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金字塔构造

金字塔构造
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2017-08-12 22:17
向量立体几何几何作图
①本范例演示了如何通过 GeoGebra 构造金字塔及其相关几何元素；②涉及的核心数学概念包括向量、球体、余弦定理、线段、极限、线段、正交线、中点、角平分线、旋转等；③作图要点提炼自构造摘要，使用了 Polygon（多边形）、Line（线）、Circle（圆）、Intersect（交点）、Segment（线段）、OrthogonalLine（正交线）、Midpoint（中点）、AngularBisector（角平分线）、Angle（角）、Rotate（旋转）、Ray（射线）等命令；④适合高中及以上学段，用于立体几何和解析几何的教学与探索。

GeoGebra 命令及参数

  1. Polygon(B, C, A) → Area, a, b, c
  2. Line(A, B) → L_c
  3. Line(B, C) → L_a
  4. Line(C, A) → L_b
  5. Circle(C, A) → CirA_C
  6. Intersect(CirA_C, L_a, 2) → pPerimeter_C
  7. Circle(B, A) → CirA_B
  8. Intersect(CirA_B, L_a, 1) → pPerimeter_B
  9. Segment(pPerimeter_C, pPerimeter_B) → Perimeter
  10. OrthogonalLine(A, L_a) → Lh_a
  11. Intersect(L_a, Lh_a) → ph_a
  12. OrthogonalLine(B, L_b) → Lh_b
  13. Intersect(L_b, Lh_b) → ph_b
  14. OrthogonalLine(C, L_c) → Lh_c
  15. Intersect(L_c, Lh_c) → ph_c
  16. Segment(A, ph_a) → ha
  17. Segment(ph_a, C) → ha_C
  18. Segment(ph_a, B) → ha_B
  19. Segment(C, ph_c) → hc
  20. Segment(A, ph_c) → hc_A
  21. Segment(B, ph_c) → hc_B
  22. Segment(B, ph_b) → hb
  23. Segment(ph_b, C) → hb_C
  24. Segment(ph_b, A) → hb_A
  25. Intersect(Lh_a, Lh_c) → pOrthocenter
  26. Midpoint(C, B) → pMidpoint_a
  27. Midpoint(A, B) → pMidpoint_c
  28. Midpoint(A, C) → pMidpoint_b
  29. Segment(A, pMidpoint_a) → Median_a
  30. Segment(B, pMidpoint_b) → Median_b
  31. Segment(C, pMidpoint_c) → Median_c
  32. Intersect(Median_a, Median_b) → pBarycenter
  33. AngularBisector(C, A, B) → LBisector_A
  34. AngularBisector(A, B, C) → LBisector_B
  35. AngularBisector(A, C, B) → LBisector_C
  36. Intersect(LBisector_A, LBisector_B) → pIncenter
  37. Intersect(LBisector_A, a) → pBisector_A
  38. Intersect(LBisector_B, b) → pBisector_B
  39. Intersect(LBisector_C, c) → pBisector_C
  40. Segment(C, pBisector_C) → Bisector_C
  41. Segment(B, pBisector_B) → Bisector_B
  42. Segment(A, pBisector_A) → Bisector_A
  43. OrthogonalLine(pIncenter, L_a) → LIncenter_a
  44. Intersect(L_a, LIncenter_a) → pIncircle_a
  45. Segment(pIncenter, pIncircle_a) → RadiusCI
  46. Circle(pIncenter, pIncircle_a) → Incircle
  47. OrthogonalLine(pIncenter, L_b) → LIncenter_b
  48. Intersect(L_b, LIncenter_b) → pIncircle_b
  49. OrthogonalLine(pIncenter, L_c) → LIncenter_c
  50. OrthogonalLine(pMidpoint_a, L_a) → LBisector_a
  51. OrthogonalLine(pMidpoint_b, L_b) → LBisector_b
