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Complex Shape Construction

Complex Shape Construction
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2019-04-06 15:26
PolygonCoordinate GeometryTransformationGeometric Construction
This example demonstrates the construction and manipulation of complex polygons and geometric transformations using GeoGebra. It involves core mathematical concepts such as polygon construction, angle bisectors, rotations, reflections, and coordinate geometry. Key GeoGebra commands and objects used include Polygon, LineBisector, Intersect, Circle, Point, Segment, Rotate, Line, and Mirror. Suitable for high school and above, this example helps learners understand geometric transformations and complex shape constructions.

GeoGebra Commands & Parameters

  1. Polygon(A, B, 8) → poly1, f, D, E, F, G, H, g, h, i, j, k, l, m, C
  2. LineBisector(G, H) → n
  3. LineBisector(H, A) → p
  4. LineBisector(A, B) → q
  5. LineBisector(B, C) → r
  6. Intersect(n, l) → I
  7. Intersect(r, k) → J
  8. Intersect(q, j) → K
  9. Intersect(p, i) → L
  10. Intersect(n, h) → M
  11. Intersect(r, g) → N
  12. Intersect(q, f) → O
  13. Intersect(p, m) → P
  14. Intersect(n, p) → Q
  15. Circle(Q, M) → c
  16. Point(n) → R
  17. Circle(Q, R) → d
  18. Intersect(d, q, 2) → S
  19. Intersect(d, n, 2) → T
  20. Intersect(d, r, 1) → U
  21. Intersect(d, q, 1) → V
  22. Intersect(d, p, 1) → W
  23. Intersect(d, r, 2) → Z
  24. Intersect(d, p, 2) → A_1
  25. Segment(G, Z) → s
  26. Segment(Z, F) → t
  27. Segment(F, S) → a
  28. Segment(S, E) → b
  29. Segment(E, A_1) → e
  30. Segment(A_1, D) → f_1
  31. Segment(T, D) → g_1
  32. Segment(T, C) → h_1
  33. Segment(U, C) → i_1
  34. Segment(U, B) → j_1
  35. Segment(V, B) → k_1
  36. Segment(V, A) → l_1
  37. Segment(W, A) → m_1
  38. Segment(H, W) → n_1
  39. Segment(R, H) → p_1
  40. Segment(R, G) → q_1
  41. Polygon(G, Z, B, V, A, W, H, R, F, S, E, A_1, D, T, C, U) → poly2, g_2, u, b_1, v, a_2, w, h_2, r_1, z_1, f_2, s_1, e_1, a_1, d_1, t_1, c_1
  42. Point(Circle[B_1, Segment[G, Z]]) → C_1
  43. Rotate(C_1 + (Segment[Z, F], 0), Angle[F - Z] + Angle[C_1 - B_1] - Angle[Z - G], C_1) → D_1
  44. Rotate(D_1 + (Segment[F, S], 0), Angle[S - F] + Angle[D_1 - C_1] - Angle[F - Z], D_1) → E_1
  45. Rotate(E_1 + (Segment[S, E], 0), Angle[E - S] + Angle[E_1 - D_1] - Angle[S - F], E_1) → F_1
  46. Rotate(F_1 + (Segment[E, A_1], 0), Angle[A_1 - E] + Angle[F_1 - E_1] - Angle[E - S], F_1) → G_1
  47. Rotate(G_1 + (Segment[A_1, D], 0), Angle[D - A_1] + Angle[G_1 - F_1] - Angle[A_1 - E], G_1) → H_1
