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Geometric Construction of Square Puzzles

Geometric Construction of Square Puzzles
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2020-09-20 18:58
CirclePolygonAngleTransformationGeometric Construction
This example demonstrates how to build a sequence of “square puzzle” configurations by combining geometric construction, intersection-based point placement, and rotation-driven replication. The key ideas include geometric construction (assembling segments and polygonal boundaries), using circles and intersection points to locate vertices precisely, constructing orthogonal lines and midpoints to establish right-angle and symmetry/reference scaffolding, and applying rotation to transform a base piece into repeated sections that fit together. Construction follows GeoGebra-style commands: Segment to form structural edges; Circle to generate guiding loci; Point to define vertices and anchors; OrthogonalLine to enforce perpendicular constraints; Intersect to compute new points from crossings; Polygon to connect vertices into evolving shapes; Line as auxiliary references; Rotate to copy/align puzzle components; and Midpoint to create central or symmetric baselines. Suitable for middle/high school geometry, transformation geometry lessons, and GeoGebra construction practice.

GeoGebra Commands & Parameters

  1. Segment(A_1, B_1) → c_1
  2. Circle(O_1, 2) → p_1
  3. Point(p_1) → P_1
  4. Segment(O_1, P_1) → n_1
  5. OrthogonalLine(O_1, n_1) → r_1
  6. Intersect(p_1, r_1, 2) → Q_1
  7. Polygon(Q_1, O_1, P_1) → poly6, p_2, q_2, o_1
  8. Circle(R_1, 2) → s_1
  9. Point(s_1) → S_1
  10. Segment(R_1, S_1) → t_1
  11. OrthogonalLine(R_1, t_1) → a_2
  12. Intersect(s_1, a_2, 2) → T_1
  13. Polygon(T_1, R_1, S_1) → poly7, s_2, t_2, r_2
  14. Circle(U_1, 2) → c_2
  15. Point(c_2) → V_1
  16. Segment(U_1, V_1) → b_2
  17. OrthogonalLine(U_1, b_2) → d_2
  18. Intersect(c_2, d_2, 2) → W_1
  19. Polygon(W_1, U_1, V_1) → poly8, v_1, w_1, u_1
  20. Circle(Z_1, 2) → e_2
  21. Point(e_2) → A_2
  22. Segment(Z_1, A_2) → f_2
  23. OrthogonalLine(Z_1, f_2) → g_2
  24. Intersect(e_2, g_2, 2) → B_2
  25. Polygon(B_2, Z_1, A_2) → poly9, a_3, b_3, z_1
  26. Circle(C_2, 5) → h_2
  27. Point(h_2) → D_2
  28. Segment(C_2, D_2) → i_2
  29. OrthogonalLine(C_2, i_2) → j_2
  30. Intersect(h_2, j_2, 1) → E_2
  31. Polygon(E_2, C_2, D_2) → poly10, d_3, e_3, c_3
  32. Circle(F_2, 2.5) → k_2
  33. Point(k_2) → G_2
  34. Segment(F_2, G_2) → l_2
  35. OrthogonalLine(G_2, l_2) → m_2
  36. OrthogonalLine(F_2, l_2) → n_2
  37. Intersect(k_2, n_2) → H_2, I_2
  38. OrthogonalLine(I_2, n_2) → f_3
  39. Intersect(m_2, f_3) → J_2
  40. Polygon(F_2, G_2, J_2, I_2) → poly11, f_4, g_3, j_3, i_3
  41. Circle(K_2, 2.5) → h_3
  42. Point(h_3) → L_2
  43. Circle(K_2, 5) → k_3
  44. Segment(K_2, L_2) → l_3
  45. OrthogonalLine(L_2, l_3) → m_3
  46. OrthogonalLine(K_2, l_3) → n_3
  47. Intersect(k_3, n_3, 2) → M_2
  48. Line(M_2, l_3) → p_3
  49. Intersect(m_3, p_3) → N_2
  50. Polygon(K_2, L_2, N_2, M_2) → poly12, k_4, l_4, n_4, m_4
  51. Circle(O_2, 2.5) → q_3
  52. Point(q_3) → P_2
  53. Circle(O_2, 5) → r_3
  54. Segment(P_2, O_2) → s_3
  55. OrthogonalLine(O_2, s_3) → t_3
  56. Intersect(r_3, t_3, 2) → Q_2
  57. Polygon(O_2, Q_2, P_2) → poly13, p_4, o_2, q_4
  58. Circle(R_2, 2.5) → c_4
  59. Circle(R_2, 5) → d_4
  60. Circle(V_2, 3) → e_4
  61. Point(e_4) → W_2
  62. Rotate(W_2, 90°, V_2) → Z_2
  63. Line(W_2, Z_2) → g_4
  64. Line(V_2, g_4) → h_4
  65. Segment(V_2, W_2) → i_4
  66. Point(g_4) → A_3
  67. Line(A_3, i_4) → j_4
