(p, m) → K, L
  30. Rotate(L, If[5 < e ≤ 6, ((-(e - 5)) * π / 6), ((-π)) / 6], A) → L'
  31. CircleSector(A, L', L) → q
  32. Rotate(q, If[7 < e ≤ 8, ((-(e - 7)) * π / 6), ((-π)) / 6], A) → q'
  33. Line(H, j'') → n
  34. Line(H, l) → r
  35. Circle(H, 2) → s
  36. Intersect(s, n, 1) → M
  37. Intersect(s, k) → N, O
  38. Angle(M, H, O) → β
  39. Dilate(A, If[9 < e ≤ 10, 10 - e, 0], H) → A'_3
  40. Vector(A, A'_3) → u
  41. Translate(q', u) → q''
  42. Rotate(q'', If[9 < e ≤ 10, ((e - 9) * β), β], A'_3) → q'''
  43. Circle(H, (0.55 * b)) → t
  44. Intersect(t, k) → P, Q
  45. Rotate(Q, π / 6, H) → Q'
  46. Segment(H, Q') → f_1
  47. Rotate(Q', (5 * π / 24), H) → Q''
  48. Rotate(Q'', (5 * π / 24), H) → Q'''
  49. Rotate(Q''', (5 * π / 24), H) → R
  50. Circle(H, (0.3 * b)) → d_1
  51. Ray(H, Q') → h_1
  52. Intersect(d_1, h_1) → S,
  53. Dilate(H, If[10 < e ≤ 11, 11 - e, 0], Q') → H'
  54. Dilate(H, If[10 < e ≤ 11, 11 - e, 0], Q'') → H'_1
  55. Dilate(H, If[10 < e ≤ 11, 11 - e, 0], Q''') → H'_2
  56. Dilate(H, If[10 < e ≤ 11, 11 - e, 0], R) → H'_3
  57. Segment(H, H') → i_1
  58. Segment(H, H'_1) → j_1
  59. Segment(H, H'_2) → k_1
  60. Segment(H, H'_3) → l_1
  61. Rotate(S, If[11 < e ≤ 12, (((e - 11) * 5) * π / 24), (5 * π / 24)], H) → S'
  62. CircleSector(H, S, S') → e_1
  63. Rotate(e_1, If[13 < e ≤ 14, (((e - 13) * 5) * π / 24), (5 * π / 24)], H) → p_1
  64. Rotate(p_1, If[15 < e ≤ 16, (((e - 15) * 5) * π / 24), (5 * π / 24)], H) → q_1
  65. Rotate(q_1, If[17 < e ≤ 18, (((e - 17) * 5) * π / 24), (5 * π / 24)], H) → r_1
  66. Dilate(C, If[20 < e ≤ 21, 21 - e, 0], E) → C'
  67. Dilate(C, If[20 < e ≤ 21, 21 - e, 0], D) → C'_1
  68. Segment(C, C'_1) → m_1
  69. Segment(C, C') → n_1
  70. Angle(D, C, A) → γ
  71. Dilate(D, If[21 < e ≤ 22, 22 - e, 0], F) → D'
  72. Segment(D, D') → s_1
  73. Dilate(F, If[22 < e ≤ 23, 23 - e, 0], A) → F'
  74. Segment(F, F') → t_1
  75. Dilate(F, If[23 < e ≤ 24, 24 - e, 0], G') → F'_1
  76. Segment(F, F'_1) → b_1
  77. Line(F, k) → f_2
  78. Point(f_2) → T
  79. Angle(T, F, G) → δ
  80. Dilate(H, If[25 < e ≤ 26, 26 - e, 0], F) → H'_4
  81. Vector(H, H'_4) → v
  82. Translate(r_1, v) → c_2
  83. Rotate(c_2, If[25 < e ≤ 26, ((e - 25) * δ), δ], H'_4) → d_2