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Marion 三维几何定理的金字塔立体演示

Marion 三维几何定理的金字塔立体演示
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2024-08-10 23:09
多边形立体几何变换几何作图
所属专题 Besondere Punkte/Special Points
该范例用三维金字塔/四面体的可视化展示与截面面积或对应边几何量相关的“Marion 三维定理”现象。核心数学概念包括：立体几何中的四面体构型、三角形与其“影子/截面”之间的面积关系、对称/等腰三角形的几何约束、以及由点的相交与线段连接形成的网格化空间关系。作图要点：根据构造摘要使用了 GeoGebra 命令 Polygon 与 Pyramid 生成多边形面与金字塔体；用 Intersect 计算点/线的相交以确定关键几何位置；用 Segment 连接顶点构成边与用于度量的线段对象。相关对象包含多个三角形面（如 面ABC、面ABD、面BCD、面ACD）以及多条棱（如 KanteAB、KanteBC、KanteAC、KanteAD、KanteBD、KanteCD），并通过额外点/变量（如 E, F, G, H 及参数 d1、b、a 等）控制构型与尺寸关系。适合中学高年级到大学基础几何课程，用于立体几何定理的直观理解、课后探索与课堂演示。

GeoGebra 命令及参数

  1. Pyramid(A, B, C, D) → pyr, FlächeABC, KanteCD, FlächeABD, FlächeBCD, FlächeACD, KanteAB, KanteBC, KanteAC, KanteAD, KanteBD
  2. Polygon(A, P, B) → d1, b, a, p
  3. Polygon(A, M, B) → d2, b_1, a_1, m
  4. Polygon(A, O, C) → d3, c, a_2, o
  5. Polygon(A, J, C) → d4, c_1, a_3, j
  6. Polygon(B, F, D) → d5, d, b_2, f
  7. Polygon(B, K, D) → d6, d_1, b_3, k
  8. Polygon(B, N, C) → d7, c_2, b_4, n
  9. Polygon(B, G, C) → d8, c_3, b_5, g
  10. Polygon(D, L, A) → d9, a_4, d_2, l
  11. Polygon(D, I, A) → d10, a_5, d_3, i
  12. Polygon(D, E, C) → d11, c_4, d_4, e
  13. Polygon(D, H, C) → d12, c_5, d_5, h
  14. Intersect(a_2, a) → Q
  15. Segment(Q, A) → q
  16. Intersect(a, a_3) → R
  17. Segment(R, A) → r
  18. Intersect(b_2, b) → S
  19. Segment(S, B) → s
  20. Intersect(s, q) → T
  21. Intersect(s, r) → U
  22. Intersect(b_3, b) → V
  23. Segment(V, B) → t
  24. Intersect(q, t) → W
  25. Intersect(r, t) → Z
  26. Intersect(b_4, b) → A_1
  27. Segment(A_1, B) → f_1
  28. Intersect(f_1, q) → B_1
  29. Intersect(f_1, r) → C_1
  30. Intersect(b_5, b) → D_1
  31. Segment(D_1, B) → g_1
  32. Intersect(q, g_1) → E_1
  33. Intersect(r, g_1) → F_1
  34. Intersect(a_4, a) → G_1
  35. Segment(G_1, A) → h_1
  36. Intersect(h_1, t) → H_1
  37. Intersect(h_1, f_1) → I_1
  38. Intersect(s, h_1) → J_1
  39. Intersect(g_1, h_1) → K_1
  40. Intersect(a_5, a) → L_1
  41. Segment(L_1, A) → i_1
  42. Intersect(t, i_1) → M_1
  43. Intersect(i_1, f_1) → N_1
  44. Intersect(s, i_1) → O_1
  45. Intersect(i_1, g_1) → P_1
  46. Segment(P, E) → j_1
  47. Intersect(r, j_1) → Q_1
  48. Intersect(t, j_1) → R_1
  49. Segment(P, H) → k_1
  50. Intersect(h_1, k_1) → S_1
  51. Intersect(g_1, k_1) → T_1
  52. Intersect(a_1, a_2) → U_1
  53. Segment(U_1, A) → l_1
  54. Intersect(a_1, a_3) → V_1
  55. Intersect(b_2, b_1) → Z_1
  56. Segment(Z_1, B) → n_1
  57. Intersect(l_1, n_1) → E_2
  58. Segment(V_1, A) → m_1
  59. Intersect(n_1, m_1) → W_1
