MathPlotAI Logo MathPlotAI

Phase F: Octagon

Phase F: Octagon
Loading the math board and drawing, please wait…
2023-01-09 22:42
PolygonAngleGeometric Construction
In collections Polygon Angles: Interactive Exploration
This example demonstrates the construction and properties of a regular octagon. Key mathematical concepts include polygons, angles, transformations, and optimization problems. Construction involves using GeoGebra commands like Ray, Circle, Intersect, and objects such as Polygon, Point. Suitable for middle school and above for geometric learning and drawing practice.

GeoGebra Commands & Parameters

  1. Ray(S_1, D) → i_5
  2. Circle(D, 3) → k
  3. Intersect(k, i_5) → U_1, V_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. Mirror(r'_2, E) → p_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. Dilate(E, If[2 < l < 3, 3 - l, 0], A_1) → E'_2
  99. Dilate(C, If[2 < l < 3, 3 - l, 0], A_1) → C'_3
  100. Dilate(C, If[3 < l < 4, 4 - l, 0], D) → C'_4
  101. Dilate(A_1, If[3 < l < 4, 4 - l, 0], D) → H_1
  102. Vector(E, A_1) → f_1
  103. Vector(C, A_1) → g_1
  104. Translate(d, g_1) → d'
  105. Vector(E, E'_2) → a_2
  106. Vector(C, C'_3) → b_2
  107. Translate(d, b_2) → d'_1
  108. CircleSector(A_1, N, M) → f_3
  109. Intersect(q_3, a_3, 1) → I_1
  110. CircleSector(A_1, M, I_1) → g_3
  111. CircleSector(D, D_1, E_1) → k_3
  112. Vector(C, D) → j_2
  113. Vector(A_1, D) → u_3
  114. Translate(g_3, u_3) → p_3
  115. Translate(t, j_2) → t'_2
  116. Mirror(p_3, D) → r_3
  117. Vector(A_1, H_1) → v_3
  118. Vector(C, C'_4) → w_3
  119. Translate(g_3, v_3) → s_3
  120. Translate(t, w_3) → t'_3
  121. Intersect(p, b_3, 2) → J_1
  122. Polygon(A, B, C) → poly1, c_4, a_4, b_4
  123. Polygon(A, C, E) → poly2, e_4, a_5, c_5
  124. Polygon(E, C, A_1) → poly3, a_6, e_5, c_6
  125. Polygon(A_1, C, D) → poly4, d_4, a_7, c_7
  126. Intersect(p, e, 1) → L_1
  127. CircleSector(E, J_1, L_1) → f_4
  128. CircleSector(A_1, B_1, N) → g_4
  129. Vector(E, C) → j_3
  130. Vector(A_1, C) → l_3
  131. Vector(D, C) → m_3
  132. Translate(f_4, j_3) → k_4
  133. Translate(g_4, l_3) → p_4
  134. CircleSector(D, V_1, D_1) → h_4
  135. Translate(h_4, m_3) → q_4
  136. Dilate(E, If[2 < o < 3, 3 - o, 0], C) → E'_3
  137. Dilate(A_1, If[3 < o < 4, 4 - o, 0], C) → M_1
  138. Dilate(D, If[4 < o < 5, 5 - o, 0], C) → D'_1
  139. Vector(E, E'_3) → n_3
  140. Vector(A_1, M_1) → u_4
  141. Vector(D, D'_1) → v_4
  142. If(2 < o < 3, Translate[f_4, n_3]) → r_4
  143. If(3 < o < 4, Translate[g_4, u_4]) → s_4
  144. If(4 < o < 5, Translate[h_4, v_4]) → t_4
  145. Ray(C, N_1) → i_4
  146. Ray(N_1, D) → l_4
  147. Circle(N_1, 3) → d_5
  148. Intersect(d_5, i_4, 2) → K_1
  149. Intersect(d_5, l_4, 1) → O_1
