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Octagon Properties: Incircle, Rotation and Symmetry Transformations

Octagon Properties: Incircle, Rotation and Symmetry Transformations
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2018-02-26 19:49
CirclePolygonAngleTransformationGeometric Construction
In collections Folding paper, doing geometry figures
This GeoGebra example dynamically explores the geometric properties of a regular octagon, including its incircle, circumcircle, axes of symmetry, and rotational symmetry. Core mathematical concepts include properties of regular polygons, relationships between incircle and circumcircle, angle calculations (e.g., central angle 45°), symmetry transformations (rotation and reflection), and vector translation. Construction highlights: the Polygon command creates the octagon, Circle commands draw incircle and circumcircle, Rotate and Mirror commands demonstrate symmetry, Midpoint and Angle commands measure key angles, Sequence generates repeated structures, and Translate with Vector duplicates shapes. Suitable for middle school to high school students learning regular polygon properties, geometric transformations, and circle geometry. It serves as an interactive tool for classroom demonstrations or self-study, visually reinforcing the symmetry and mathematical patterns of a regular octagon.

GeoGebra Commands & Parameters

  1. Polygon(A, C, D, B) → vierhoek1, a, c, d, b
  2. Segment(DD, E) → f
  3. Vector((0, -0.15)) → u
  4. Polygon(B, DD, E, D) → vierhoek2, b_1, dd, e, d_1
  5. Polygon(A, C, E, DD) → vierhoek3, a_1, c_1, e_1, dd_1
  6. If(s < 1, vierhoek3, If[1 < s ≤ 2, Polygon[Translate[A, Vector[((s - 1) * u)]], Translate[C, Vector[((s - 1) * u)]], Translate[E, Vector[((s - 1) * u)]], Translate[DD, Vector[((s - 1) * u)]]], Polygon[Translate[A, u], Translate[C, u], Translate[E, u], Translate[DD, u]]]) → vierhoek4
  7. Vector(D, B) → v
  8. Translate(afbeelding1, Vector[(1.25 * ((s - 1) * v))]) → afbeelding1'
  9. Segment(B, E) → g
  10. CircleArc(B, DD, D) → h
  11. Segment(DD, D) → i
  12. Polygon(A2, A1, D1, D2) → vierhoek5, a2, a1, d1, d2
  13. Mirror(D, d1) → DB
  14. Mirror(DD, a2) → DDB
  15. CircleArc(D2, DB, D) → k
  16. CircleArc(A2, DD, DDB) → p
  17. Polygon(B, A2, A1, E, D1, D2) → veelhoek1, b_2, a2_1, a1_1, e_2, d1_1, d2_1
  18. Polygon(D1, D, D2) → driehoek1, d2_2, d1_2, d_2
  19. Rotate(D1, (((-(s - 3)) * 180))°, d1) → D1'
  20. Rotate(D, (((-(s - 3)) * 180))°, d1) → D'
  21. Rotate(D2, (((-(s - 3)) * 180))°, d1) → D2'
  22. Polygon(D1', D', D2') → driehoek1', d2', d1', d'
  23. Polygon(D2, DB, D1) → driehoek2, d1_3, d2_3, db
  24. Midpoint(B, E) → M
  25. Rotate(D2, 180°, M) → D2'_1