  60. Intersect(b_1, b_3) → A_2
  61. Segment(A_2, B) → p_1
  62. Intersect(l_1, p_1) → B_2
  63. Intersect(m_1, p_1) → C_2
  64. Intersect(b_4, b_1) → D_2
  65. Segment(D_2, B) → q_1
  66. Intersect(l_1, q_1) → F_2
  67. Intersect(q_1, m_1) → G_2
  68. Intersect(b_1, b_5) → H_2
  69. Segment(H_2, B) → r_1
  70. Intersect(r_1, m_1) → I_2
  71. Intersect(l_1, r_1) → J_2
  72. Intersect(a_1, a_4) → K_2
  73. Segment(K_2, A) → s_1
  74. Intersect(q_1, s_1) → L_2
  75. Intersect(p_1, s_1) → M_2
  76. Intersect(s_1, r_1) → N_2
  77. Intersect(s_1, n_1) → O_2
  78. Intersect(a_5, d10) → P_2, Q_2
  79. Intersect(a_5, a_1) → R_2
  80. Segment(R_2, A) → t_1
  81. Intersect(t_1, n_1) → S_2
  82. Intersect(r_1, t_1) → T_2
  83. Intersect(t_1, q_1) → U_2
  84. Intersect(t_1, p_1) → V_2
  85. Intersect(a_4, d10) → W_2
  86. Segment(M, E) → e_1
  87. Intersect(q_1, e_1) → Z_2
  88. Intersect(e_1, t_1) → A_3
  89. Segment(M, H) → f_2
  90. Intersect(f_2, n_1) → B_3
  91. Intersect(f_2, l_1) → C_3
  92. Segment(O, F) → g_2
  93. Segment(O, K) → h_2
  94. Intersect(c_2, c) → D_3
  95. Segment(D_3, C) → i_2
  96. Intersect(h_2, i_2) → E_3
  97. Intersect(c, c_3) → F_3
  98. Segment(F_3, C) → j_2
  99. Intersect(g_2, j_2) → G_3
  100. Intersect(i_2, g_2) → H_3
  101. Intersect(h_2, j_2) → I_3
  102. Intersect(a_4, a_2) → J_3
  103. Segment(J_3, A) → k_2
  104. Intersect(a_5, a_2) → K_3
  105. Segment(K_3, A) → l_2
  106. Intersect(h_2, l_2) → L_3
  107. Intersect(j_2, l_2) → M_3
  108. Intersect(c, c_4) → N_3
  109. Segment(N_3, C) → m_2
  110. Intersect(l_2, m_2) → O_3
  111. Intersect(c_5, c) → P_3
  112. Segment(P_3, C) → n_2
  113. Intersect(n_2, g_2) → Q_3
  114. Intersect(n_2, l_2) → R_3
  115. Intersect(n_2, k_2) → S_3
  116. Intersect(m_2, k_2) → T_3
  117. Segment(J, F) → p_2
  118. Segment(J, K) → q_2
  119. Intersect(c_2, c_1) → U_3
  120. Segment(U_3, C) → r_2
  121. Intersect(r_2, p_2) → V_3
  122. Intersect(q_2, r_2) → W_3
  123. Intersect(c_1, c_3) → Z_3
  124. Segment(Z_3, C) → s_2
  125. Intersect(s_2, p_2) → A_4
  126. Intersect(q_2, s_2) → B_4
  127. Intersect(a_3, a_4) → C_4
  128. Segment(C_4, A) → t_2
  129. Intersect(p_2, t_2) → D_4
  130. Intersect(r_2, t_2) → E_4
  131. Intersect(a_5, a_3) → H_4
  132. Segment(H_4, A) → e_2
  133. Intersect(c_1, c_4) → I_4
  134. Segment(I_4, C) → f_3
  135. Intersect(e_2, f_3) → J_4
  136. Intersect(q_2, f_3) → K_4
  137. Intersect(t_2, f_3) → L_4
  138. Intersect(c_5, c_1) → M_4
  139. Segment(M_4, C) → g_3
  140. Intersect(g_3, e_2) → N_4
  141. Intersect(g_3, t_2) → O_4
  142. Intersect(b_2, b_4) → P_4
  143. Segment(P_4, B) → h_3
