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凸多边形外角和的无字证明

凸多边形外角和的无字证明
几何画板及绘图内容加载中，请稍等……
2015-07-30 12:32
多边形角变换几何作图
所属专题 Polygons & Angles Polygons & Angles Polygons & Angles ( s n attar) Polygons & Angles Polygons & Angles Polygons & Angles Polygons & Angles Geometry: Polygons & Angles Geometry: Polygons & Angles
本范例通过动态交互演示了凸多边形所有外角之和恒等于 360 度的几何现象。其核心数学概念在于多边形外角定义及其度量之和的恒定性。在作图实现上，利用了 Ray（射线）构建外角，通过 Vector（向量）和 Translate（平移）将各外角统一平移至同一顶点，并结合 Segment（线段）、Point（点）以及 Intersect（交点）等命令构建闭合的圆周结构，从而直观地将外角之和转化为一个完整的圆。该资源非常适合初中几何教学，用于引导学生通过观察而非繁琐的代数推导来理解多边形性质。

GeoGebra 命令及参数

  1. Ray(A, B) → a
  2. Ray(B, C) → b
  3. Ray(C, A) → c
  4. Vector(B, D) → u
  5. Vector(C, D) → v
  6. Vector(A, D) → w
  7. Point(a) → E
  8. Point(b) → F
  9. Point(b) → G
  10. Point(c) → H
  11. Point(c) → I
  12. Point(a) → J
  13. Segment(B, E) → e
  14. Segment(B, F) → f
  15. Segment(C, G) → g
  16. Segment(C, H) → h
  17. Segment(A, I) → i
  18. Segment(A, J) → j
  19. Translate(B, u) → B'_1
  20. Distance(B, D) → distanceBD
  21. Point(a) → K
  22. Point(b) → L
  23. Segment(B, K) → l
  24. Segment(B, L) → m
  25. Translate(B, u) → B'
  26. Translate(K, u) → K'
  27. Segment(B', K') → l'
  28. Translate(B, u) → B'_2
  29. Translate(L, u) → L'
  30. Segment(B'_2, L') → m'
  31. Translate(C, v) → C'
  32. Translate(G, v) → G'
  33. Segment(C', G') → g'
  34. Translate(C, v) → C'_1
  35. Translate(H, v) → H'
  36. Segment(C'_1, H') → h'
  37. Translate(A, w) → A'
  38. Translate(I, w) → I'
  39. Segment(A', I') → i'
  40. Translate(A, w) → A'_1
  41. Translate(J, w) → J'
  42. Segment(A'_1, J') → j'
  43. Ray(D, H') → p_1
  44. Circle(D, H') → q_1
  45. Ray(D, G') → r_1
  46. Intersect(q_1, r_1) → M,
  47. Ray(D, K') → s_1
  48. Intersect(q_1, s_1) → N,
  49. CircleSector(D, G', H') → t
  50. Circle(D, M) → d_1
  51. Circle(A, 1) → f_1
  52. Circle(B, 1) → g_1
  53. Circle(C, 1) → h_1
  54. Intersect(h_1, b, 2) → O
  55. Intersect(h_1, c, 1) → P
  56. Intersect(f_1, c, 2) → Q
  57. Intersect(f_1, a, 1) → R
  58. Intersect(g_1, a, 2) → S
  59. Intersect(g_1, b, 1) → T
  60. CircleSector(A, Q, R) → ext_3
  61. CircleSector(B, S, T) → ext_1
  62. CircleSector(C, O, P) → ext_2
  63. Translate(ext_3, w) → k'
  64. Translate(ext_1, u) → k_1
  65. Translate(ext_2, v) → ext_4
  66. Ray(U, V) → a_1