  68. Intersect(h_4, j_4) → B_3
  69. OrthogonalLine(B_3, i_4) → t_4
  70. Intersect(g_4, t_4) → C_3
  71. Polygon(C_3, B_3, V_2, W_2) → poly15, c_5, b_5, v_2, w_2
  72. Circle(D_3, 2.5) → d_5
  73. Point(d_5) → E_3
  74. Segment(E_3, D_3) → a_5
  75. OrthogonalLine(D_3, a_5) → e_5
  76. OrthogonalLine(E_3, a_5) → f_5
  77. Intersect(d_5, e_5, 2) → F_3
  78. Circle(E_3, 5) → g_5
  79. Intersect(g_5, f_5, 2) → G_3
  80. Polygon(D_3, E_3, G_3, F_3) → poly16, d_6, e_6, g_6, f_6
  81. Circle(H_3, 2.5) → k_5
  82. Point(k_5) → I_3
  83. Rotate(I_3, 90°, H_3) → J_3
  84. Circle(I_3, 5) → p_5
  85. Segment(H_3, I_3) → j_5
  86. OrthogonalLine(I_3, j_5) → l_5
  87. Intersect(p_5, l_5, 2) → K_3
  88. Polygon(H_3, I_3, K_3, J_3) → poly17, h_6, i_6, k_6, j_6
  89. Circle(L_3, 5) → q_5
  90. Point(q_5) → M_3
  91. Rotate(M_3, 90°, L_3) → N_3
  92. Segment(N_3, M_3) → m_5
  93. OrthogonalLine(L_3, m_5) → n_5
  94. Intersect(n_5, m_5) → O_3
  95. Polygon(L_3, M_3, O_3) → poly18, o_3, l_6, m_6
  96. Circle(P_3, 3) → r_5
  97. Point(r_5) → Q_3
  98. Rotate(Q_3, 90°, P_3) → R_3
  99. Circle(V_3, 2.5) → c_6
  100. Point(c_6) → W_3
  101. Rotate(W_3, 90°, V_3) → Z_3
  102. Segment(V_3, W_3) → a_6
  103. Line(Z_3, a_6) → b_6
  104. Rotate(W_3, 270°, V_3) → A_4
  105. Line(A_4, b_6) → n_6
  106. OrthogonalLine(W_3, b_6) → s_6
  107. Rotate(W_3, 180°, V_3) → B_4
  108. OrthogonalLine(B_4, b_6) → t_6
  109. Intersect(n_6, t_6) → C_4
  110. Circle(C_4, 2) → c_7
  111. Intersect(c_7, t_6) → D_4, E_4
  112. Intersect(c_7, n_6) → F_4, G_4
  113. Intersect(b_6, t_6) → H_4
  114. Intersect(n_6, s_6) → I_4
  115. Polygon(V_3, Z_3, H_4, E_4, G_4, I_4, W_3) → poly20, v_3, z_3, h_7, e_7, g_7, i_7, w_3
  116. Circle(J_4, 3) → d_7
  117. Point(d_7) → K_4
  118. Rotate(K_4, 90°, J_4) → L_4
  119. Line(J_4, L_4) → a_7
  120. OrthogonalLine(J_4, a_7) → b_7
  121. Circle(J_4, 5) → f_7
  122. Intersect(f_7, b_7, 1) → M_4
  123. Rotate(M_4, 90°, J_4) → N_4
  124. Line(N_4, b_7) → j_7
  125. Line(M_4, a_7) → k_7
  126. Intersect(j_7, k_7) → O_4
  127. Rotate(K_4, 270°, M_4) → P_4
  128. Rotate(L_4, 90°, N_4) → Q_4
  129. Polygon(J_4, K_4, P_4, O_4, Q_4, L_4) → poly21, j_8, k_8, p_7, o_4, q_7, l_7
  130. Circle(R_4, 5) → s_5
  131. Point(s_5) → S_4
  132. Rotate(S_4, 90°, R_4) → T_4
  133. Midpoint(R_4, T_4) → U_4
  134. Polygon(R_4, S_4, U_4) → poly19, u_4, r_6, s_7
  135. Polygon(A, B, 4) → poly1, a_1, b_1, f, g, C, D
  136. Polygon(E, F, 4) → poly2, e_1, f_1, h, i, G, H
  137. Polygon(I, J, 4) → poly3, i_1, j, k, l, K, L
  138. Polygon(M, N, 4) → poly4, m, n, p, q, O, P
  139. Polygon(Q, R, 4) → poly5, q_1, r, s, t, S, T

AI Drawing Prompt Reference

On a plane, construct several square‑puzzle style cells starting from multiple center points: draw circles of given radii, then draw radius line segments from each circle center to points on the circles. Through each center (or its associated point), draw a perpendicular line to a relevant segment, compute intersection points between the circle and the perpendicular line, and connect these vertices with polygon edges to form small triangular or quadrilateral boundary pieces. Repeat the same circle–point–perpendicular–intersection–polygon process for successive circle centers to build neighboring units. Finally, introduce additional circles and perpendicular guides, determine their intersections, and connect the resulting vertices into a final quadrilateral polygon, keeping vertices fixed by perpendicular constraints and circle intersections.
The prompt above was compiled from the original drawing commands and is for reference only.