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Phase E: Heptagon

Phase E: Heptagon
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2023-01-09 22:40
PolygonAngleVectorGeometric Construction
In collections Polygon Angles: Interactive Exploration
This example demonstrates constructing a heptagon using GeoGebra and highlighting its geometric properties. Key mathematical concepts include polygons, angles, vectors, and transformations. Construction involves commands like Ray, Circle, Intersect, and Polygon, along with objects such as Point and Midpoint. Suitable for middle school students learning geometric constructions and parametric equations.

GeoGebra Commands & Parameters

  1. Ray(N_1, D) → l_4
  2. Circle(D, 3) → k
  3. Intersect(k, l_4) → W_1, Z_1
  4. Ray(D, A_1) → a_3
  5. Intersect(k, a_3, 1) → D_1
  6. Segment(C, A_1) → i_3
  7. Circle(C, 3) → h
  8. Intersect(h, i_3) → L,
  9. Ray(C, D) → c
  10. Intersect(h, c, 1) → P
  11. Ray(A, B) → a
  12. Ray(B, C) → b
  13. Ray(E, A) → e
  14. Circle(A, 3) → f
  15. Circle(B, 3) → g
  16. Circle(E, 3) → p
  17. Segment(C, A) → i
  18. Segment(C, E) → j
  19. Intersect(f, a, 1) → F
  20. Intersect(f, i, 1) → G
  21. Intersect(f, e, 1) → H
  22. Intersect(f, e, 2) → I
  23. Intersect(p, e) → J,
  24. Intersect(p, j) → K,
  25. Intersect(k, c, 2) → O
  26. Intersect(h, j, 1) → Q
  27. Intersect(h, i, 1) → R
  28. Intersect(h, b, 1) → S
  29. Intersect(g, a, 2) → T
  30. Intersect(g, b, 1) → U
  31. Intersect(g, a, 1) → V
  32. CircleSector(A, F, G) → q
  33. CircleSector(A, G, H) → r
  34. CircleSector(B, U, V) → s
  35. CircleSector(C, Q, R) → c_1
  36. CircleSector(C, R, S) → d_1
  37. Intersect(k, c, 1) → W
  38. CircleSector(E, J, K) → h_1
  39. CircleSector(A, I, F) → p_1
  40. CircleSector(B, T, U) → q_1
  41. Intersect(h, b, 2) → Z
  42. Vector(A, B) → u
  43. Vector(C, B) → v
  44. Vector(E, D) → w
  45. Vector(C, D) → u_1
  46. Vector(C, E) → v_1
  47. Vector(A, E) → w_1
  48. Translate(d_1, v) → s_1
  49. Dilate(A, If[0 < l ≤ 1, 1 - l, 0], B) → A'
  50. Dilate(C, If[0 < l ≤ 1, 1 - l, 0], B) → C'
  51. Dilate(A, If[1 < l ≤ 2, 2 - l, 0], E) → A'_1
  52. Dilate(C, If[1 < l ≤ 2, 2 - l, 0], E) → C'_1
  53. Dilate(C, If[2 < l ≤ 3, 3 - l, 0], D) → C'_2
  54. Dilate(E, If[2 < l ≤ 3, 3 - l, 0], D) → E'
  55. Vector(A, B) → a_1
  56. Vector(C, C') → b_1
  57. Vector(A, A'_1) → i_1
  58. Vector(E, E') → j_1
  59. Vector(C, C'_2) → l_1
  60. Vector(C, C'_1) → m_1
  61. Translate(d_1, b_1) → c_2
  62. Translate(r, i_1) → r'_1
  63. Translate(r, w_1) → r'_2
  64. Translate(c_1, v_1) → f_2
  65. Translate(c_1, m_1) → g_2
  66. CircleSector(C, P, L) → t
  67. Translate(t, u_1) → t'
  68. Translate(t, l_1) → t'_1
  69. Translate(q, a_1) → r'
  70. Translate(d_1, b_1) → h_2
  71. Vector(A, A') → n_1
  72. Ray(C, E) → i_2
  73. Vector(D, C) → u_2
  74. Vector(E, C) → v_2
  75. Vector(A, C) → w_2
  76. Vector(B, C) → l_2
  77. Dilate(B, If[0 < o < 1, 1 - o, 0], C) → B'
  78. Dilate(A, If[1 < o < 2, 2 - o, 0], C) → A'_2
  79. Dilate(E, If[2 < o < 3, 3 - o, 0], C) → E'_1
  80. Dilate(D, If[3 < o < 4, 4 - o, 0], C) → D'
  81. Vector(B, B') → m_2
  82. Vector(A, A'_2) → n_2
  83. Vector(E, E'_1) → s_2