  67. Ray(V, W) → b_1
  68. Ray(W, Z) → c_1
  69. Ray(Z, U) → e_1
  70. Circle(V, 1) → t_1
  71. Circle(W, 1) → c_2
  72. Circle(Z, 1) → d_2
  73. Circle(U, 1) → e_2
  74. Intersect(t_1, a_1, 2) → A_1
  75. Intersect(t_1, b_1, 1) → B_1
  76. Intersect(c_2, b_1, 2) → C_1
  77. Intersect(c_2, c_1, 1) → D_1
  78. Intersect(d_2, c_1, 2) → E_1
  79. Intersect(d_2, e_1, 1) → F_1
  80. Intersect(e_2, e_1, 2) → G_1
  81. Intersect(e_2, a_1, 1) → H_1
  82. CircleSector(V, A_1, B_1) → f_2
  83. CircleSector(W, C_1, D_1) → g_2
  84. CircleSector(Z, E_1, F_1) → h_2
  85. CircleSector(U, G_1, H_1) → k_2
  86. Vector(U, I_1) → u_1
  87. Vector(V, I_1) → v_1
  88. Vector(W, I_1) → w_1
  89. Vector(Z, I_1) → i_1
  90. Translate(k_2, u_1) → p_2
  91. Translate(f_2, v_1) → q_2
  92. Translate(g_2, w_1) → r_2
  93. Translate(h_2, i_1) → s_2
  94. Ray(J_1, K_1) → d
  95. Ray(K_1, L_1) → a_2
  96. Ray(L_1, M_1) → b_2
  97. Ray(M_1, N_1) → i_2
  98. Ray(N_1, J_1) → j_2
  99. Circle(J_1, 1) → t_2
  100. Circle(K_1, 1) → c_3
  101. Circle(L_1, 1) → d_3
  102. Circle(M_1, 1) → e_3
  103. Circle(N_1, 1) → f_3
  104. Intersect(t_2, j_2, 2) → O_1
  105. Intersect(t_2, d, 1) → P_1
  106. Intersect(c_3, d, 2) → Q_1
  107. Intersect(c_3, a_2, 1) → R_1
  108. Intersect(d_3, a_2, 2) → S_1
  109. Intersect(d_3, b_2, 1) → T_1
  110. Intersect(e_3, b_2, 2) → U_1
  111. Intersect(e_3, i_2, 1) → V_1
  112. Intersect(f_3, i_2, 2) → W_1
  113. Intersect(f_3, j_2, 1) → Z_1
  114. CircleSector(J_1, O_1, P_1) → g_3
  115. CircleSector(K_1, Q_1, R_1) → h_3
  116. CircleSector(L_1, S_1, T_1) → k_3
  117. CircleSector(M_1, U_1, V_1) → p_3
  118. CircleSector(N_1, W_1, Z_1) → q_3
  119. Vector(J_1, A_2) → u_2
  120. Vector(K_1, A_2) → v_2
  121. Vector(L_1, A_2) → w_2
  122. Vector(M_1, A_2) → l_2
  123. Vector(N_1, A_2) → m_2
  124. Translate(g_3, u_2) → r_3
  125. Translate(h_3, v_2) → s_3
  126. Translate(k_3, w_2) → t_3
  127. Translate(q_3, m_2) → c_4
  128. Translate(p_3, l_2) → d_4
  129. Ray(B_2, C_2) → l_3
  130. Ray(C_2, D_2) → m_3
  131. Ray(D_2, E_2) → q_6
  132. Ray(E_2, F_2) → a_4
  133. Ray(F_2, G_2) → b_4
  134. Ray(G_2, B_2) → e_4
  135. Circle(B_2, 1) → f_4
  136. Circle(C_2, 1) → g_4
  137. Circle(D_2, 1) → h_4
  138. Circle(E_2, 1) → k_4
  139. Circle(F_2, 1) → p_4
  140. Circle(G_2, 1) → q_4
