积极 AI Logo 积极 AI

可重叠直角三角形组合的几何构造

可重叠直角三角形组合的几何构造
几何画板及绘图内容加载中，请稍等……
2017-07-16 00:32
测试改名0圆角变换几何作图
① 本范例演示了多个直角三角形通过平移、旋转、镜像等变换后相互重叠、拼合形成的复合图形，重点展示三角形全等（可重叠）关系及其在平面上的位置变化。② 核心数学概念包括三角形全等、几何变换（平移、旋转、镜像）、圆与圆弧的辅助构造、以及分段函数与参数曲线在描述边界时的应用。③ 作图要点：从构造摘要可见，使用了Semicircle（半圆）、Point（点）、Polygon（多边形）、Segment（线段）、Circle（圆）、Intersect（交点）、Line（直线）、OrthogonalLine（垂线）、Angle（角）、CircleArc（圆弧）、Rotate（旋转）、Ray（射线）、Vector（向量）、Translate（平移）、CircleSector（扇形）、Midpoint（中点）、Mirror（镜像）等GeoGebra命令，对象包括多个三角形、圆、线段及辅助点，通过变换命令生成可重叠的直角三角形组合。④ 适合中学几何教学或大学几何变换入门，可用于直观理解全等三角形的判定与变换性质，也可作为动态几何探究的示例。

GeoGebra 命令及参数

  1. Semicircle(O_3, R_3) → s_6
  2. Point(s_6) → S_3
  3. Polygon(O_3, S_3, R_3) → poly11, r_7, o_3, s_7
  4. Point(Segment[Pos_D, W_1], v - 7) → D_9
  5. Segment(W_1, D_2) → l_5
  6. Segment(B_1, C_1) → a_2
  7. Circle(E_1, a_2) → e_3
  8. Point(e_3) → G_1
  9. Semicircle(E_1, G_1) → h_3
  10. Semicircle(B_1, C_1) → t_2
  11. Point(t_2) → D_1
  12. Segment(D_1, C_1) → f_3
  13. Circle(G_1, f_3) → k_3
  14. Intersect(h_3, k_3, 1) → H_1
  15. Segment(D_1, H_1) → a_3
  16. Circle(D, 6) → c_1
  17. Line(C, D) → s
  18. Intersect(c_1, s, 1) → F
  19. Circle(D, 7) → t_1
  20. Point(Segment[C, D], n) → N
  21. Segment(C, N) → f_1
  22. OrthogonalLine(D, f_1) → s_1
  23. Intersect(t_1, s_1, 1) → R
  24. Line(L, J) → g
  25. Segment(L, J) → h
  26. Circle(L, (m * h)) → d
  27. Intersect(d, g, 2) → G
  28. Segment(L, G) → i
  29. OrthogonalLine(J, h) → j
  30. Point(j) → H
  31. Segment(J, H) → l
  32. Circle(J, ((m - 1) * l)) → e
  33. Intersect(e, j) → I, K
  34. Segment(J, I) → k
  35. Angle(L, J, H) → γ
  36. Circle(J, 6) → t
  37. Circle(J, 4) → h_1
  38. Intersect(t, h, 1) → A
  39. Circle(J, 4) → c
  40. Intersect(c, l, 1) → B
  41. Point(Segment[J, A], m - 2) → M
  42. Segment(J, M) → f
  43. Point(Segment[J, B], m - 3) → M_1
  44. Segment(J, M_1) → q
  45. Point(Segment[A, B], m - 4) → M_2
  46. Segment(A, M_2) → r
  47. Polygon(A, J, B) → poly1, b, a, j_1
  48. OrthogonalLine(D, f_1) → g_1
  49. Point(Segment[D, R], n - 1) → N_1
  50. Segment(D, N_1) → i_1
  51. Intersect(c_1, f_1, 1) → F_1
  52. Point(Segment[D, F_1], n - 2) → N_2
  53. Segment(D, N_2) → k_1
  54. Circle(F, 7.5) → d_1
  55. Point(d_1) → O
  56. Intersect(d_1, i_1, 1) → P
  57. Circle(P, O) → e_1
  58. Intersect(d_1, e_1, 2) → Q
  59. Point(Arc[d_1, O, Q], n - 4) → N_4
  60. CircleArc(F_1, O, N_4) → p_1
  61. Point(Segment[F_1, O], n - 3) → N_3
  62. Segment(F_1, N_3) → l_1
  63. Segment(F_1, N_4) → m_1