  26. Rotate(DB, 180°, M) → DB'
  27. Rotate(D1, 180°, M) → D1'_1
  28. Polygon(D2'_1, DB', D1'_1) → driehoek2', d1'_1, d2'_1, db'
  29. Polygon(B, B1, B2) → driehoek4, b2, b_3, b1
  30. Polygon(A2, DD, A1) → driehoek3, a1_2, a2_2, dd_2
  31. Rotate(A2, (((-(s - 3)) * 180))°, a2) → A2'
  32. Rotate(DD, (((-(s - 3)) * 180))°, a2) → DD'
  33. Rotate(A1, (((-(s - 3)) * 180))°, a2) → A1'
  34. Polygon(A2', DD', A1') → driehoek3', a1', a2', dd'
  35. Polygon(B2, B1, A2, A1, C1, C2, D1, D2) → veelhoek2, b2_1, b1_1, a2_3, a1_3, c1, c2, d1_4, d2_4
  36. Polygon(B1, C1, C2, B2) → vierhoek6, b1_2, c1_1, c2_1, b2_2
  37. Segment(DDB, DB) → j
  38. Rotate(B, (((s - 5) * 180))°, b_3) → B'
  39. Rotate(B1, (((s - 5) * 180))°, b_3) → B1'
  40. Rotate(B2, (((s - 5) * 180))°, b_3) → B2'
  41. Polygon(B', B1', B2') → driehoek4', b2', b', b1'
  42. Polygon(B2, B1, A2, A1, C1, C2, D1, D2) → veelhoek3, b2_3, b1_3, a2_4, a1_4, c1_2, c2_2, d1_5, d2_5
  43. Rotate(C1, (((-(s - 5)) * 180))°, c1) → C1'
  44. Rotate(E, (((-(s - 5)) * 180))°, c1) → E'
  45. Rotate(C2, (((-(s - 5)) * 180))°, c1) → C2'
  46. Polygon(C1', E', C2') → driehoek5', c2', c1', e'
  47. Rotate(DDB, 90°, M) → DDB'
  48. Rotate(DB, 90°, M) → DB'_1
  49. Segment(DDB', DB'_1) → j'
  50. Polygon(B1, DB'_1, B2) → driehoek5, b2_4, b1_4, db'_1
  51. Polygon(C1, DDB', C2) → driehoek6, c2_3, c1_3, ddb'
  52. Rotate(B1, (((-(s - 6)) * 45))°, M) → B1'_1
  53. Rotate(C1, (((-(s - 6)) * 45))°, M) → C1'_1
  54. Rotate(C2, (((-(s - 6)) * 45))°, M) → C2'_1
  55. Rotate(B2, (((-(s - 6)) * 45))°, M) → B2'_1
  56. Polygon(B1'_1, C1'_1, C2'_1, B2'_1) → vierhoek6', b1'_1, c1'_1, c2'_1, b2'_1
  57. Mirror(B2, M) → E1
  58. Mirror(B1, M) → E2
  59. Mirror(B, b2_2) → BB
  60. Mirror(BB, M) → EE
  61. CircleArc(B1, BB, B) → q
  62. CircleArc(C1, E, EE) → r
  63. Segment(D2, D) → l
  64. Segment(D1, D) → m
  65. Segment(DD, A1) → n
  66. Segment(DD, A2) → t
  67. Segment(B, B2) → f_1
  68. Segment(B, B1) → g_1
  69. Segment(E, C1) → h_1
  70. Segment(E, C2) → i_1
  71. Angle(D1, D2, D) → α
  72. Angle(A2, A1, DD) → β
  73. Angle(B, B2, B1) → γ
  74. Angle(E, C1, C2) → δ
  75. Angle(B1, B2, D2) → hoek
  76. Angle(B2, D2, D1) → ε
  77. Angle(D2, D1, C2) → ζ
  78. Angle(D1, C2, C1) → η
  79. Angle(C2, C1, A1) → θ
  80. Angle(C1, A1, A2) → ι
  81. Angle(A1, A2, B1) → κ
  82. Angle(A2, B1, B2) → λ
  83. Circle(M, B1) → k_1
  84. Point(k_1) → H
  85. Mirror(H, M) → H'
  86. Segment(H, H') → j_1
  87. Circle(M, 0.6) → p_1
  88. Line(B, EE) → l_1
  89. Intersect(p_1, l_1, 1) → I
  90. Intersect(p_1, l_1, 2) → J
  91. Segment(I, J) → m_1
  92. Sequence(Segment[Rotate[I, ((nn * 22.5))°, M], Rotate[J, ((nn * 22.5))°, M]], nn, 1, 8) → lijst_1