  48. Rotate(H_1 + (Segment[D, T], 0), Angle[T - D] + Angle[H_1 - G_1] - Angle[D - A_1], H_1) → I_1
  49. Rotate(I_1 + (Segment[T, C], 0), Angle[C - T] + Angle[I_1 - H_1] - Angle[T - D], I_1) → J_1
  50. Rotate(J_1 + (Segment[C, U], 0), Angle[U - C] + Angle[J_1 - I_1] - Angle[C - T], J_1) → K_1
  51. Rotate(K_1 + (Segment[U, B], 0), Angle[B - U] + Angle[K_1 - J_1] - Angle[U - C], K_1) → L_1
  52. Rotate(L_1 + (Segment[B, V], 0), Angle[V - B] + Angle[L_1 - K_1] - Angle[B - U], L_1) → M_1
  53. Rotate(M_1 + (Segment[V, A], 0), Angle[A - V] + Angle[M_1 - L_1] - Angle[V - B], M_1) → N_1
  54. Rotate(N_1 + (Segment[A, W], 0), Angle[W - A] + Angle[N_1 - M_1] - Angle[A - V], N_1) → O_1
  55. Rotate(O_1 + (Segment[W, H], 0), Angle[H - W] + Angle[O_1 - N_1] - Angle[W - A], O_1) → P_1
  56. Rotate(P_1 + (Segment[H, R], 0), Angle[R - H] + Angle[P_1 - O_1] - Angle[H - W], P_1) → Q_1
  57. Polygon(A_3, B_2, 8) → poly1_1, f_4, D_2, E_2, F_2, G_2, H_2, g_4, h_4, i_2, j_2, k_2, l_2, m_2, C_2
  58. LineBisector(G_2, H_2) → n_2
  59. LineBisector(H_2, A_3) → p_2
  60. LineBisector(A_3, B_2) → q_2
  61. LineBisector(B_2, C_2) → r_3
  62. Intersect(n_2, p_2) → Q_2
  63. Point(n_2) → R_1
  64. Circle(Q_2, R_1) → d_3
  65. Intersect(d_3, q_2, 2) → S_1
  66. Intersect(d_3, n_2, 2) → T_1
  67. Intersect(d_3, r_3, 1) → U_1
  68. Intersect(d_3, q_2, 1) → V_1
  69. Intersect(d_3, p_2, 1) → W_1
  70. Intersect(d_3, r_3, 2) → Z_1
  71. Intersect(d_3, p_2, 2) → A_2
  72. Segment(G_2, Z_1) → s_3
  73. Segment(Z_1, F_2) → t_3
  74. Segment(F_2, S_1) → a_5
  75. Segment(S_1, E_2) → b_3
  76. Segment(E_2, A_2) → e_3
  77. Segment(A_2, D_2) → f_5
  78. Segment(T_1, D_2) → g_5
  79. Segment(T_1, C_2) → h_5
  80. Segment(U_1, C_2) → i_3
  81. Segment(U_1, B_2) → j_3
  82. Segment(V_1, B_2) → k_3
  83. Segment(V_1, A_3) → l_3
  84. Segment(W_1, A_3) → m_3
  85. Segment(H_2, W_1) → n_3
  86. Segment(R_1, H_2) → p_3
  87. Segment(R_1, G_2) → q_3
  88. Polygon(G_2, Z_1, B_2, V_1, A_3, W_1, H_2, R_1, F_2, S_1, E_2, A_2, D_2, T_1, C_2, U_1) → poly2_1, g_3, u_1, b_2, v_1, a_4, w_1, h_3, r_2, z_2, f_3, s_2, e_2, a_3, d_2, t_2, c_2
  89. Intersect(f_1, a_1) → I_2
  90. Intersect(f_1, a_1) → J_2
  91. Rotate(G, 22.5°, Q) → G'
  92. Rotate(Z, 22.5°, Q) → Z'
  93. Rotate(F, 22.5°, Q) → F'
  94. Rotate(S, 22.5°, Q) → S'
  95. Rotate(E, 22.5°, Q) → E'
  96. Rotate(A_1, 22.5°, Q) → K_2
  97. Rotate(D, 22.5°, Q) → D'
  98. Rotate(T, 22.5°, Q) → T'
  99. Rotate(C, 22.5°, Q) → C'
  100. Rotate(U, 22.5°, Q) → U'
  101. Rotate(B, 22.5°, Q) → B'
  102. Rotate(V, 22.5°, Q) → V'
  103. Rotate(A, 22.5°, Q) → A'
  104. Rotate(W, 22.5°, Q) → W'
  105. Rotate(H, 22.5°, Q) → H'
  106. Rotate(R, 22.5°, Q) → R'
