MathPlotAI Logo MathPlotAI

Dynamic & Modifiable Geometric Problem Illustration

Dynamic & Modifiable Geometric Problem Illustration
Loading the math board and drawing, please wait…
2017-03-09 09:50
PolygonVectorCoordinate GeometryTransformationGeometric Construction
This example dynamically demonstrates a geometric problem posed by Alexander Bogomolny, illustrating relationships between geometric transformations, special points, and linear equations through interactive graphics. Core math concepts include polygon construction, linear equations, transformations (rotation, dilation), vectors, and locus. Construction highlights GeoGebra commands/objects like Polygon, Dilate, Segment, Rotate, Circle, Intersect, CircleSector, Vector, Translate, Point, Angle, Ray, and Line. Suitable for high school students and math enthusiasts exploring geometric transformations, special points, and linear equations dynamically.

GeoGebra Commands & Parameters

  1. Polygon(A, B, 4) → poly1, a, b, c, d, C, D
  2. Dilate(A, If[0 < e ≤ 1, 1 - e, 0], B) → A'
  3. Segment(A, A') → f
  4. Rotate(A, If[1 < e ≤ 2, ((-(e - 1)) * π / 2), -1.5707963267948966], B) → A'_1
  5. Rotate(A', If[1 < e ≤ 2, ((-(e - 1)) * π / 2), -1.5707963267948966], B) → A''
  6. Segment(A'_1, A'') → f'
  7. Segment(A, B) → g
  8. Segment(B, C) → h
  9. Segment(C, D) → i
  10. Segment(D, A) → j
  11. Circle(C, (0.15 * j)) → k
  12. Circle(B, (0.15 * j)) → p
  13. Circle(A, (0.15 * j)) → q
  14. Circle(D, (0.15 * j)) → r
  15. Segment(A, C) → l
  16. Segment(D, B) → m
  17. Intersect(r, j, 1) → F
  18. Intersect(r, m, 1) → G
  19. Intersect(r, i, 1) → H
  20. Intersect(k, i, 1) → I
  21. Intersect(k, l, 1) → J
  22. Intersect(k, h, 1) → K
  23. Intersect(p, h, 1) → L
  24. Intersect(p, m, 1) → M
  25. Intersect(p, g, 1) → N
  26. Intersect(q, g, 1) → O
  27. Intersect(q, l, 1) → P
  28. Intersect(q, j, 1) → Q
  29. Rotate(A'_1, If[2 < e ≤ 3, ((-(e - 2)) * π / 2), -1.5707963267948966], C) → R
  30. Rotate(A'', If[2 < e ≤ 3, ((-(e - 2)) * π / 2), -1.5707963267948966], C) → A'''
  31. Segment(R, A''') → f''
  32. CircleSector(B, L, N) → s
  33. Rotate(R, If[3 < e ≤ 4, ((-(e - 3)) * π / 2), -1.5707963267948966], D) → R'
  34. Rotate(A''', If[3 < e ≤ 4, ((-(e - 3)) * π / 2), -1.5707963267948966], D) → S
  35. Segment(R', S) → f'''
  36. Dilate(B, If[4 < e ≤ 5, 5 - e, 0], C) → B'
  37. Vector(B, B') → u
  38. Translate(s, u) → s'
  39. Rotate(s', If[5 < e ≤ 6, ((e - 5) * π / 2), 1.5707963267948966], C) → s''
  40. Dilate(C, If[6 < e ≤ 7, 7 - e, 0], D) → C'
  41. Vector(C, C') → v
  42. Translate(s'', v) → s'''
  43. Rotate(s''', If[7 < e ≤ 8, ((e - 7) * π / 2), 1.5707963267948966], D) → t
  44. Dilate(D, If[8 < e ≤ 9, 9 - e, 0], A) → D'
  45. Vector(D, D') → w
  46. Translate(t, w) → t'
  47. Rotate(t', If[9 < e ≤ 10, ((e - 9) * π / 2), 1.5707963267948966], A) → t''
  48. Dilate(B, If[18 < e ≤ 19, 19 - e, 0], C) → B'_1
  49. Dilate(B, If[18 < e ≤ 19, 19 - e, 0], A) → B'_2
  50. Dilate(B, If[18 < e ≤ 19, 19 - e, 0], D) → B'_3
  51. Vector(B, B'_1) → v_1
  52. Vector(B, B'_3) → w_1
  53. Vector(B, B'_2) → i_1
  54. Point(f'') → V
  55. Segment(A, V) → n
  56. Angle(B, A, V) → α
  57. Rotate(B, α / 2, A) → B'_4
  58. Angle(B, A, B'_4) → β
  59. Ray(A, B'_4) → j_1
  60. Intersect(f', j_1) → W
  61. Segment(A, W) → l_1
  62. Intersect(q, l_1) → A_1,
  63. Dilate(A, If[13 < e ≤ 14, 14 - e, 0], W) → A'_3
  64. Segment(A, A'_3) → m_1
  65. Segment(A''', V) → n_1
  66. Circle(V, n_1) → d_1
  67. Intersect(d_1, n, 1) → B_1
  68. Intersect(q, n, 1) → E
  69. CircleSector(A, A_1, E) → c_1
  70. Dilate(A, If[11 < e ≤ 12, 12 - e, 0], B_1) → A'_2
  71. Segment(A, A'_2) → f_1
  72. Dilate(B_1, If[12 < e ≤ 13, 13 - e, 0], V) → U
  73. Segment(B_1, U) → g_1
  74. Rotate(c_1, If[14.5 < e ≤ 15.5, ((-(e - 14.5)) * β), (-β)], A) → e_1
  75. Dilate(A''', If[15.5 < e ≤ 16.5, 16.5 - e, 0], V) → T
  76. Dilate(B, If[15.5 < e ≤ 16.5, 16.5 - e, 0], W) → B'_5
  77. Segment(A''', T) → a_1
  78. Segment(B, B'_5) → b_1
  79. Rotate(A''', If[17 < e ≤ 18, ((2 * (e - 17)) * β), (2 * β)], V) → Z
  80. Rotate(T, If[17 < e ≤ 18, ((2 * (e - 17)) * β), (2 * β)], V) → T'
  81. Segment(Z, T') → h_1
  82. Line(B, f_1) → k_1
  83. Circle(B, W) → p_1
  84. Intersect(p_1, k_1, 2) → C_1
  85. Angle(C_1, B, W) → γ
  86. Rotate(B, If[18 < e ≤ 19, ((-(e - 18)) * γ), (-γ)], B) → B'_6
  87. Rotate(B'_5, If[18 < e ≤ 19, ((-(e - 18)) * γ), (-γ)], B) → D_1
  88. Segment(B'_6, D_1) → q_1
  89. Dilate(B, If[19 < e ≤ 20, 20 - e, 0], A) → B'_7
  90. Vector(B, B'_7) → u_1
  91. Translate(B'_6, u_1) → E_1
  92. Translate(D_1, u_1) → F_1
  93. Segment(E_1, F_1) → r_1