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Geometric construction between pentagons

Geometric construction between pentagons
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2023-03-13 17:54
CirclePolygonAngleTransformationGeometric Construction
In collections geometriadinamica.es
This example demonstrates how to construct an embedded, chain-like geometric configuration between multiple pentagons using circle intersections, rays, and a sequence of transformations. It highlights key ideas such as polygon vertex construction (pentagons), how circle–line/ray intersections determine critical points, and how angles and symmetries guide the layout. The construction uses GeoGebra commands including Circle, Ray, Intersect, Rotate, LineBisector (angle bisector), Mirror (reflection), Polygon, Vector, Line, Midpoint, and Translate. A set of geometric objects (e.g., points labeled like a, A, B, C, D, e–h and their indexed counterparts such as a_1, b_1, c_1, d_1, e_1, f_1) is created and connected so that each transformed pentagon shares consistent geometric relationships with the next. Suitable for secondary-school geometry, teaching transformation-based construction, and for students learning rigorous geometric modeling.

GeoGebra Commands & Parameters

  1. Circle(A, i) → k
  2. Circle(A, c) → d
  3. Ray(A, B) → b
  4. Intersect(d, b, 1) → C
  5. Rotate(A, 252°, C) → D
  6. Circle(C, A) → e
  7. Circle(A, f) → g
  8. LineBisector(A, C) → h
  9. Rotate(C, 252°, D) → E
  10. Mirror(D, h) → F
  11. Polygon(A, F, E, D, C) → P, a_1, f_1, e_1, d_1, c_1
  12. Intersect(k, c_1, 1) → H
  13. Vector(C, H) → z_1
  14. Circle(C, H) → h_1
  15. Intersect(h_1, d_1) → W,
  16. Vector(W, C) → v_3
  17. Circle(E, j) → p
  18. Intersect(k, a_1, 1) → G
  19. Intersect(p, f_1, 1) → I
  20. Intersect(p, e_1, 1) → J
  21. Line(J, d_1) → l
  22. Line(I, a_1) → m
  23. Circle(I, n) → q
  24. Intersect(q, m, 2) → K
  25. Line(K, c_1) → r
  26. Intersect(r, l) → L
  27. Polygon(I, K, L, J, E) → Q, i_1, k_1, l_1, j_1, e_2
  28. Midpoint(G, K) → M
  29. Midpoint(G, M) → N
  30. Midpoint(M, K) → O
  31. Line(N, a_1) → t
  32. Vector(G, N) → u
  33. Translate(F, u) → R
  34. Translate(R, u) → S
  35. Translate(S, u) → T
  36. Translate(T, u) → U
  37. Polygon(F, G, R, N, S, M, T, O, U, I) → V, f_2, i_2, g_1, r_1, n_1, s, m_1, t_1, o, u_1
  38. Midpoint(J, W) → Y
  39. Midpoint(Y, W) → Z
  40. Midpoint(Y, J) → A_1
  41. Line(A_1, e_1) → b_1
  42. Intersect(b_1, l) → B_1
  43. Vector(J, A_1) → v
  44. Translate(B_1, v) → C_1
  45. Translate(C_1, v) → D_1
  46. Translate(D_1, v) → E_1
  47. Polygon(D, W, E_1, Z, D_1, Y, C_1, A_1, B_1, J) → P_1, d_2, j_2, w, e_3, z, d_3, y, c_2, a_2, b_2
  48. Vector(W, E_1) → v_1
  49. Vector(L, B_1) → w_1
  50. Translate(H, v_1) → F_1
  51. Translate(F_1, w_1) → G_1
  52. Translate(G_1, z_1) → H_1
  53. Translate(H_1, v_1) → I_1
  54. Translate(I_1, w_1) → J_1
  55. Translate(J_1, z_1) → K_1
  56. Translate(K_1, v_1) → L_1
  57. Translate(L_1, w_1) → M_1
  58. Translate(M_1, z_1) → N_1
  59. Translate(N_1, v_1) → O_1
  60. Translate(O_1, w_1) → Q_1
  61. Midpoint(Q_1, L) → R_1
  62. Line(N_1, e_2) → q_1
  63. Line(R_1, a_1) → p_1
  64. Intersect(p_1, q_1) → S_1
  65. Polygon(Q_1, O_1, N_1, S_1, R_1) → T_1, q_2, o_1, n_2, s_1, r_2
  66. Polygon(N_1, M_1, C, W, E_1, Z, D_1, Y, C_1, A_1, B_1, L, L_1, R_1, S_1, K_1, J_1, I_1, H_1, G_1, F_1, H) → U_1, n_3, h_3, c_3, w_2, e_4, z_2, d_4, y_1, c_4, a_3, b_3, m_2, l_3, r_3, s_2, l_2, k_2, j_3, i_3, h_2, g_2, f_3