  107. Polygon(G', Z', B', V', A', W', H', R', F', S', E', K_2, D', T', C', U') → poly2', g', u', b', v', a', w', h', r', z', f', s', e', c_3, d', t', c'
  108. Line(L_2, M_2) → i_4
  109. Mirror(G_2, i_4) → N_2
  110. Mirror(Z_1, i_4) → O_2
  111. Mirror(F_2, i_4) → P_2
  112. Mirror(S_1, i_4) → R_2
  113. Mirror(E_2, i_4) → S_2
  114. Mirror(A_2, i_4) → T_2
  115. Mirror(D_2, i_4) → U_2
  116. Mirror(T_1, i_4) → V_2
  117. Mirror(C_2, i_4) → W_2
  118. Mirror(U_1, i_4) → Z_2
  119. Mirror(B_2, i_4) → B_3
  120. Mirror(V_1, i_4) → C_3
  121. Mirror(A_3, i_4) → D_3
  122. Mirror(W_1, i_4) → E_3
  123. Mirror(H_2, i_4) → F_3
  124. Mirror(R_1, i_4) → G_3
  125. Polygon(N_2, O_2, B_3, C_3, D_3, E_3, F_3, G_3, P_2, R_2, S_2, T_2, U_2, V_2, W_2, Z_2) → poly3, j_4, p_4, q_4, r_4, s_4, t_4, b_4, c_4, o_2, k_4, l_4, m_4, n_4, u_2, v_2, w_2
  126. Segment(H_3, I_3) → d_4
  127. Mirror(N_2, d_4) → J_3
  128. Mirror(O_2, d_4) → K_3
  129. Mirror(P_2, d_4) → L_3
  130. Mirror(R_2, d_4) → M_3
  131. Mirror(S_2, d_4) → N_3
  132. Mirror(T_2, d_4) → O_3
  133. Mirror(U_2, d_4) → P_3
  134. Mirror(V_2, d_4) → Q_3
  135. Mirror(W_2, d_4) → R_3
  136. Mirror(Z_2, d_4) → S_3
  137. Mirror(B_3, d_4) → T_3
  138. Mirror(C_3, d_4) → U_3
  139. Mirror(D_3, d_4) → V_3
  140. Mirror(E_3, d_4) → W_3
  141. Mirror(F_3, d_4) → Z_3
  142. Mirror(G_3, d_4) → A_4
  143. Polygon(J_3, K_3, T_3, U_3, V_3, W_3, Z_3, A_4, L_3, M_3, N_3, O_3, P_3, Q_3, R_3, S_3) → poly3', e_4, q_5, r_5, u_3, v_3, w_3, z_3, s_5, i_5, j_5, k_5, l_5, o_3, m_5, n_5, p_5
  144. Mirror(G_2, d_4) → B_4
  145. Mirror(Z_1, d_4) → C_4
  146. Mirror(F_2, d_4) → D_4
  147. Mirror(S_1, d_4) → E_4
  148. Mirror(E_2, d_4) → F_4
  149. Mirror(A_2, d_4) → G_4
  150. Mirror(D_2, d_4) → H_4
  151. Mirror(T_1, d_4) → I_4
  152. Mirror(C_2, d_4) → J_4
  153. Mirror(U_1, d_4) → K_4
  154. Mirror(B_2, d_4) → L_4
  155. Mirror(V_1, d_4) → M_4
  156. Mirror(A_3, d_4) → N_4
  157. Mirror(W_1, d_4) → O_4
  158. Mirror(H_2, d_4) → P_4
  159. Mirror(R_1, d_4) → Q_4
  160. Polygon(B_4, C_4, L_4, M_4, N_4, O_4, P_4, Q_4, D_4, E_4, F_4, G_4, H_4, I_4, J_4, K_4) → poly4, t_5, i_6, j_6, k_6, l_6, o_4, m_6, n_6, b_5, c_5, d_5, e_5, f_6, g_6, ı_4, h_6

AI Drawing Prompt Reference

Draw a complex polygon by constructing angle bisectors and rotating lines. Start by drawing an octagon, then find intersections using angle bisectors, construct circles and segments with these intersections, and finally generate new points and segments through rotations to form a complex polygon.
The prompt above was compiled from the original drawing commands and is for reference only.