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Dissect one square into three squares

Dissect one square into three squares
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2019-03-21 00:41
CirclePolygonVectorTransformationGeometric Construction
In collections Cutting up squares
This example demonstrates how to dissect a single square into three smaller squares whose areas sum to the original square's area. The core mathematical concepts involve planar geometry, area conservation, properties of squares, and geometric construction techniques for partitioning and reassembly. The construction uses GeoGebra commands such as Point, Intersect, Segment, Circle, Line, OrthogonalLine, Polygon, Vector, Translate, and CircleArc, along with multiple auxiliary points (A, B, C, D, etc.) and segments, to achieve the dissection through translations and circular arcs. This example is suitable for middle or high school geometry lessons, helping students understand area decomposition and geometric construction, and can also serve as material for mathematical exploration or competition training.

GeoGebra Commands & Parameters

  1. Point(yAxis) → A
  2. Intersect(xAxis, yAxis) → B
  3. Segment(A, B) → f
  4. Segment(B, C) → g
  5. Segment(C, D) → h
  6. Segment(D, A) → i
  7. Point(f) → E
  8. Segment(E, F) → j
  9. Point(h) → G
  10. Point(f) → H
  11. Segment(G, H) → k
  12. Point(i) → I
  13. Point(g) → J
  14. Segment(I, J) → l
  15. Point(g) → K
  16. Point(i) → L
  17. Segment(K, L) → m
  18. Circle(J, C) → c
  19. Intersect(c, f, 1) → M
  20. Circle(L, A) → d
  21. Intersect(d, h, 1) → N
  22. Line(B, C) → n
  23. Circle(B, (3 * sqrt(3))) → e
  24. Intersect(e, n, 2) → O
  25. OrthogonalLine(M, f) → p
  26. OrthogonalLine(N, h) → q
  27. Line(N, A) → r
  28. Polygon(A, D, N) → t1, n_1, a, d_1
  29. Vector(N, O) → u
  30. Translate(A, u) → A'
  31. Translate(D, u) → D'
  32. Translate(N, u) → N'
  33. Polygon(A', D', N') → t1', n', a', d'
  34. Polygon(A, M, A') → t2, a'_1, a_1, m_1
  35. Vector(A, N) → v
  36. Translate(A, v) → A'_1
  37. Translate(M, v) → M'
  38. Translate(A', v) → A''
  39. Polygon(A'_1, M', A'') → t2', a'', a'_2, m'
  40. Polygon(M, B, C, N, A') → poly1, m_2, b, c_1, n_2, a'_3
  41. Intersect(p, h) → P
  42. Segment(M, P) → s
  43. Intersect(q, f) → Q
  44. Intersect(a'_3, d_1) → R
  45. Translate(A, u) → A'_2
  46. Translate(D, u) → D'_1
  47. Translate(N, u) → N'_1
  48. Translate(A, w) → A'_3
  49. Translate(D, w) → D'_2
  50. Translate(N, w) → N'_2
  51. Polygon(A'_3, D'_2, N'_2) → t1'_1, n'_2, a'_4, d'_2
  52. Translate(A, u) → A'_4
  53. Translate(D, u) → D'_3
  54. Translate(N, u) → N'_3
  55. Polygon(A'_4, D'_3, N'_3) → t1'_2, n'_3, a'_5, d'_3
  56. Translate(A, t) → A'_5
  57. Translate(M, t) → M'_1
  58. Translate(A', t) → A''_1
  59. Polygon(A'_5, M'_1, A''_1) → t2'_1, a''_1, a'_6, m'_1
  60. Translate(A, v) → A'_6
  61. Translate(M, v) → M'_2
  62. Translate(A', v) → A''_2
  63. Polygon(A'_6, M'_2, A''_2) → t2'_2, a''_2, a'_7, m'_2
  64. Intersect(f, g) → S
  65. Intersect(f, a'_3) → T
  66. Circle(S, T) → e_1
  67. Circle(U, V) → f_1
  68. Line(S, U) → g_1
  69. Intersect(e_1, g_1) → W, Z
  70. Intersect(f_1, g_1) → A_1, B_1
  71. Point(e_1) → C_1
  72. Point(e_1) → D_1
  73. Point(f_1) → E_1
  74. Point(f_1) → F_1
  75. CircleArc(S, D_1, C_1) → h_1
  76. CircleArc(U, F_1, E_1) → k_1
  77. OrthogonalLine(Z, g) → i_1
  78. OrthogonalLine(A_1, g) → j_1
  79. Intersect(i_1, poly1) → G_1, H_1
  80. Intersect(j_1, t1'_2) → I_1, J_1
  81. Segment(H_1, Z) → l_1
  82. Segment(J_1, A_1) → p_1
  83. Segment(T, K_1) → q_1
  84. Segment(K_1, O) → r_1
  85. Segment(O, B) → s_1

AI Drawing Prompt Reference

Draw a square and dissect it into three smaller squares, demonstrating area conservation. Start by drawing a square, then use geometric constructions involving points, lines, circles, and vectors to partition the square into three smaller squares whose total area equals the original square. The process includes circular arcs and orthogonal lines, showcasing geometric transformations and area decomposition.
The prompt above was compiled from the original drawing commands and is for reference only.