  67. Polygon(A, H, N_1, O_1, Q_1, K, U, O, T, M, S, N, F_1, R, G, G_1, H_1, I_1, J_1, K_1, L_1, M_1) → V_1, a_4, m_3, n_4, o_2, q_3, k_4, u_2, o_3, t_2, m_4, s_3, h_4, n_5, r_4, g_4, f_4, g_3, h_5, i_4, j_4, k_3, l_4
  68. Rotate(I, ((-a) * 180°), D) → I'
  69. Rotate(K, ((-a) * 180°), D) → K'
  70. Rotate(L, ((-a) * 180°), D) → L'
  71. Rotate(J, ((-a) * 180°), D) → J'
  72. Rotate(E, ((-a) * 180°), D) → E'
  73. Polygon(I', K', L', J', E') → Q', i', k', l', j', e'
  74. Rotate(I', ((-a) * 36°), E') → I''
  75. Rotate(K', ((-a) * 36°), E') → K''
  76. Rotate(L', ((-a) * 36°), E') → L''
  77. Rotate(J', ((-a) * 36°), E') → J''
  78. Rotate(E', ((-a) * 36°), E') → E''
  79. Polygon(I'', K'', L'', J'', E'') → Q'', i'', k'', l'', j'', e''
  80. Translate(F, v_2) → F'
  81. Translate(G, v_2) → G'
  82. Translate(R, v_2) → R'
  83. Translate(N, v_2) → N'
  84. Translate(S, v_2) → S'
  85. Translate(M, v_2) → M'
  86. Translate(T, v_2) → T'
  87. Translate(O, v_2) → O'
  88. Translate(U, v_2) → U'
  89. Translate(I, v_2) → I'_1
  90. Polygon(F', G', R', N', S', M', T', O', U', I'_1) → V', f', i'_1, g', r', n', s', m', t', o', u'
  91. Midpoint(I'_1, S_1) → W_1
  92. Translate(N_1, p_2) → N_2
  93. Translate(M_1, p_2) → M_1'
  94. Translate(L_1, p_2) → L_1'
  95. Translate(K_1, p_2) → K_1'
  96. Translate(J_1, p_2) → J_1'
  97. Translate(I_1, p_2) → I_1'
  98. Translate(H_1, p_2) → H_1'
  99. Translate(G_1, p_2) → G_1'
  100. Translate(F_1, p_2) → F_1'
  101. Translate(H, p_2) → H'
  102. Translate(C, p_2) → C'
  103. Translate(W, p_2) → W'
  104. Translate(E_1, p_2) → E_1'
  105. Translate(Z, p_2) → Z'
  106. Translate(D_1, p_2) → D_1'
  107. Translate(Y, p_2) → Y'
  108. Translate(C_1, p_2) → C_1'
  109. Translate(A_1, p_2) → A_1'
  110. Translate(B_1, p_2) → B_1'
  111. Translate(L, p_2) → L'_1
  112. Translate(R_1, p_2) → R_2
  113. Translate(S_1, p_2) → S_2
  114. Polygon(N_2, M_1', C', W', E_1', Z', D_1', Y', C_1', A_1', B_1', L'_1, L_1', R_2, S_2, K_1', J_1', I_1', H_1', G_1', F_1', H') → U_1', n_6, h', c', w', e_1', z', d_1', y', c_1', a_1', b_1', m_1', l'_1, r_5, s_4, l_1', k_1', j_1', i_1', h_1', g_1', f_1'
  115. Translate(D, u_3) → D'
  116. Translate(W, u_3) → W'_1
  117. Translate(E_1, u_3) → E_2
  118. Translate(Z, u_3) → Z'_1
  119. Translate(D_1, u_3) → D_2
  120. Translate(Y, u_3) → Y'_1
  121. Translate(C_1, u_3) → C_2
  122. Translate(A_1, u_3) → A_2
  123. Translate(B_1, u_3) → B_2
  124. Translate(J, u_3) → J'_1
  125. Polygon(D', W'_1, E_2, Z'_1, D_2, Y'_1, C_2, A_2, B_2, J'_1) → P_1', d', j'_1, w'_1, e_5, z'_1, d_5, y'_1, c_5, a_5, b_4
  126. Rotate(Q_1, ((-a) * 180°), W_1) → Q_1'
  127. Rotate(O_1, ((-a) * 180°), W_1) → O_1'
  128. Rotate(N_1, ((-a) * 180°), W_1) → N_1'
  129. Rotate(S_1, ((-a) * 180°), W_1) → S_1'
  130. Rotate(R_1, ((-a) * 180°), W_1) → R_1'
  131. Polygon(Q_1', O_1', N_1', S_1', R_1') → T_1', q_1', o_1', n_1', s_1', r_1'

AI Drawing Prompt Reference

Draw a chain-like geometric figure composed of multiple pentagons. Each pentagon is positioned using circle intersections, rays, and rotations. Symmetry is ensured through angle bisectors and reflections. The construction order is: first create circles and rays to find intersection points and rotate them; then use angle bisectors and reflections to generate new points; next construct the first pentagon with these points. Then use vectors and lines to find other intersections and construct the second pentagon. Finally, use midpoints and translations to generate larger pentagons. The final figure connects pentagons through shared geometric relationships, forming a coherent chain.
The prompt above was compiled from the original drawing commands and is for reference only.