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根脉轮：基于勾股定理的几何构造

根脉轮：基于勾股定理的几何构造
几何画板及绘图内容加载中，请稍等……
2020-03-22 15:11
圆多边形解析几何变换几何作图
① 本范例通过GeoGebra展示了一个名为“根脉轮”的几何图形，它由一系列线段、圆和交点构成，形成类似曼陀罗的对称图案，并隐含勾股定理的几何关系。② 核心数学概念包括勾股定理（Pythagorean Theorem）、单位圆、线段与四边形的构造，以及圆与直线的交点运算。③ 作图要点：使用Circle命令创建多个圆，Intersect命令求圆与直线或圆的交点，Line命令绘制辅助线段；对象列表包含A、B、C等大量点，以及c、d、e等圆和线段，通过逐步相交生成复杂网络。④ 适合中学阶段学习几何作图、勾股定理的直观验证，以及探索几何图案的对称性与美学，也可用于冥想或艺术数学的跨学科教学。

GeoGebra 命令及参数

  1. Circle(A, B) → c
  2. Circle(B, A) → d
  3. Intersect(c, d, 1) → C
  4. Circle(C, B) → e
  5. Intersect(c, e, 2) → D
  6. Circle(D, C) → f
  7. Intersect(c, f, 2) → E
  8. Circle(E, D) → g
  9. Intersect(c, g, 2) → F
  10. Circle(F, E) → h
  11. Intersect(c, d, 2) → G
  12. Circle(G, F) → k
  13. Line(E, G) → i
  14. Intersect(k, i, 2) → H
  15. Circle(H, G) → p
  16. Intersect(d, p, 2) → I
  17. Circle(I, H) → q
  18. Intersect(d, q, 1) → J
  19. Circle(J, I) → r
  20. Intersect(e, r, 1) → K
  21. Circle(K, J) → s
  22. Intersect(e, s, 1) → L
  23. Circle(L, K) → t
  24. Intersect(f, t, 1) → M
  25. Circle(M, L) → c_1
  26. Intersect(f, c_1, 1) → N
  27. Circle(N, M) → d_1
  28. Intersect(g, d_1, 1) → O
  29. Circle(O, N) → e_1
  30. Intersect(g, e_1, 1) → P
  31. Circle(P, O) → f_1
  32. Intersect(h, f_1, 1) → Q
  33. Circle(Q, P) → g_1
  34. Intersect(h, g_1, 1) → R
  35. Circle(R, Q) → h_1
  36. Intersect(k, p, 2) → S
  37. Circle(S, R) → k_1
  38. Intersect(p, k_1, 1) → B_1
  39. Circle(B_1, H) → d_2
  40. Intersect(p, q, 2) → C_1
  41. Circle(C_1, I) → e_2
  42. Intersect(q, r, 2) → D_1
  43. Circle(D_1, J) → f_2
  44. Intersect(r, s, 2) → E_1
  45. Circle(E_1, K) → g_2
  46. Intersect(s, t, 2) → F_1
  47. Circle(F_1, L) → h_2
  48. Intersect(t, c_1, 2) → G_1
  49. Circle(G_1, M) → k_2
  50. Intersect(c_1, d_1, 2) → H_1
  51. Circle(H_1, N) → p_2
  52. Intersect(d_1, e_1, 2) → I_1
  53. Circle(I_1, O) → q_2
  54. Intersect(e_1, f_1, 2) → J_1
  55. Circle(J_1, P) → r_2
  56. Intersect(f_1, g_1, 2) → K_1
  57. Circle(K_1, Q) → s_2
  58. Intersect(g_1, h_1, 2) → L_1
  59. Circle(L_1, R) → t_2
  60. Intersect(h_1, k_1, 2) → M_1
  61. Circle(M_1, S) → c_3
  62. Intersect(d_2, c_3, 1) → N_1
  63. Circle(N_1, B_1) → d_3
  64. Intersect(d_2, e_2, 2) → O_1
  65. Circle(O_1, C_1) → e_3
  66. Intersect(e_2, f_2, 2) → P_1
  67. Circle(P_1, D_1) → f_3
  68. Intersect(f_2, g_2, 2) → Q_1
  69. Circle(Q_1, E_1) → g_3
  70. Intersect(g_2, h_2, 2) → R_1
  71. Circle(R_1, F_1) → h_3