  150. Intersect(d_5, i_4, 1) → P_1
  151. Intersect(h, i_4, 1) → Q_1
  152. Intersect(k, l_4, 1) → R_1
  153. Dilate(D, If[4 < l < 5, 5 - l, 0], N_1) → D'_2
  154. Dilate(C, If[4 < l < 5, 5 - l, 0], N_1) → C'_5
  155. Vector(C, N_1) → w_4
  156. Vector(D, N_1) → j_4
  157. Vector(D, D'_2) → m_4
  158. Vector(C, C'_5) → n_4
  159. Ray(N_1, S_1) → b_5
  160. Segment(D, C) → j_5
  161. Intersect(d_5, b_5) → T_1,
  162. Dilate(D, If[5 < l < 6, 6 - l, 0], S_1) → D'_3
  163. Dilate(N_1, If[5 < l < 6, 6 - l, 0], S_1) → W_1
  164. Vector(D, D'_3) → u_5
  165. Vector(N_1, W_1) → v_5
  166. Vector(D, S_1) → w_5
  167. Vector(N_1, S_1) → m_5
  168. Circle(S_1, 3) → r_6
  169. Intersect(r_6, i_5, 1) → Z_1
  170. Intersect(r_6, b_5) → A_2, B_2
  171. Polygon(A, B, C, N_1, S_1, D, A_1, E) → poly7, a_8, b_6, c_9, n_7, s_7, d_8, a_9, e_8
  172. CircleSector(C, Z, Q_1) → r_1
  173. CircleSector(W_1, B_2, Z_1) → h_7
  174. CircleSector(N_1, K_1, T_1) → k_7
  175. Vector(W_1, C) → u_6
  176. Vector(N_1, C) → v_6
  177. Translate(h_7, u_6) → p_7
  178. Vector(D, C) → w_6
  179. Translate(h_4, w_6) → q_7
  180. Dilate(W_1, If[5 < o < 6, 6 - o, 0], C) → C_2
  181. Vector(W_1, C_2) → i_6
  182. Translate(h_7, i_6) → r_7
  183. Dilate(N_1, If[6 < o < 7, 7 - o, 0], C) → D_2
  184. Vector(N_1, D_2) → j_6
  185. Translate(k_7, j_6) → t_7
  186. Vector(N_1, C) → l_6
  187. Translate(k_7, l_6) → f_8
  188. Ray(C, D) → m_6
  189. Ray(C, A_1) → b_7
  190. Ray(C, E) → i_7
  191. Point(i_7) → E_2
  192. Point(b_7) → F_2
  193. Point(m_6) → G_2
  194. Ray(E, E_2) → j_7
  195. Ray(A_1, F_2) → l_7
  196. Ray(D, G_2) → m_7
  197. Segment(C, S_1) → l_5
  198. Intersect(k, i_5, 1) → H_2
  199. Intersect(d_5, b_5, 1) → I_2
  200. CircleSector(N_1, I_2, P_1) → g_5
  201. Intersect(r_6, l_5) → J_2,
  202. Intersect(k, j_5) → K_2,
  203. CircleSector(S_1, Z_1, J_2) → f_7
  204. CircleSector(S_1, J_2, A_2) → f_5
  205. CircleSector(D, K_2, H_2) → h_5
  206. Intersect(h, j_5) → L_2,
  207. Intersect(h, l_5) → M_2,
  208. Intersect(h, i_4) → N_2,
  209. CircleSector(C, M_2, L_2) → k_5
  210. CircleSector(C, N_2, M_2) → p_5
  211. Dilate(D, If[4 < l < 5, 5 - l, 0], S_1) → D'_4
  212. Dilate(C, If[4 < l < 5, 5 - l, 0], S_1) → C'_6
  213. Vector(D, S_1) → n_5
  214. Vector(C, S_1) → q_5
  215. Translate(h_5, n_5) → r_5
  216. Vector(D, D'_4) → t_5
  217. Translate(h_5, t_5) → d_6
  218. Translate(k_5, q_5) → f_6
  219. Vector(C, C'_6) → g_6
  220. Translate(k_5, g_6) → h_6
  221. Dilate(S_1, If[5 < l < 6, 6 - l, 0], N_1) → O_2
  222. Dilate(C, If[5 < l < 6, 6 - l, 0], N_1) → C'_7
  223. Vector(C, N_1) → k_6
  224. Vector(C, C'_7) → n_6
  225. Vector(S_1, N_1) → s_6
  226. Translate(f_5, s_6) → t_6
