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Rafael's Desposorios: Polygon in Circle with Similarity Transformation

Rafael's Desposorios: Polygon in Circle with Similarity Transformation
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2023-03-20 19:44
CirclePolygonTransformationGeometric ConstructionLocus
In collections geometriadinamica.es
This GeoGebra example demonstrates constructing multiple similar polygons inside a circle and applying dilation transformations. The core mathematical concepts include properties of cyclic polygons, similarity transformations, point symmetry, and geometric relationships between midpoints and intersections. Construction key points: Using Polygon command to create polygons, Circle command for the base circle, CircleArc for arcs, Midpoint command to find edge midpoints, Line and OrthogonalLine for lines and perpendiculars, Intersect to find intersections, and Dilate for similarity transformations. If commands are also used for dynamic control. This example is suitable for secondary school geometry teaching, helping students understand relationships between circles and polygons, applications of similarity transformations, and logical reasoning in geometric constructions.

GeoGebra Commands & Parameters

  1. If(paso > 0 ∧ paso < 7, 1, 0) → t_1
  2. If(paso > 1 ∧ paso < 7, 1, 0) → t_2
  3. If(paso > 2 ∧ paso < 7, 1, 0) → t_3
  4. If(paso > 3 ∧ paso < 7, 1, 0) → t_4
  5. If(paso > 4 ∧ paso < 7, 1, 0) → t_5
  6. If(paso > 5 ∧ paso < 7, 1, 0) → t_6
  7. Line(H1, H2) → a
  8. Line(A, a) → b
  9. OrthogonalLine(A, a) → c
  10. Point(b) → B
  11. Dilate(B, pr, A) → Dil
  12. Circle(A, Dil) → d
  13. Intersect(d, c) → D5, C
  14. OrthogonalLine(C, c) → e
  15. OrthogonalLine(B, b) → f
  16. Intersect(f, e) → D
  17. Polygon(A, B, D, C) → P, a_1, b_1, d_1, c_1
  18. Midpoint(A, B) → E
  19. OrthogonalLine(B_1, a_1) → g
  20. Intersect(g, d_1) → F
  21. Intersect(g, a_1) → G
  22. Segment(G, F) → h
  23. Circle(F, C) → k
  24. OrthogonalLine(B_2, h) → i
  25. Intersect(h, i) → H
  26. Circle(H, F) → p
  27. Segment(B_3, A) → j
  28. Point(j) → I
  29. CircleArc(I, B, A) → q
  30. Segment(B, B_4) → l
  31. Point(l) → K
  32. Point(l) → M
  33. Circle(K, M) → r
  34. Circle(M, K) → s
  35. Point(r) → J
  36. Point(r) → L
  37. Point(s) → N
  38. Point(s) → O
  39. CircleArc(M, N, O) → t
  40. Point(q) → Q
  41. CircleArc(Q, L, J) → e_1
  42. Intersect(t, e_1) → R, , ,
  43. Polygon(M, R, K) → S, k_1, m, r_1
  44. OrthogonalLine(B_5, b_1) → n
  45. Point(n) → T
  46. Point(n) → U
  47. Point(h) → V
  48. Conic(T, V, U, W, Y) → f_1
  49. OrthogonalLine(B_6, c_1) → g_1
  50. Point(g_1) → Z
  51. Point(c_1) → C_1
  52. Segment(Z, C_1) → h_1
  53. Point(c_1) → D_1
  54. Segment(Z, D_1) → i_1
  55. Point(b_1) → F_1
  56. Segment(Z, F_1) → l_1
  57. Point(b_1) → G_1
  58. Segment(Z, G_1) → m_1
  59. Point(b_1) → H_1
  60. Segment(Z, H_1) → n_1
  61. Point(b_1) → I_1
  62. Segment(Z, I_1) → p_1
  63. Point(c_1) → J_1
  64. Segment(Z, J_1) → q_1
  65. Point(c_1) → K_1
  66. Segment(Z, K_1) → s_1
  67. Point(b_1) → E_1
  68. Segment(Z, E_1) → j_1
  69. Point(c_1) → L_1
  70. Segment(Z, L_1) → a_2

AI Drawing Prompt Reference

Construct a quadrilateral inside a circle, with vertices determined by intersections of a line and perpendiculars with the circle. From midpoints of its sides, draw perpendiculars to intersect sides, creating new points. Use these points as centers for circles and arcs. Apply a similarity transformation (dilation) to generate an inner similar polygon, revealing construction steps progressively. Finally, combine multiple arcs and intersection points to form a complex polygon, and add a curve passing through several points. The overall figure illustrates similarity transformations and geometric relationships within a circle.
The prompt above was compiled from the original drawing commands and is for reference only.