  64. Point(Segment[F_1, P], n - 5) → N_5
  65. Segment(F_1, N_5) → n_1
  66. Point(Segment[D, P], n - 6) → N_6
  67. Segment(D, N_6) → q_1
  68. Polygon(F_1, D, P) → poly2, p_2, f_2, d_2
  69. Segment(F_1, P) → r_1
  70. Angle(F_1, D, R) → α
  71. OrthogonalLine(D, s) → a_1
  72. Line(E, S) → b_1
  73. Rotate(S, 50°, E) → S'
  74. Angle(S, E, S') → β
  75. Line(E, S') → g_2
  76. Circle(E, 7) → c_2
  77. Intersect(c_2, b_1, 2) → T
  78. OrthogonalLine(T, g_2) → h_2
  79. Intersect(g_2, h_2) → U
  80. Point(Segment[E, T], p) → P_1
  81. Segment(E, P_1) → j_2
  82. Ray(E, T) → k_2
  83. Point(k_2) → W
  84. Segment(E, W) → l_2
  85. Circle(E, W) → e_2
  86. Intersect(e_2, g_2, 2) → Z
  87. Point(Arc[e_2, W, Z], p - 1) → P_2
  88. Segment(E, P_2) → m_2
  89. Angle(W, E, P_2) → δ
  90. Segment(E, W) → i_2
  91. Point(h_2) → A_1
  92. Segment(T, A_1) → n_2
  93. Angle(E, U, T) → ε
  94. Point(Segment[T, U], p - 3) → P_3
  95. Segment(T, P_3) → q_2
  96. Point(Segment[E, U], p - 4) → P_4
  97. Segment(E, P_4) → r_2
  98. Point(Segment[T, A_1], p - 2) → P_{21}
  99. Segment(T, P_{21}) → s_2
  100. Angle(E, U, T) → ζ
  101. Segment(B_1, D_1) → b_2
  102. Polygon(B_1, D_1, C_1) → poly3, c_3, b_3, d_3
  103. Segment(E_1, G_1) → g_3
  104. Segment(E_1, H_1) → i_3
  105. Segment(H_1, G_1) → j_3
  106. Polygon(E_1, H_1, G_1) → poly4, g_4, e_4, h_4
  107. Line(C_1, D_1) → l_3
  108. Line(G_1, H_1) → m_3
  109. Angle(l_3, m_3) → θ
  110. Segment(D_1, H_1) → n_3
  111. Line(D_1, H_1) → p_3
  112. Circle(D_1, (u * a_3)) → q_3
  113. Intersect(q_3, p_3) → I_1, J_1
  114. Vector(D_1, J_1) → w_5
  115. Translate(B_1, w_5) → K_1
  116. Translate(D_1, w_5) → L_1
  117. Translate(C_1, w_5) → O_1
  118. Polygon(K_1, L_1, O_1) → poly3', o_1, k_4, l_4
  119. Segment(J_1, O_1) → r_3
  120. Segment(O_1, K_1) → s_3
  121. Segment(K_1, J_1) → t_3
  122. Vector(D_1, H_1) → u_1
  123. Translate(B_1, u_1) → Q_1
  124. Translate(D_1, u_1) → R_1
  125. Translate(C_1, u_1) → S_1
  126. Polygon(Q_1, R_1, S_1) → poly3'_1, s_4, q_4, r_4
  127. Rotate(Q_1, ((u - 1) * θ), H_1) → T_1
  128. Rotate(R_1, ((u - 1) * θ), H_1) → U_1
  129. Rotate(S_1, ((u - 1) * θ), H_1) → V_1
  130. Polygon(T_1, U_1, V_1) → poly5, v_5, t_4, u_2
  131. Segment(H_1, T_1) → f_4
  132. Segment(T_1, V_1) → i_4
  133. Segment(V_1, H_1) → j_4
  134. Angle(K_1, J_1, O_1) → η
  135. Angle(T_1, H_1, V_1) → ι
  136. Angle(E_1, H_1, G_1) → κ
  137. Polygon(E, U, T) → poly6, t_5, e_5, u_3
  138. Semicircle(W_1, Z_1) → c_4