  144. Intersect(b_2, b_5) → Q_4
  145. Segment(Q_4, B) → i_3
  146. Intersect(d, d_2) → R_4
  147. Segment(R_4, D) → j_3
  148. Intersect(j_3, h_3) → S_4
  149. Intersect(d_3, d) → T_4
  150. Segment(T_4, D) → k_3
  151. Intersect(d, d_4) → U_4
  152. Segment(U_4, D) → l_3
  153. Intersect(l_3, h_3) → V_4
  154. Intersect(i_3, l_3) → W_4
  155. Intersect(d, d_5) → Z_4
  156. Segment(Z_4, D) → m_3
  157. Intersect(h_3, m_3) → A_5
  158. Intersect(m_3, i_3) → B_5
  159. Intersect(b_4, b_3) → C_5
  160. Segment(C_5, B) → n_3
  161. Intersect(b_5, b_3) → D_5
  162. Segment(D_5, B) → p_3
  163. Intersect(d_1, d_2) → E_5
  164. Segment(E_5, D) → q_3
  165. Intersect(d_3, d_1) → F_5
  166. Segment(F_5, D) → r_3
  167. Intersect(r_3, p_3) → G_5
  168. Intersect(d_4, d_1) → H_5
  169. Segment(H_5, W_2) → s_3
  170. Intersect(s_3, p_3) → I_5
  171. Intersect(s_3, n_3) → J_5
  172. Intersect(d_5, d_1) → K_5
  173. Segment(K_5, D) → t_3
  174. Intersect(t_3, p_3) → L_5
  175. Intersect(t_3, n_3) → N_5
  176. Segment(N, L) → e_3
  177. Segment(N, Q_2) → f_4
  178. Intersect(c_4, c_2) → O_5
  179. Segment(O_5, C) → g_4
  180. Intersect(g_4, f_4) → P_5
  181. Intersect(c_5, c_2) → Q_5
  182. Segment(Q_5, C) → h_4
  183. Segment(G, L) → i_4
  184. Segment(G, Q_2) → j_4
  185. Intersect(c_3, c_4) → R_5
  186. Segment(R_5, C) → k_4
  187. Intersect(c_3, c_5) → S_5
  188. Segment(S_5, C) → l_4
  189. Intersect(l_4, i_4) → T_5
  190. Intersect(d_4, d_2) → U_5
  191. Segment(U_5, D) → m_4
  192. Intersect(i_2, m_4) → V_5
  193. Intersect(d_5, d_2) → W_5
  194. Segment(W_5, D) → n_4
  195. Intersect(d_4, d_3) → Z_5
  196. Segment(Z_5, D) → p_4
  197. Intersect(d_5, d_3) → A_6
  198. Segment(A_6, D) → q_4
  199. Polygon(N_5, C_2, M_2, E_3) → v1, n_5, r_4, s_4, t_4
  200. Polygon(E_3, M_2, N_2, T_3) → v2, e_4, f_5, g_5, h_5
  201. Polygon(N_1, N_4, C_2, N_5) → v3, i_5, j_5, k_5, l_5
  202. Polygon(C_2, I_2, N_2, M_2) → v4, m_5, p_5, q_5, r_5
  203. Polygon(N_4, A_4, I_2, C_2) → v5, s_5, t_5, e_5, f_6
  204. Polygon(T_3, N_2, W_4, T) → v6, g_6, h_6, w_4, t_6
  205. Polygon(T, W_4, A_4, O_1) → v7, t_7, i_6, j_6, o_1
  206. Polygon(O_1, A_4, N_4, N_1) → v8, k_6, l_6, m_6, n_6
  207. Polygon(N_2, I_2, A_4, W_4) → v9, p_6, q_6, r_6, s_6
  208. Polygon(E_3, T_3, T, B_1) → v11, d_6, e_6, t_9, f_7
  209. Polygon(N_1, N_5, E_3, B_1) → v12, g_7, h_7, i_7, j_7

AI 绘图提示语参考

在三维空间中画一个四面体，由四个点作为顶点构成四个三角形面，并将四个面分别显示。先建立四面体，随后在两条与两底边相关的几何路径上取多组位于同一直线/同平面的点形成三角形轮廓：在连接顶点的平面内构造若干三角形边段与其对应交点，并用这些交点把不同面上的关键点连成一组折线/线段。最后确保出现多个面之间的交汇点与由交点生成的连接边，用于观察各三角形面之间随边长与位置变化的几何关系。
以上提示语为根据原始绘图命令整理而来，仅供参考。