  72. Intersect(h_2, k_2, 2) → S_1
  73. Circle(S_1, G_1) → k_3
  74. Intersect(k_2, p_2, 2) → T_1
  75. Circle(T_1, H_1) → p_3
  76. Intersect(p_2, q_2, 2) → U_1
  77. Circle(U_1, I_1) → q_3
  78. Intersect(q_2, r_2, 2) → V_1
  79. Circle(V_1, J_1) → r_3
  80. Intersect(r_2, s_2, 2) → W_1
  81. Circle(W_1, K_1) → s_3
  82. Intersect(s_2, t_2, 2) → Z_1
  83. Circle(Z_1, L_1) → t_3
  84. Intersect(t_2, c_3, 2) → A_2
  85. Circle(A_2, M_1) → c_4
  86. Intersect(c_3, d_3, 2) → B_2
  87. Circle(B_2, N_1) → d_4
  88. Intersect(d_2, d_3, 1) → C_2
  89. Circle(C_2, O_1) → e_4
  90. Intersect(e_2, e_3, 1) → D_2
  91. Circle(D_2, P_1) → f_4
  92. Intersect(f_2, f_3, 1) → E_2
  93. Circle(E_2, Q_1) → g_4
  94. Intersect(g_2, g_3, 1) → F_2
  95. Circle(F_2, R_1) → h_4
  96. Intersect(h_2, h_3, 1) → G_2
  97. Circle(G_2, S_1) → k_4
  98. Intersect(k_2, k_3, 1) → H_2
  99. Circle(H_2, T_1) → p_4
  100. Intersect(p_2, p_3, 1) → I_2
  101. Circle(I_2, U_1) → q_4
  102. Intersect(q_2, q_3, 1) → J_2
  103. Circle(J_2, V_1) → r_4
  104. Intersect(r_2, r_3, 1) → K_2
  105. Circle(K_2, W_1) → s_4
  106. Intersect(s_2, s_3, 1) → L_2
  107. Circle(L_2, Z_1) → t_4
  108. Intersect(t_3, c_4, 2) → M_2
  109. Circle(M_2, A_2) → c_5
  110. Intersect(c_4, d_4, 2) → N_2
  111. Circle(N_2, B_2) → d_5
  112. Intersect(d_3, e_4, 2) → O_2
  113. Circle(O_2, C_2) → e_5
  114. Intersect(e_3, f_4, 2) → P_2
  115. Circle(P_2, D_2) → f_5
  116. Intersect(f_3, g_4, 2) → Q_2
  117. Circle(Q_2, E_2) → g_5
  118. Intersect(g_3, h_4, 2) → R_2
  119. Circle(R_2, F_2) → h_5
  120. Intersect(h_3, k_4, 2) → S_2
  121. Circle(S_2, G_2) → k_5
  122. Intersect(k_3, p_4, 2) → T_2
  123. Circle(T_2, H_2) → p_5
  124. Intersect(p_3, q_4, 2) → U_2
  125. Circle(U_2, I_2) → q_5
  126. Intersect(q_3, r_4, 2) → V_2
  127. Circle(V_2, J_2) → r_5
  128. Intersect(r_3, s_4, 2) → W_2
  129. Circle(W_2, K_2) → s_5
  130. Intersect(s_3, t_4, 2) → Z_2
  131. Circle(Z_2, L_2) → t_5
  132. Intersect(t_3, c_5, 2) → A_3
  133. Circle(A_3, M_2) → c_6
  134. Intersect(c_5, d_5, 2) → B_3
  135. Circle(B_3, N_2) → d_6
  136. Intersect(d_4, e_5, 2) → C_3
  137. Circle(C_3, O_2) → e_6
  138. Intersect(e_4, f_5, 2) → D_3
  139. Circle(D_3, P_2) → f_6
  140. Intersect(f_4, g_5, 2) → E_3
  141. Circle(E_3, Q_2) → g_6
  142. Intersect(g_4, h_5, 2) → F_3
  143. Circle(F_3, R_2) → h_6
  144. Intersect(h_4, k_5, 2) → G_3
  145. Circle(G_3, S_2) → k_6
  146. Intersect(k_4, p_5, 2) → H_3
  147. Circle(H_3, T_2) → p_6
