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Chord Polygons Preserve a Ratio: A GeoGebra Construction

Chord Polygons Preserve a Ratio: A GeoGebra Construction
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2023-03-19 02:30
CirclePolygonTrigonometrySequenceGeometric Construction
In collections geometriadinamica.es
This example demonstrates how to construct a chord-based polygon (with vertices tied to a common circular configuration) and visually verify that a key ratio remains invariant across different polygon instances. The phenomenon highlights the stability of geometric relationships under construction changes. The core ideas include circle–chord geometry, polygon angle/segment relationships, measurement via lengths (distances), and the typical reasoning pathway from triangle/trigonometric structure to an invariant ratio (with an intuition connected to sequences/series behavior). Construction uses GeoGebra tools reflected in the command summary: Line, Point, Polygon, Midpoint, Segment, Intersect, and Distance. The objects poly1, poly4, and poly11 represent different polygon stages/sizes, while named points (e.g., A/A', B/B', C/C') and computed quantities (e.g., via midpoints and intersection points) define chord and partition positions, enabling the ratio to be computed and compared. Suitable for secondary-school geometry exploration, high-school geometry problem solving, and classroom demonstrations with GeoGebra for invariant-property investigations.

GeoGebra Commands & Parameters

  1. Line(A, A') → h
  2. Point(h) → A''
  3. Line(B, h) → i
  4. Point(i) → B'
  5. Polygon(B, A, 5) → poly1, a, b, c, d, e, E1, D, C
  6. Line(C, h) → m
  7. Point(m) → C_2
  8. Line(D, h) → l
  9. Point(l) → D'
  10. Line(E1, h) → j
  11. Point(j) → E'_1
  12. Polygon(A'', E'_1, D', C_2, B') → poly4, a', e_2, d_2, c_2, b_2
  13. Midpoint(b_2) → F''
  14. Midpoint(a') → R''
  15. Segment(E'_1, B') → f_2
  16. Midpoint(e) → J
  17. Midpoint(d) → I
  18. Midpoint(c) → H
  19. Midpoint(a) → F
  20. Midpoint(b) → R
  21. Polygon(R, F, J, I, H) → poly11, r_1, f_3, j_6, i_2, h_6
  22. Segment(E1, B) → k_2
  23. Intersect(f_3, k_2) → M
  24. Intersect(h_6, k_2) → G
  25. Line(G, i) → m_2
  26. Line(M, m_2) → p_2
  27. Intersect(p_2, f_2) → L
  28. Intersect(m_2, f_2) → K
  29. Polygon(A', B', C_2, D', E'_1) → poly2, a_1, b_1, c_1, d_1, e_1
  30. Midpoint(a_1) → F'
  31. Midpoint(e_1) → R'
  32. Midpoint(b_1) → J'
  33. Midpoint(d_1) → H'
  34. Midpoint(c_1) → I'
  35. Polygon(R', F', J', I', H') → poly3, g_1, f_1, j_1, i_1, h_1
  36. Segment(H, R) → q
  37. Line(R, h) → n_1
  38. Line(F, h) → p_1
  39. Line(H, p_1) → a_2
  40. Polygon(K, R', F', L) → poly8, s, t_{3}, k_1, l_1
  41. Line(J, p_1) → n_2
  42. Polygon(K, R'', F'', L) → poly9, m_{3}, q_1, s_1, t_1
  43. Polygon(H, I, J) → poly14, v_1, t_2, u
  44. Segment(F, J) → t
  45. Distance(R', R'') → a3
  46. Segment(R'', H') → g_2
  47. Segment(R', H') → h_2
  48. Line(D, F) → m_1
  49. Segment(E'_1, C_2) → j_2
  50. Segment(E1, C) → l_2
  51. Intersect(j_6, l_2) → N
  52. Line(N, i) → q_2
  53. Intersect(h_6, l_2) → O
  54. Segment(J', F') → n
  55. Segment(F'', J') → p
  56. Line(O, i) → r
  57. Intersect(j_2, r) → E
  58. Intersect(j_2, q_2) → P
  59. Polygon(P, J', L, K, E) → poly5, p_3, j', l_3, k_3, e_3
  60. Polygon(E, H', I', P) → poly6, e_4, h', i', p_4
  61. Polygon(O, G, M, J, N) → poly7, o, g_3, m_3, j_3, n_3

AI Drawing Prompt Reference

Draw a circle and construct multiple polygons by connecting vertices on the circle. Show the relationships formed by lines connecting the vertices to the circle's center. Verify that a key ratio of lengths remains invariant across different polygons by measuring distances. Illustrate the construction using midpoints, intersection points, and lines formed by these points to create new polygons, ultimately displaying a collection of multiple polygons and auxiliary shapes.
The prompt above was compiled from the original drawing commands and is for reference only.