  139. Point(c_4) → C_2
  140. Polygon(W_1, C_2, Z_1) → poly7, z_1, w_1, c_5
  141. Point(Circle[D_2, c_5]) → E_2
  142. Segment(D_2, E_2) → m_4
  143. Semicircle(D_2, E_2) → d_4
  144. Circle(D_2, z_1) → p_4
  145. Intersect(d_4, p_4, 1) → F_2
  146. Polygon(D_2, F_2, E_2) → poly8, e_6, d_5, f_5
  147. Segment(W_1, Z_1) → n_4
  148. Segment(Z_1, E_2) → b_4
  149. Circle(Z_1, (v * b_4)) → g_5
  150. Vector(Z_1, E_2) → w_2
  151. Translate(W_1, w_2) → J_2
  152. Translate(Z_1, w_2) → K_2
  153. Vector(Z_1, E_2) → v_3
  154. Translate(W_1, v_3) → L_2
  155. Translate(Z_1, v_3) → O_2
  156. Segment(L_2, O_2) → h_5
  157. Angle(L_2, E_2, D_2) → λ
  158. Rotate(L_2, ((v - 1) * λ), E_2) → R_2
  159. Rotate(O_2, ((v - 1) * λ), E_2) → S_2
  160. Segment(R_2, S_2) → i_5
  161. Point(poly7) → U_2
  162. CircleSector(W_1, U_2, C_2) → k_5
  163. Segment(W_1, U_2) → j_5
  164. Circle(D_2, j_5) → p_5
  165. Intersect(m_4, f_5) → V_2
  166. Intersect(m_4, f_5) → W_2
  167. Intersect(m_4, f_5) → Z_2
  168. Intersect(p_5, f_5) → A_3, J_3
  169. CircleSector(D_2, A_3, F_2) → q_5
  170. Segment(E_2, D_2) → m_5
  171. Vector(W_1, D_2) → u_4
  172. Translate(k_5, u_4) → c_6
  173. Rotate(c_6, ((v - 3) * λ), D_2) → d_6
  174. Angle(W_1, C_2, Z_1) → μ
  175. Angle(D_2, F_2, E_2) → ν
  176. Segment(W_1, D_2) → n_5
  177. Line(W_1, D_2) → s_5
  178. Point(Segment[W_1, D_2], v - 2) → A_4
  179. Vector(W_1, A_4) → w_3
  180. Translate(k_5, w_3) → r_5
  181. Midpoint(i_5) → B_3
  182. Midpoint(m_4) → D_3
  183. Point(r_5) → E_3
  184. Point(d_6) → F_3
  185. Angle(J_1, O_1, K_1) → ξ
  186. Angle(H_1, G_1, E_1) → ο
  187. Angle(H_1, V_1, T_1) → ρ
  188. Point(Segment[Z_1, E_2], v) → A_5
  189. Vector(Z_1, A_5) → v_2
  190. Translate(W_1, v_2) → G_2
  191. Translate(Z_1, v_2) → H_2
  192. Segment(G_2, H_2) → a_4
  193. Midpoint(a_4) → I_2
  194. Line(D_2, c_5) → a_5
  195. Circle(D_2, E_2) → f_6
  196. Intersect(f_6, a_5, 1) → T_2
  197. Ray(D_2, F_2) → b_5
  198. Point(b_5) → C_3
  199. Point(Segment[D_2, C_3], v - 4) → A_6
  200. Segment(D_2, A_6) → g_6
  201. Vector(W_1, A_4) → v_4
  202. Translate(k_5, v_4) → h_6
  203. Point(Segment[S_2, F_2], v - 5) → A_7
  204. Segment(S_2, A_7) → i_6
  205. Point(q_5) → G_3
  206. Angle(D_2, F_2, E_2) → σ
  207. Point(Segment[D_2, F_2], v - 6) → A_8
  208. Segment(D_2, A_8) → j_6
  209. Polygon(W_1, C_2, Z_1) → poly9, z_2, w_4, c_7
  210. Polygon(D_2, F_2, E_2) → poly10, e_7, d_7, f_7
  211. Rotate(Z_1, λ, W_1) → I_3
  212. Circle(W_1, I_3) → k_6