  148. Intersect(p_4, q_5, 2) → I_3
  149. Circle(I_3, U_2) → q_6
  150. Intersect(q_4, r_5, 2) → J_3
  151. Circle(J_3, V_2) → r_6
  152. Intersect(r_4, s_5, 2) → K_3
  153. Circle(K_3, W_2) → s_6
  154. Intersect(s_4, t_5, 2) → L_3
  155. Circle(L_3, Z_2) → t_6
  156. Intersect(t_4, c_6, 2) → M_3
  157. Circle(M_3, A_3) → c_7
  158. Intersect(c_6, d_6, 2) → N_3
  159. Circle(N_3, B_3) → d_7
  160. Intersect(d_5, e_6, 2) → O_3
  161. Circle(O_3, C_3) → e_7
  162. Intersect(e_5, f_6, 2) → P_3
  163. Circle(P_3, D_3) → f_7
  164. Intersect(f_5, g_6, 2) → Q_3
  165. Circle(Q_3, E_3) → g_7
  166. Intersect(g_5, h_6, 2) → R_3
  167. Circle(R_3, F_3) → h_7
  168. Intersect(h_5, k_6, 2) → S_3
  169. Circle(S_3, G_3) → k_7
  170. Intersect(k_5, p_6, 2) → T_3
  171. Circle(T_3, H_3) → p_7
  172. Intersect(p_5, q_6, 2) → U_3
  173. Circle(U_3, I_3) → q_7
  174. Intersect(q_5, r_6, 2) → V_3
  175. Circle(V_3, J_3) → r_7
  176. Intersect(r_5, s_6, 2) → W_3
  177. Circle(W_3, K_3) → s_7
  178. Intersect(s_5, t_6, 2) → Z_3
  179. Circle(Z_3, L_3) → t_7
  180. Intersect(t_5, c_7, 2) → A_4
  181. Circle(A_4, M_3) → c_8
  182. Intersect(c_6, c_7, 1) → B_4
  183. Circle(B_4, N_3) → d_8
  184. Intersect(d_5, d_6, 1) → C_4
  185. Circle(C_4, O_3) → e_8
  186. Intersect(e_5, e_6, 1) → D_4
  187. Circle(D_4, P_3) → f_8
  188. Intersect(f_5, f_6, 1) → E_4
  189. Circle(E_4, Q_3) → g_8
  190. Intersect(g_5, g_6, 1) → F_4
  191. Circle(F_4, R_3) → h_8
  192. Intersect(h_5, h_6, 1) → G_4
  193. Circle(G_4, S_3) → k_8
  194. Intersect(k_5, k_6, 1) → H_4
  195. Circle(H_4, T_3) → p_8
  196. Intersect(p_5, p_6, 1) → I_4
  197. Circle(I_4, U_3) → q_8
  198. Intersect(q_5, q_6, 1) → J_4
  199. Circle(J_4, V_3) → r_8
  200. Intersect(r_5, r_6, 1) → K_4
  201. Circle(K_4, W_3) → s_8
  202. Intersect(s_5, s_6, 1) → L_4
  203. Circle(L_4, Z_3) → t_8
  204. Intersect(t_5, t_6, 1) → M_4
  205. Circle(M_4, A_4) → c_9
  206. Intersect(c_7, c_8, 1) → N_4
  207. Circle(N_4, B_4) → d_9
  208. Intersect(d_7, e_8, 2) → O_4
  209. Circle(O_4, C_4) → e_9
  210. Intersect(e_6, e_7, 1) → P_4
  211. Circle(P_4, D_4) → f_9
  212. Intersect(f_6, f_7, 1) → Q_4
  213. Circle(Q_4, E_4) → g_9
  214. Intersect(g_6, g_7, 1) → R_4
  215. Circle(R_4, F_4) → h_9
  216. Intersect(h_6, h_7, 1) → S_4
  217. Circle(S_4, G_4) → k_9
  218. Intersect(k_6, k_7, 1) → T_4
  219. Circle(T_4, H_4) → p_9
  220. Intersect(p_6, p_7, 1) → U_4
  221. Circle(U_4, I_4) → q_9
  222. Intersect(q_6, q_7, 1) → V_4