  84. Vector(D, D') → t_2
  85. Translate(q_1, m_2) → c_3
  86. Translate(q_1, l_2) → d_3
  87. Translate(p_1, w_2) → e_3
  88. If(1 < o < 2, Translate[p_1, n_2]) → h_3
  89. Ray(A_1, E) → b_3
  90. Circle(A_1, 3) → q_3
  91. Intersect(q_3, i_3) → M,
  92. Intersect(q_3, b_3, 1) → N
  93. Intersect(q_3, a_3, 2) → B_1
  94. Intersect(k, c, 2) → C_1
  95. Intersect(k, c, 1) → E_1
  96. CircleSector(C, L, Q) → d
  97. Intersect(p, b_3, 1) → F_1
  98. Intersect(p, i_2) → G_1, H_2
  99. CircleSector(E, G_1, N) → e_1
  100. Dilate(E, If[2 < l < 3, 3 - l, 0], A_1) → E'_2
  101. Dilate(C, If[2 < l < 3, 3 - l, 0], A_1) → C'_3
  102. Dilate(C, If[3 < l < 4, 4 - l, 0], D) → C'_4
  103. Dilate(A_1, If[3 < l < 4, 4 - l, 0], D) → H_1
  104. Vector(E, A_1) → f_1
  105. Vector(C, A_1) → g_1
  106. Translate(e_1, f_1) → k_1
  107. Translate(d, g_1) → d'
  108. Vector(E, E'_2) → a_2
  109. Vector(C, C'_3) → b_2
  110. Translate(e_1, a_2) → d_2
  111. Translate(d, b_2) → d'_1
  112. CircleSector(A_1, N, M) → f_3
  113. Intersect(q_3, a_3, 1) → I_1
  114. CircleSector(A_1, M, I_1) → g_3
  115. CircleSector(D, D_1, E_1) → k_3
  116. Vector(C, D) → j_2
  117. Vector(A_1, D) → u_3
  118. Translate(g_3, u_3) → p_3
  119. Translate(t, j_2) → t'_2
  120. Vector(A_1, H_1) → v_3
  121. Vector(C, C'_4) → w_3
  122. Translate(g_3, v_3) → s_3
  123. Translate(t, w_3) → t'_3
  124. Intersect(p, b_3, 2) → J_1
  125. Polygon(A, B, C) → poly1, c_4, a_4, b_4
  126. Polygon(A, C, E) → poly2, e_4, a_5, c_5
  127. Polygon(E, C, A_1) → poly3, a_6, e_5, c_6
  128. Polygon(A_1, C, D) → poly4, d_4, a_7, c_7
  129. Intersect(p, e, 1) → L_1
  130. CircleSector(E, J_1, L_1) → f_4
  131. CircleSector(A_1, B_1, N) → g_4
  132. Vector(E, C) → j_3
  133. Vector(A_1, C) → l_3
  134. Vector(D, C) → m_3
  135. Translate(f_4, j_3) → k_4
  136. Translate(g_4, l_3) → p_4
  137. CircleSector(D, Z_1, D_1) → h_4
  138. Translate(h_4, m_3) → q_4
  139. Dilate(E, If[2 < o < 3, 3 - o, 0], C) → E'_3
  140. Dilate(A_1, If[3 < o < 4, 4 - o, 0], C) → M_1
  141. Dilate(D, If[4 < o < 5, 5 - o, 0], C) → D'_1
  142. Vector(E, E'_3) → n_3
  143. Vector(A_1, M_1) → u_4
  144. Vector(D, D'_1) → v_4
  145. If(2 < o < 3, Translate[f_4, n_3]) → r_4
  146. If(3 < o < 4, Translate[g_4, u_4]) → s_4
  147. If(4 < o < 5, Translate[h_4, v_4]) → t_4
  148. Ray(C, N_1) → i_4
  149. Circle(N_1, 3) → d_5
  150. Intersect(d_5, i_4, 2) → K_1
  151. Intersect(d_5, l_4, 1) → O_1
  152. Intersect(d_5, i_4, 1) → P_1
  153. Intersect(h, i_4, 1) → Q_1
  154. CircleSector(C, Q_1, P) → f_5
  155. CircleSector(N_1, O_1, P_1) → g_5
  156. Intersect(k, l_4, 1) → R_1
  157. CircleSector(D, E_1, R_1) → h_5
  158. Dilate(D, If[4 < l < 5, 5 - l, 0], N_1) → D'_2
  159. Dilate(C, If[4 < l < 5, 5 - l, 0], N_1) → C'_5
  160. Vector(C, N_1) → w_4
  161. Vector(D, N_1) → j_4
  162. Translate(h_5, j_4) → p_5
  163. Vector(D, D'_2) → m_4
  164. Vector(C, C'_5) → n_4
  165. Translate(h_5, m_4) → r_5
  166. Polygon(D, C, N_1) → poly5, n_5, d_6, c_8
  167. Segment(D, C) → b_5
  168. Ray(C, E) → i_5
  169. Ray(C, A_1) → j_5
  170. Ray(C, D) → l_5
  171. Point(j_5) → T_1
  172. Point(l_5) → U_1
  173. Ray(A_1, T_1) → m_5
  174. Ray(D, U_1) → b_6