  213. Rotate(D_2, (-(((v - 8) * λ))), W_1) → K_3
  214. Rotate(F_2, (-(((v - 8) * λ))), W_1) → L_3
  215. Rotate(E_2, (-(((v - 8) * λ))), W_1) → M_3
  216. Polygon(K_3, L_3, M_3) → poly10', m_6, k_7, l_6
  217. Segment(S_2, D_2) → n_6
  218. Segment(W_1, M_3) → p_6
  219. Segment(L_3, M_3) → q_6
  220. Segment(L_3, W_1) → r_6
  221. Angle(W_1, L_3, M_3) → τ
  222. Angle(M_3, W_1, L_3) → υ
  223. Segment(S_3, O_3) → b_6
  224. Circle(Q_3, s_7) → t_6
  225. Point(t_6) → T_3
  226. Semicircle(Q_3, T_3) → g_7
  227. Circle(Q_3, r_7) → h_7
  228. Intersect(g_7, h_7, 1) → V_3
  229. Polygon(Q_3, V_3, T_3) → poly12, t_7, q_7, v_1
  230. Point(Segment[S_3, V_3], vo) → B_4
  231. Vector(S_3, B_4) → u_5
  232. Translate(O_3, u_5) → W_3
  233. Translate(S_3, u_5) → Z_3
  234. Segment(W_3, Z_3) → a_6
  235. Line(S_3, t_7) → i_7
  236. Intersect(i_7, s_7) → C_4
  237. Angle(O_3, S_3, C_4) → ϕ
  238. Rotate(W_3, ((vo - 1) * ϕ), V_3) → D_4
  239. Rotate(Z_3, ((vo - 1) * ϕ), V_3) → E_4
  240. Segment(D_4, E_4) → j_7
  241. Ray(V_3, T_3) → l_7
  242. Point(l_7) → F_4
  243. Point(Segment[V_3, F_4], vo - 2) → B_5
  244. Segment(V_3, B_5) → m_7
  245. Angle(Q_3, V_3, T_3) → χ
  246. Point(Segment[O_3, Q_3], vo - 3) → B_6
  247. Vector(O_3, B_6) → u_6
  248. Translate(R_3, u_6) → G_4
  249. Translate(O_3, u_6) → H_4
  250. Segment(G_4, H_4) → n_7
  251. Line(Q_3, s_7) → p_7
  252. Intersect(t_6, p_7, 1) → I_4
  253. Circle(T_3, I_4) → c_8
  254. Intersect(t_6, c_8, 2) → J_4
  255. CircleArc(Q_3, I_4, J_4) → d_8
  256. Point(d_8, vo - 4) → B_7
  257. Segment(Q_3, B_7) → a_7
  258. CircleArc(Q_3, I_4, B_7) → e_8
  259. Point(Segment[Q_3, T_3], vo - 5) → B_8
  260. Segment(Q_3, B_8) → b_7
  261. Point(Segment[V_3, T_3], vo - 6) → B_9
  262. Segment(V_3, B_9) → f_8
  263. Point(Segment[V_3, S_3], vot) → C_5
  264. Vector(V_3, C_5) → v_6
  265. Translate(Q_3, v_6) → K_4
  266. Translate(V_3, v_6) → L_4
  267. Translate(T_3, v_6) → M_4
  268. Polygon(K_4, L_4, M_4) → poly12', m_8, k_8, l_8
  269. Rotate(K_4, (-(((vot - 1) * ϕ))), C_5) → O_4
  270. Rotate(L_4, (-(((vot - 1) * ϕ))), C_5) → Q_4
  271. Rotate(M_4, (-(((vot - 1) * ϕ))), C_5) → R_4
  272. Polygon(O_4, Q_4, R_4) → poly12'', r_8, o_4, q_8
  273. Segment(C_5, O_4) → g_8
  274. Angle(O_3, S_3, R_3) → ψ
  275. Angle(K_4, C_5, M_4) → ω
  276. Angle(O_4, C_5, R_4) → α_1
  277. Midpoint(a_6) → S_4
  278. Midpoint(t_7) → T_4
  279. Midpoint(H_4, G_4) → U_4
  280. Segment(Q_3, T_3) → h_8
  281. Midpoint(Q_3, T_3) → V_4