  223. Circle(V_4, J_4) → r_9
  224. Intersect(r_6, r_7, 1) → W_4
  225. Circle(W_4, K_4) → s_9
  226. Intersect(s_6, s_7, 1) → Z_4
  227. Circle(Z_4, L_4) → t_9
  228. Intersect(t_6, t_7, 1) → A_5
  229. Circle(A_5, M_4) → c_{10}
  230. Intersect(c_8, d_9, 2) → B_5
  231. Circle(B_5, N_4) → d_{10}
  232. Intersect(d_7, e_9, 2) → C_5
  233. Circle(C_5, O_4) → e_{10}
  234. Intersect(e_7, f_9, 2) → D_5
  235. Circle(D_5, P_4) → f_{10}
  236. Intersect(f_7, g_9, 2) → E_5
  237. Circle(E_5, Q_4) → g_{10}
  238. Intersect(g_7, h_9, 2) → F_5
  239. Circle(F_5, R_4) → h_{10}
  240. Intersect(h_7, k_9, 2) → G_5
  241. Circle(G_5, S_4) → k_{10}
  242. Intersect(k_7, p_9, 2) → H_5
  243. Circle(H_5, T_4) → p_{10}
  244. Intersect(p_7, q_9, 2) → I_5
  245. Circle(I_5, U_4) → q_{10}
  246. Intersect(q_7, r_9, 2) → J_5
  247. Circle(J_5, V_4) → r_{10}
  248. Intersect(r_7, s_9, 2) → K_5
  249. Circle(K_5, W_4) → s_{10}
  250. Intersect(s_7, t_9, 2) → L_5
  251. Circle(L_5, Z_4) → t_{10}
  252. Intersect(t_7, c_{10}, 2) → M_5
  253. Circle(M_5, A_5) → c_{11}
  254. Intersect(c_9, d_{10}, 2) → T
  255. Circle(T, B_5) → p_1
  256. Intersect(d_8, e_{10}, 2) → U
  257. Circle(U, C_5) → q_1
  258. Intersect(e_8, f_{10}, 2) → V
  259. Circle(V, D_5) → r_1
  260. Intersect(f_8, g_{10}, 2) → W
  261. Circle(W, E_5) → s_1
  262. Intersect(g_8, h_{10}, 2) → Z
  263. Circle(Z, F_5) → t_1
  264. Intersect(h_8, k_{10}, 2) → A_1
  265. Circle(A_1, G_5) → c_2
  266. Intersect(k_8, p_{10}, 2) → N_5
  267. Circle(N_5, H_5) → d_{11}
  268. Intersect(p_8, q_{10}, 2) → O_5
  269. Circle(O_5, I_5) → e_{11}
  270. Intersect(q_8, r_{10}, 2) → P_5
  271. Circle(P_5, J_5) → f_{11}
  272. Intersect(r_8, s_{10}, 2) → Q_5
  273. Circle(Q_5, K_5) → g_{11}
  274. Intersect(s_8, t_{10}, 2) → R_5
  275. Circle(R_5, L_5) → h_{11}
  276. Intersect(t_8, c_{11}, 2) → S_5
  277. Circle(S_5, M_5) → k_{11}
  278. Intersect(c_{10}, p_1, 2) → T_5
  279. Circle(T_5, T) → p_{11}
  280. Intersect(d_9, q_1, 2) → U_5
  281. Circle(U_5, U) → q_{11}
  282. Intersect(e_9, r_1, 2) → V_5
  283. Circle(V_5, V) → r_{11}
  284. Intersect(f_9, s_1, 2) → W_5
  285. Circle(W_5, W) → s_{11}
  286. Intersect(g_9, t_1, 2) → Z_5
  287. Circle(Z_5, Z) → t_{11}
  288. Intersect(h_9, c_2, 2) → A_6
  289. Circle(A_6, A_1) → c_{12}
  290. Intersect(k_9, d_{11}, 2) → B_6
  291. Circle(B_6, N_5) → d_{12}
  292. Intersect(p_9, e_{11}, 2) → C_6
  293. Circle(C_6, O_5) → e_{12}
  294. Intersect(q_9, f_{11}, 2) → D_6