  141. Intersect(q_4, b_4, 2) → H_2
  142. Intersect(q_4, e_4, 1) → I_2
  143. Intersect(f_4, e_4, 2) → J_2
  144. Intersect(f_4, l_3, 1) → K_2
  145. Intersect(g_4, l_3, 2) → L_2
  146. Intersect(g_4, m_3, 1) → M_2
  147. Intersect(h_4, m_3, 2) → N_2
  148. Intersect(h_4, q_6, 1) → O_2
  149. Intersect(k_4, q_6, 2) → P_2
  150. Intersect(k_4, a_4, 1) → Q_2
  151. Intersect(p_4, a_4, 2) → R_2
  152. Intersect(p_4, b_4, 1) → S_2
  153. CircleSector(F_2, R_2, S_2) → r_4
  154. CircleSector(G_2, H_2, I_2) → s_4
  155. CircleSector(B_2, J_2, K_2) → t_4
  156. CircleSector(C_2, L_2, M_2) → c_5
  157. CircleSector(D_2, N_2, O_2) → d_5
  158. CircleSector(E_2, P_2, Q_2) → e_5
  159. Vector(B_2, T_2) → u_3
  160. Vector(C_2, T_2) → v_3
  161. Vector(D_2, T_2) → w_3
  162. Vector(E_2, T_2) → n_3
  163. Vector(F_2, T_2) → u_4
  164. Vector(G_2, T_2) → v_4
  165. Translate(s_4, v_4) → f_5
  166. Translate(t_4, u_3) → g_5
  167. Translate(c_5, v_3) → h_5
  168. Translate(d_5, w_3) → k_5
  169. Translate(e_5, n_3) → p_5
  170. Translate(r_4, u_4) → q_5
  171. Ray(U_2, V_2) → i_4
  172. Ray(V_2, W_2) → j_4
  173. Ray(W_2, Z_2) → l_4
  174. Ray(Z_2, A_3) → m_4
  175. Ray(A_3, B_3) → n_4
  176. Ray(B_3, C_3) → a_5
  177. Ray(C_3, U_2) → b_5
  178. Circle(B_3, 1) → r_5
  179. Circle(C_3, 1) → s_5
  180. Circle(U_2, 1) → t_5
  181. Circle(V_2, 1) → c_6
  182. Circle(W_2, 1) → d_6
  183. Circle(Z_2, 1) → e_6
  184. Circle(A_3, 1) → f_6
  185. Intersect(r_5, n_4, 2) → D_3
  186. Intersect(r_5, a_5, 1) → E_3
  187. Intersect(s_5, a_5, 2) → F_3
  188. Intersect(s_5, b_5, 1) → G_3
  189. Intersect(t_5, b_5, 2) → H_3
  190. Intersect(t_5, i_4, 1) → I_3
  191. Intersect(c_6, i_4, 2) → J_3
  192. Intersect(c_6, j_4, 1) → K_3
  193. Intersect(d_6, j_4, 2) → L_3
  194. Intersect(d_6, l_4, 1) → M_3
  195. Intersect(e_6, l_4, 2) → N_3
  196. Intersect(e_6, m_4, 1) → O_3
  197. Intersect(f_6, m_4, 2) → P_3
  198. Intersect(f_6, n_4, 1) → Q_3
  199. CircleSector(B_3, D_3, E_3) → g_6
  200. CircleSector(C_3, F_3, G_3) → h_6
  201. CircleSector(U_2, H_3, I_3) → k_6
  202. CircleSector(V_2, J_3, K_3) → p_6
  203. CircleSector(W_2, L_3, M_3) → r_6
  204. CircleSector(Z_2, N_3, O_3) → s_6
  205. CircleSector(A_3, P_3, Q_3) → t_6
  206. Vector(U_2, R_3) → w_4
  207. Vector(V_2, R_3) → u_5
  208. Vector(W_2, R_3) → v_5
