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3D Theorem von Marion: a Pyramid Solid Demonstration

3D Theorem von Marion: a Pyramid Solid Demonstration
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2024-08-10 23:09
PolygonSolid GeometryTransformationGeometric Construction
In collections Besondere Punkte/Special Points
This GeoGebra example demonstrates a “Theorem von Marion” in 3D by constructing a pyramid/tetrahedron configuration and visualizing the geometric relationships suggested by its theorem (e.g., via corresponding triangular faces and their area/edge-dependent relations). The main mathematical ideas are solid geometry with tetrahedra, relationships among the triangular faces (and their intersections), symmetry-type constraints (often connected to isosceles triangle behavior in the setup), and the role of constructed intersection points and segment connections in determining the final geometry. Construction-wise, it uses Polygon and Pyramid to build the faces and the 3D solid, Intersect to compute key intersection points/constraints, and Segment to define the connecting edges used for measurements and verification. The model includes multiple face objects such as PlaneABC, PlaneABD, PlaneBCD, PlaneACD and the corresponding edges (KanteAB, KanteBC, KanteAC, KanteAD, KanteBD, KanteCD), together with auxiliary points (E–P) and parameters (e.g., d1, d2, d3, d4, a, b, p) to control the configuration. Suitable for late secondary school through early university geometry, especially for classroom demonstrations and exploratory learning of 3D theorem statements.

GeoGebra Commands & Parameters

  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 Drawing Prompt Reference

Draw a 3D tetrahedron using four vertices and show its four triangular faces. Construct the solid first, then create several auxiliary points on lines/planes determined by the tetrahedron’s edges, forming multiple triangular edge configurations. Compute key intersection points where the constructed segments meet, then connect those intersection points back to the relevant tetrahedron vertices to form defining edges across different faces. Ensure the drawing contains clear face planes, intersection points, and connecting segments so the geometric relationships among the triangular faces can be visually compared as the auxiliary points vary.
The prompt above was compiled from the original drawing commands and is for reference only.