  295. Circle(D_6, P_5) → f_{12}
  296. Intersect(r_9, g_{11}, 2) → E_6
  297. Circle(E_6, Q_5) → g_{12}
  298. Intersect(s_9, h_{11}, 2) → F_6
  299. Circle(F_6, R_5) → h_{12}
  300. Intersect(t_9, k_{11}, 2) → G_6
  301. Circle(G_6, S_5) → k_{12}
  302. Intersect(c_{11}, p_{11}, 2) → H_6
  303. Circle(H_6, T_5) → p_{12}
  304. Intersect(d_{10}, q_{11}, 2) → I_6
  305. Circle(I_6, U_5) → q_{12}
  306. Intersect(e_{10}, r_{11}, 2) → J_6
  307. Circle(J_6, V_5) → r_{12}
  308. Intersect(f_{10}, s_{11}, 2) → K_6
  309. Circle(K_6, W_5) → s_{12}
  310. Intersect(g_{10}, t_{11}, 2) → L_6
  311. Circle(L_6, Z_5) → t_{12}
  312. Intersect(h_{10}, c_{12}, 2) → M_6
  313. Circle(M_6, A_6) → c_{13}
  314. Intersect(k_{10}, d_{12}, 2) → N_6
  315. Circle(N_6, B_6) → d_{13}
  316. Intersect(p_{10}, e_{12}, 2) → O_6
  317. Circle(O_6, C_6) → e_{13}
  318. Intersect(q_{10}, f_{12}, 2) → P_6
  319. Circle(P_6, D_6) → f_{13}
  320. Intersect(r_{10}, g_{12}, 2) → Q_6
  321. Circle(Q_6, E_6) → g_{13}
  322. Intersect(s_{10}, h_{12}, 2) → R_6
  323. Circle(R_6, F_6) → h_{13}
  324. Intersect(t_{10}, k_{12}, 2) → S_6
  325. Circle(S_6, G_6) → k_{13}
  326. Intersect(c_{11}, k_{11}, 1) → T_6
  327. Circle(T_6, H_6) → p_{13}
  328. Intersect(d_{10}, p_1, 1) → U_6
  329. Circle(U_6, I_6) → q_{13}
  330. Intersect(e_{10}, q_1, 1) → V_6
  331. Circle(V_6, J_6) → r_{13}
  332. Intersect(f_{10}, r_1, 1) → W_6
  333. Circle(W_6, K_6) → s_{13}
  334. Intersect(g_{10}, s_1, 1) → Z_6
  335. Circle(Z_6, L_6) → t_{13}
  336. Intersect(h_{10}, t_1, 1) → A_7
  337. Circle(A_7, M_6) → c_{14}
  338. Intersect(k_{10}, c_2, 1) → B_7
  339. Circle(B_7, N_6) → d_{14}
  340. Intersect(p_{10}, d_{11}, 1) → C_7
  341. Circle(C_7, O_6) → e_{14}
  342. Intersect(q_{10}, e_{11}, 1) → D_7
  343. Circle(D_7, P_6) → f_{14}
  344. Intersect(r_{10}, g_{13}, 2) → E_7
  345. Circle(E_7, Q_6) → g_{14}
  346. Intersect(s_{10}, g_{11}, 1) → F_7
  347. Circle(F_7, R_6) → h_{14}
  348. Intersect(t_{10}, h_{11}, 1) → G_7
  349. Circle(G_7, S_6) → k_{14}
  350. Intersect(k_{11}, k_{12}, 1) → H_7
  351. Circle(H_7, T_6) → p_{14}
  352. Intersect(p_1, p_{11}, 1) → I_7
  353. Circle(I_7, U_6) → q_{14}
  354. Intersect(q_1, q_{11}, 1) → J_7
  355. Circle(J_7, V_6) → r_{14}
  356. Intersect(r_1, r_{11}, 1) → K_7
  357. Circle(K_7, W_6) → s_{14}
  358. Intersect(s_1, s_{11}, 1) → L_7
  359. Circle(L_7, Z_6) → t_{14}
  360. Intersect(t_1, t_{11}, 1) → M_7
  361. Circle(M_7, A_7) → c_{15}