  209. Vector(Z_2, R_3) → w_5
  210. Vector(A_3, R_3) → i_5
  211. Vector(B_3, R_3) → j_5
  212. Vector(C_3, R_3) → l_5
  213. Translate(g_6, j_5) → c_7
  214. Translate(h_6, l_5) → d_7
  215. Translate(k_6, w_4) → e_7
  216. Translate(p_6, u_5) → f_7
  217. Translate(r_6, v_5) → g_7
  218. Translate(s_6, w_5) → h_7
  219. Translate(t_6, i_5) → k_7
  220. Ray(S_3, T_3) → m_5
  221. Ray(T_3, U_3) → n_5
  222. Ray(U_3, V_3) → a_6
  223. Ray(V_3, W_3) → b_6
  224. Ray(W_3, Z_3) → i_6
  225. Ray(Z_3, A_4) → j_6
  226. Ray(A_4, B_4) → l_6
  227. Ray(B_4, S_3) → m_6
  228. Circle(A_4, 1) → p_7
  229. Circle(B_4, 1) → q_7
  230. Circle(S_3, 1) → r_7
  231. Circle(T_3, 1) → s_7
  232. Circle(U_3, 1) → t_7
  233. Circle(V_3, 1) → c_8
  234. Circle(W_3, 1) → d_8
  235. Circle(Z_3, 1) → e_8
  236. Intersect(q_7, l_6, 2) → C_4
  237. Intersect(q_7, m_6, 1) → D_4
  238. Intersect(r_7, m_6, 2) → E_4
  239. Intersect(r_7, m_5, 1) → F_4
  240. Intersect(s_7, m_5, 2) → G_4
  241. Intersect(s_7, n_5, 1) → H_4
  242. Intersect(t_7, n_5, 2) → I_4
  243. Intersect(t_7, a_6, 1) → J_4
  244. Intersect(c_8, a_6, 2) → K_4
  245. Intersect(c_8, b_6, 1) → L_4
  246. Intersect(d_8, b_6, 2) → M_4
  247. Intersect(d_8, i_6, 1) → N_4
  248. Intersect(e_8, i_6, 2) → O_4
  249. Intersect(e_8, j_6, 1) → P_4
  250. Intersect(p_7, j_6, 2) → Q_4
  251. Intersect(p_7, l_6, 1) → R_4
  252. CircleSector(A_4, Q_4, R_4) → f_8
  253. CircleSector(B_4, C_4, D_4) → g_8
  254. CircleSector(S_3, E_4, F_4) → h_8
  255. CircleSector(T_3, G_4, H_4) → k_8
  256. CircleSector(U_3, I_4, J_4) → p_8
  257. CircleSector(V_3, K_4, L_4) → q_8
  258. CircleSector(W_3, M_4, N_4) → r_8
  259. CircleSector(Z_3, O_4, P_4) → s_8
  260. Vector(S_3, S_4) → u_6
  261. Vector(T_3, S_4) → v_6
  262. Vector(U_3, S_4) → w_6
  263. Vector(V_3, S_4) → n_6
  264. Vector(W_3, S_4) → u_7
  265. Vector(Z_3, S_4) → v_7
  266. Vector(A_4, S_4) → w_7
  267. Vector(B_4, S_4) → a_7
  268. Translate(k_8, v_6) → t_8
  269. Translate(p_8, w_6) → c_9
  270. Translate(q_8, n_6) → d_9
  271. Translate(r_8, u_7) → e_9
  272. Translate(s_8, v_7) → f_9
  273. Translate(f_8, w_7) → g_9
  274. Translate(g_8, a_7) → h_9
  275. Translate(h_8, u_6) → k_9

AI 绘图提示语参考

请绘制一个凸多边形，展示其外角。将每个外角平移到一个共同顶点，形成一个完整的圆。通过动态可视化，体现外角和恒为360度。图形应清晰展示多边形与圆的关系，帮助理解几何定理。
以上提示语为根据原始绘图命令整理而来，仅供参考。