  175. Polygon(A_1, E, A, B, C, N_1, D) → poly6, a_8, e_6, a_9, b_7, c_9, n_6, d_7
  176. Intersect(d_5, l_4, 1) → S_1
  177. Intersect(d_5, i_4, 2) → V_1
  178. CircleSector(N_1, V_1, S_1) → f_6
  179. CircleSector(C, Z, Q_1) → r_1
  180. Vector(N_1, C) → u_5
  181. Translate(f_6, u_5) → g_6
  182. Dilate(N_1, If[5 < o < 6, 6 - o, 0], C) → A_2
  183. Vector(N_1, A_2) → v_5
  184. Translate(f_6, v_5) → h_6
  185. Vector(D, C) → w_5
  186. Translate(h_4, w_5) → k_6
  187. Segment(E, A) → i_6
  188. Segment(A, B) → j_6
  189. Segment(B, C) → l_6
  190. Segment(C, N_1) → m_6
  191. Segment(N_1, D) → p_6
  192. Segment(D, A_1) → q_6
  193. Segment(A_1, E) → r_6
  194. Midpoint(i_6) → B_2
  195. Midpoint(j_6) → C_2
  196. Midpoint(l_6) → D_2
  197. Midpoint(m_6) → E_2
  198. Midpoint(p_6) → F_2
  199. Midpoint(q_6) → G_2
  200. Midpoint(r_6) → I_2
  201. Rotate(d_1, If[0 < l ≤ 1, ((-l) * 180°), -180°], D_2) → t_1
  202. Rotate(C, If[0 < l ≤ 1, ((-l) * 180°), -180°], D_2) → C'_6
  203. Segment(D_2, C'_6) → k_2
  204. Rotate(A, If[0 < l ≤ 1, (l * 180°), 180°], C_2) → A'_3
  205. Segment(C_2, A'_3) → s_6
  206. Rotate(q, If[0 < l ≤ 1, (l * 180°), 180°], C_2) → q'
  207. Rotate(R, 180°, C_2) → R'
  208. Rotate(R, 180°, D_2) → R'_1
  209. Line(B, R'_1) → t_6
  210. Rotate(A, If[1 < l ≤ 2, ((-(l - 1)) * 180°), 180°], B_2) → A'_4
  211. Rotate(r, If[1 < l ≤ 2, ((-(l - 1)) * 180°), 180°], B_2) → r'_3
  212. Segment(B_2, A'_4) → q_2
  213. Ray(E, H_2) → e_7
  214. Rotate(e_1, If[2 < l ≤ 3, ((-(l - 2)) * 180°), 180°], I_2) → e_2
  215. Rotate(E, If[2 < l ≤ 3, ((-(l - 2)) * 180°), 180°], I_2) → E'_4
  216. Segment(I_2, E'_4) → f_7
  217. Rotate(g_3, If[3 < l ≤ 4, ((-(l - 3)) * 180°), 180°], G_2) → t_3
  218. Rotate(A_1, If[3 < l ≤ 4, ((-(l - 3)) * 180°), 180°], G_2) → J_2
  219. Segment(G_2, J_2) → g_7
  220. Rotate(h_5, If[4 < l ≤ 5, ((-(l - 4)) * 180°), 180°], F_2) → t_5
  221. Rotate(D, If[4 < l ≤ 5, ((-(l - 4)) * 180°), 180°], F_2) → D'_3
  222. Segment(F_2, D'_3) → h_7
  223. Rotate(f_5, If[4 < l ≤ 5, ((l - 4) * 180°), 180°], E_2) → s_5
  224. Rotate(C, If[4 < l ≤ 5, ((l - 4) * 180°), 180°], E_2) → C'_7
  225. Segment(E_2, C'_7) → i_7
  226. Rotate(W, 180°, F_2) → W'
  227. Line(N_1, W') → k_5

AI Drawing Prompt Reference

Create a geometric construction of a heptagon in the plane by using several congruent circles centered at existing key points. Start with a base circle centered at D, and use rays from D through given directions to get intersection points on that circle. Introduce additional auxiliary circles centered at C and at other provided points (such as A, B, and E) and compute their intersections with rays or segments to obtain the required vertices. Mark the construction’s crucial angles by drawing several circle-sector regions: use sectors around A, B, and C formed by specific intersection points, and also around E to emphasize the intended angle partition and rotational relationships. Connect the resulting key points on the relevant circles to form the heptagon boundary, and apply a translation to one marked sector to visually reinforce the symmetry/shift between configurations.
The prompt above was compiled from the original drawing commands and is for reference only.