  362. Intersect(c_2, c_{12}, 1) → N_7
  363. Circle(N_7, B_7) → d_{15}
  364. Intersect(d_{11}, d_{12}, 1) → O_7
  365. Circle(O_7, C_7) → e_{15}
  366. Intersect(e_{11}, e_{12}, 1) → P_7
  367. Circle(P_7, D_7) → f_{15}
  368. Intersect(f_{11}, f_{12}, 1) → Q_7
  369. Circle(Q_7, E_7) → g_{15}
  370. Intersect(g_{11}, g_{12}, 1) → R_7
  371. Circle(R_7, F_7) → h_{15}
  372. Intersect(h_{11}, h_{12}, 1) → S_7
  373. Circle(S_7, G_7) → k_{15}
  374. Intersect(p_{11}, p_{12}, 1) → T_7
  375. Circle(T_7, I_7) → p_{15}
  376. Intersect(q_{11}, q_{12}, 1) → U_7
  377. Circle(U_7, J_7) → q_{15}
  378. Intersect(r_{11}, r_{12}, 1) → V_7
  379. Circle(V_7, K_7) → r_{15}
  380. Intersect(s_{11}, s_{12}, 1) → W_7
  381. Circle(W_7, L_7) → s_{15}
  382. Intersect(t_{11}, t_{12}, 1) → Z_7
  383. Circle(Z_7, M_7) → t_{15}
  384. Intersect(c_{12}, c_{13}, 1) → A_8
  385. Circle(A_8, N_7) → c_{16}
  386. Intersect(d_{12}, d_{13}, 1) → B_8
  387. Circle(B_8, O_7) → d_{16}
  388. Intersect(e_{12}, e_{13}, 1) → C_8
  389. Circle(C_8, P_7) → e_{16}
  390. Intersect(f_{12}, f_{13}, 1) → D_8
  391. Circle(D_8, Q_7) → f_{16}
  392. Intersect(g_{12}, g_{13}, 1) → E_8
  393. Circle(E_8, R_7) → g_{16}
  394. Intersect(h_{12}, h_{13}, 1) → F_8
  395. Circle(F_8, S_7) → h_{16}
  396. Intersect(k_{12}, k_{13}, 1) → G_8
  397. Circle(G_8, H_7) → k_{16}
  398. Intersect(p_{12}, p_{15}, 2) → H_8
  399. Circle(H_8, T_7) → p_{16}
  400. Intersect(q_{12}, q_{15}, 2) → I_8
  401. Circle(I_8, U_7) → q_{16}
  402. Intersect(r_{12}, r_{15}, 2) → J_8
  403. Circle(J_8, V_7) → r_{16}
  404. Intersect(s_{12}, s_{15}, 2) → K_8
  405. Circle(K_8, W_7) → s_{16}
  406. Intersect(t_{12}, t_{15}, 2) → L_8
  407. Circle(L_8, Z_7) → t_{16}
  408. Intersect(c_{13}, c_{16}, 2) → M_8
  409. Circle(M_8, A_8) → c_{17}
  410. Intersect(d_{13}, d_{16}, 2) → N_8
  411. Circle(N_8, B_8) → d_{17}
  412. Intersect(e_{13}, e_{16}, 2) → O_8
  413. Circle(O_8, C_8) → e_{17}
  414. Intersect(f_{13}, f_{16}, 2) → P_8
  415. Circle(P_8, D_8) → f_{17}
  416. Intersect(g_{13}, g_{16}, 2) → Q_8
  417. Circle(Q_8, E_8) → g_{17}
  418. Intersect(h_{13}, h_{16}, 2) → R_8
  419. Circle(R_8, F_8) → h_{17}
  420. Intersect(k_{13}, k_{16}, 2) → S_8
  421. Circle(S_8, G_8) → k_{17}

AI 绘图提示语参考

绘制一个基于勾股定理的对称曼陀罗式几何图案，称为“根脉轮”。从两个等半径圆相交开始，以交点为圆心不断作等半径圆，沿圆周生成一系列交点与圆，形成内外嵌套的花环结构。连接特定交点作直线，再以直线与圆的交点为圆心继续作圆，逐步扩展出多层对称的圆弧与交点网络。整体图形由多个相交圆、线段和交点构成，呈现高度对称的几何美感，隐含勾股定理的几何关系。
以上提示语为根据原始绘图命令整理而来，仅供参考。