  227. Vector(S_1, O_2) → u_7
  228. Translate(f_5, u_7) → e_7
  229. Ray(C, S_1) → b_8
  230. Polygon(D, C, S_1) → poly5, s_8, d_9, c_8
  231. Polygon(S_1, C, N_1) → poly6, n_8, s_9, c_{10}
  232. Segment(A, B) → g_8
  233. Segment(B, C) → h_8
  234. Segment(C, N_1) → i_8
  235. Segment(N_1, S_1) → j_8
  236. Segment(S_1, D) → k_8
  237. Segment(D, A_1) → l_8
  238. Segment(A_1, E) → m_8
  239. Segment(E, A) → p_8
  240. Midpoint(g_8) → P_2
  241. Midpoint(h_8) → Q_2
  242. Midpoint(i_8) → R_2
  243. Midpoint(j_8) → S_2
  244. Midpoint(p_8) → T_2
  245. Midpoint(m_8) → U_2
  246. Midpoint(l_8) → V_2
  247. Midpoint(k_8) → W_2
  248. Rotate(q, If[0 < l ≤ 1, (l * 180°), 180°], P_2) → q'_1
  249. Rotate(A, If[0 < l ≤ 1, (l * 180°), 180°], P_2) → A'_3
  250. Segment(P_2, A'_3) → k_2
  251. Rotate(d_1, If[0 < l ≤ 1, ((-l) * 180°), -180°], Q_2) → t_1
  252. Rotate(C, If[0 < l ≤ 1, ((-l) * 180°), -180°], Q_2) → C'_8
  253. Segment(Q_2, C'_8) → q_8
  254. Rotate(R, 3.141592653589793, Q_2) → R'
  255. Line(B, R') → r_8
  256. Rotate(r, If[1 < l ≤ 2, ((-(l - 1)) * 180°), -180°], T_2) → r'_3
  257. Rotate(A, If[1 < l ≤ 2, ((-(l - 1)) * 180°), -180°], T_2) → A'_4
  258. Segment(T_2, A'_4) → e_1
  259. CircleSector(E, K, F_1) → k_1
  260. Rotate(k_1, If[2 < l ≤ 3, ((-(l - 2)) * 180°), -180°], U_2) → d_2
  261. Rotate(E, If[2 < l ≤ 3, ((-(l - 2)) * 180°), -180°], U_2) → E'_4
  262. Segment(U_2, E'_4) → e_2
  263. Rotate(g_3, If[3 < l ≤ 4, ((-(l - 3)) * 180°), -180°], V_2) → q_2
  264. Rotate(A_1, If[3 < l ≤ 4, ((-(l - 3)) * 180°), -180°], V_2) → G_1
  265. Segment(V_2, G_1) → i_2
  266. Intersect(l_8, i_2) → Z_2
  267. Intersect(l_8, i_2) → A_3
  268. Rotate(h_5, If[4 < l ≤ 5, ((-(l - 4)) * 180°), -180°], W_2) → r_2
  269. Rotate(D, If[4 < l ≤ 5, ((-(l - 4)) * 180°), -180°], W_2) → D'_5
  270. Segment(W_2, D'_5) → t_3
  271. Translate(M_2, g_6) → B_3
  272. Ray(S_1, B_3) → s_5
  273. Rotate(p_5, If[5 < l ≤ 6, ((l - 5) * 180°), 180°], R_2) → e_6
  274. Rotate(C, If[5 < l ≤ 6, ((l - 5) * 180°), 180°], R_2) → C'_9
  275. Segment(R_2, C'_9) → p_6
  276. Rotate(f_5, If[5 < l ≤ 6, ((-(l - 5)) * 180°), -180°], S_2) → q_6
  277. Rotate(S_1, If[5 < l ≤ 6, ((-(l - 5)) * 180°), -180°], S_2) → C_3
  278. Segment(S_2, C_3) → d_7
  279. Rotate(J_2, 3.141592653589793, S_2) → D_3
  280. Line(N_1, D_3) → g_7

AI Drawing Prompt Reference

Draw a regular octagon by first drawing a circle centered at point D, then finding intersections with rays to get points U and V. Continue this process using points and rays to construct other points, and connect them to form the octagon. Pay attention to angles and symmetry.
The prompt above was compiled from the original drawing commands and is for reference only.