  282. Line(W_4, Z_4) → i_8
  283. Line(D_5, i_8) → j_8
  284. Segment(Z_4, D_5) → n_8
  285. Line(W_4, n_8) → p_8
  286. Intersect(j_8, p_8) → E_5
  287. Polygon(W_4, Z_4, D_5, E_5) → poly13, w_6, z_4, d_9, e_9
  288. Line(E_5, Z_4) → s_8
  289. Segment(W_4, D_5) → t_8
  290. OrthogonalLine(W_4, s_8) → a_8
  291. OrthogonalLine(D_5, s_8) → b_8
  292. Intersect(s_8, a_8) → F_5
  293. Intersect(s_8, b_8) → G_5
  294. Segment(E_5, Z_4) → f_9
  295. Intersect(t_8, f_9) → H_5
  296. Polygon(W_4, F_5, H_5) → poly14, h_9, w_7, f_{10}
  297. Polygon(D_5, H_5, G_5) → poly15, g_9, d_{10}, h_{10}
  298. Rotate(W_4, (-((ap12 * π))), H_5) → I_5
  299. Rotate(F_5, (-((ap12 * π))), H_5) → J_5
  300. Rotate(H_5, (-((ap12 * π))), H_5) → K_5
  301. Polygon(I_5, J_5, K_5) → poly14', k_9, i_9, j_9
  302. Angle(H_5, F_5, W_4) → β_1
  303. Angle(H_5, G_5, D_5) → γ_1
  304. Circle(M_5, O_5) → c_9
  305. Point(c_9) → P_5
  306. Polygon(M_5, O_5, P_5) → poly16, p_9, m_9, o_5
  307. OrthogonalLine(O_5, p_9) → l_9
  308. OrthogonalLine(P_5, o_5) → n_9
  309. Intersect(l_9, n_9) → Q_5
  310. Polygon(M_5, O_5, Q_5) → poly17, q_9, m_{10}, o_6
  311. Polygon(Q_5, P_5, M_5) → poly18, m_{11}, q_{10}, p_{10}
  312. Segment(O_5, P_5) → r_9
  313. Ray(P_5, Q_5) → s_9
  314. Point(s_9) → R_5
  315. Segment(P_5, R_5) → t_9
  316. Ray(O_5, Q_5) → a_9
  317. Point(a_9) → S_5
  318. Segment(O_5, S_5) → b_9
  319. Angle(Q_5, P_5, M_5) → δ_1
  320. Angle(M_5, O_5, Q_5) → ε_1
  321. Rotate(O_5, (ap22 * π), O_5) → T_5
  322. Rotate(Q_5, (ap22 * π), O_5) → U_5
  323. Segment(O_5, U_5) → i_{10}
  324. Mirror(Q_5, O_5) → V_5
  325. Segment(O_5, V_5) → g_{10}
  326. Polygon(M_5, O_5, V_5) → poly19, v_7, m_{12}, o_7
  327. Point(Segment[M_5, V_5], ap22 - 1) → C_6
  328. Segment(M_5, C_6) → j_{10}
  329. Angle(P_5, M_5, Q_5) → ζ_1
  330. Rotate(M_5, (((ap22 - 1) * 2) * ζ_1), M_5) → W_5
  331. Rotate(M_5, (-((((ap22 - 1) * 2) * ζ_1))), M_5) → Z_5
  332. Rotate(M_5, (-((((ap22 - 2) * 2) * ζ_1))), M_5) → D_6
  333. Rotate(O_5, (-((((ap22 - 3) * 2) * ζ_1))), M_5) → E_6
  334. Rotate(V_5, (-((((ap22 - 3) * 2) * ζ_1))), M_5) → F_6
  335. Polygon(D_6, E_6, F_6) → poly19', f_{11}, d_{11}, e_{10}
  336. Segment(M_5, E_6) → k_{10}
  337. Angle(K_5, J_5, I_5) → η_1

AI 绘图提示语参考

绘制一个由多个可重合的直角三角形组成的复合图形。首先画一个半圆，在其上取一点，与半圆两端构成一个直角三角形。再画另一个半圆和圆，通过交点构造出第二个直角三角形。接着画一条线段，以其为半径作圆，与直线相交得到点，构造第三个直角三角形。然后画两条相交线段，以其中一条为半径作圆，与另一条线段相交，构造出第四个直角三角形。最后画两个同心圆，与线段相交，形成第五个直角三角形。所有三角形通过平移、旋转或镜像可以重合，展示全等三角形的性质。
以上提示语为根据原始绘图命令整理而来，仅供参考。