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Rotating Intersections

Rotating Intersections
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2019-09-30 08:20
Coordinate GeometryTransformationOptimizationGeometric ConstructionLocus
This example demonstrates the dynamic geometric phenomenon of observing the locus of intersection points of quadrilaterals through rotation and mirroring operations. Core math concepts include intersections, dilation, geometric transformations, and coordinate geometry. Construction highlights the use of GeoGebra commands like Rotate, Mirror, Polygon, Segment, Intersect, and Locus. Suitable for high school and above students to understand geometric transformations, locus construction, and optimization problems.

GeoGebra Commands & Parameters

  1. Rotate(B, (2 * π / n)) → B'
  2. Mirror(B, A) → B'_1
  3. Mirror(B', A) → B''
  4. Polygon(B, B', n) → poly1, f, q, r, s, t, a, b, c, d, e, f_1, g, g_1, C, D, E, F, G, H, I, J, K, h, L, M, N, O, P, Q, R, S, T, i, j, k, l, m, p
  5. Mirror(C, A) → C'
  6. Mirror(D, A) → D'
  7. Mirror(E, A) → E'
  8. Mirror(F, A) → F'
  9. Mirror(G, A) → G'
  10. Mirror(H, A) → H'
  11. Mirror(I, A) → I'
  12. Mirror(J, A) → J'
  13. Segment(B, B'_1) → h_1
  14. Segment(B', B'') → i_1
  15. Segment(C, C') → j_1
  16. Segment(D, D') → k_1
  17. Segment(E, E') → l_1
  18. Segment(F, F') → m_1
  19. Segment(G, G') → n_1
  20. Segment(H, H') → p_1
  21. Segment(I, I') → q_1
  22. Segment(J, J') → r_1
  23. Rotate(B, aa + ss) → B'_2
  24. Rotate(B'_1, aa + ss) → U
  25. Segment(B'_2, U) → s_1
  26. Rotate(B', aa + ss) → B''_1
  27. Rotate(B'', aa + ss) → B'''
  28. Segment(B''_1, B''') → t_1
  29. Rotate(D, aa + ss) → D'_1
  30. Rotate(D', aa + ss) → D''
  31. Segment(D'_1, D'') → a_1
  32. Rotate(E, aa + ss) → E'_1
  33. Rotate(E', aa + ss) → E''
  34. Segment(E'_1, E'') → b_1
  35. Rotate(C, aa + ss) → C'_1
  36. Rotate(C', aa + ss) → C''
  37. Segment(C'_1, C'') → c_1
  38. Rotate(F, aa + ss) → F'_1
  39. Rotate(F', aa + ss) → F''
  40. Segment(F'_1, F'') → d_1
  41. Rotate(G, aa + ss) → G'_1
  42. Rotate(G', aa + ss) → G''
  43. Segment(G'_1, G'') → e_1
  44. Rotate(H, aa + ss) → H'_1
  45. Rotate(H', aa + ss) → H''
  46. Segment(H'_1, H'') → f_2
  47. Rotate(I, aa + ss) → I'_1
  48. Rotate(I', aa + ss) → I''
  49. Segment(I'_1, I'') → g_2
  50. Rotate(J, aa + ss) → J'_1
  51. Rotate(J', aa + ss) → J''
  52. Segment(J'_1, J'') → h_2
  53. Intersect(p_2, h_2) → V
  54. Intersect(p_2, g_2) → W
  55. Intersect(p_2, f_2) → Z
  56. Intersect(p_2, e_1) → A_1
  57. Intersect(p_2, d_1) → B_1
  58. Intersect(p_2, b_1) → C_1
  59. Intersect(p_2, a_1) → D_1
  60. Intersect(p_2, c_1) → E_1
  61. Intersect(p_2, t_1) → F_1
  62. Intersect(p_2, s_1) → G_1
  63. Locus(A_1, aa) → loc1
  64. Locus(B_1, aa) → loc2
  65. Locus(C_1, aa) → loc3
  66. Locus(D_1, aa) → loc4
  67. Locus(E_1, aa) → loc5
  68. Locus(F_1, aa) → loc6
  69. Locus(G_1, aa) → loc7
  70. Locus(V, aa) → loc8
  71. Locus(W, aa) → loc9
  72. Locus(Z, aa) → loc10
  73. Polygon(H_1, I_1, 4) → poly2, i_2, j_2, k_2, l_2, J_1, K_1

AI Drawing Prompt Reference

Draw a regular polygon with vertices rotated around a central point. First, construct the polygon, then mirror each vertex across the center point to create symmetric counterparts, and connect each original vertex to its mirrored one with a segment. Next, rotate the entire polygon and its mirrored vertices by a dynamic angle around the center, generating new rotated vertices, and connect corresponding rotated vertices to form another set of segments. Finally, observe the locus of intersection points of these segments, illustrating the dynamic geometric pattern formed by rotation and mirroring operations.
The prompt above was compiled from the original drawing commands and is for reference only.