  52. OrthogonalLine(pMidpoint_c, L_c) → LBisector_c
  53. Intersect(LBisector_c, LBisector_b) → pCircumcenter
  54. Segment(pMidpoint_a, pCircumcenter) → Bisector_a
  55. Segment(pMidpoint_b, pCircumcenter) → Bisector_b
  56. Segment(pMidpoint_c, pCircumcenter) → Bisector_c
  57. Angle(C, A, B) → α
  58. Angle(A, B, C) → β
  59. Angle(B, C, A) → γ
  60. Circle(pCircumcenter, C) → Circumcircle
  61. Midpoint(A, pOrthocenter) → pMidpointO_A
  62. Segment(ph_a, pOrthocenter) → Oh_a
  63. Segment(ph_b, pOrthocenter) → Oh_b
  64. Segment(ph_c, pOrthocenter) → Oh_c
  65. Circle(ph_a, ph_b, ph_c) → C9Points
  66. Line(pBarycenter, pOrthocenter) → LEuler
  67. Intersect(C9Points, LEuler) → p9PEuler, p9PEuler'
  68. Midpoint(p9PEuler, p9PEuler') → pCenter9P
  69. Segment(pCenter9P, p9PEuler) → Radius9P
  70. Angle(C, ph_a, A) → ha_A
  71. Angle(C, ph_c, B) → hc_C
  72. Angle(B, ph_b, C) → hb_B
  73. Polygon(pMidpoint_a, pMidpoint_b, pMidpoint_c) → TInner, TInner_c, TInner_a, TInner_b
  74. Rotate(C, -0.904556894302383, B) → aux_B
  75. Angle(aux_B, B, C) → aux_β
  76. Ray(B, aux_B) → aux_b
  77. Rotate(B, 0.9045568943023763, C) → aux_C
  78. Angle(B, C, aux_C) → aux_γ
  79. Ray(C, aux_C) → aux_c
  80. Intersect(aux_b, aux_c) → aux_A
  81. Polygon(B_2, C_2, A_2) → Area_2, a_2, b_2, c_2
  82. Line(A_2, B_2) → L2_c
  83. Line(B_2, C_2) → L2_a
  84. Line(C_2, A_2) → L2_b
  85. OrthogonalLine(A_2, L2_a) → Lh2_a
  86. Intersect(L2_a, Lh2_a) → ph2_a
  87. OrthogonalLine(B_2, L2_b) → Lh2_b
  88. Intersect(L2_b, Lh2_b) → ph2_b
  89. OrthogonalLine(C_2, L2_c) → Lh2_c
  90. Intersect(L2_c, Lh2_c) → ph2_c
  91. Segment(A_2, ph2_a) → ha_2
  92. Segment(ph2_a, C_2) → ha2_C
  93. Segment(ph2_a, B_2) → ha2_B
  94. Segment(C_2, ph2_c) → hc_2
  95. Segment(A_2, ph2_c) → hc2_A
  96. Segment(B_2, ph2_c) → hc2_B
  97. Segment(B_2, ph2_b) → hb_2
  98. Segment(ph2_b, C_2) → hb2_C
  99. Segment(ph2_b, A_2) → hb2_A
  100. Intersect(Lh2_a, Lh2_c) → Orthocenter
  101. Midpoint(C_2, B_2) → pMidpoint2_a
  102. Midpoint(A_2, B_2) → pMidpoint2_c
  103. Midpoint(A_2, C_2) → pMidpoint2_b
  104. Segment(A_2, pMidpoint2_a) → Median2_a
  105. Segment(B_2, pMidpoint2_b) → Median2_b
  106. Segment(C_2, pMidpoint2_c) → Median2_c
  107. Intersect(Median2_a, Median2_b) → Barycenter
  108. AngularBisector(C_2, A_2, B_2) → LBisector2_A
  109. AngularBisector(A_2, B_2, C_2) → LBisector2_B
  110. AngularBisector(A_2, C_2, B_2) → LBisector2_C
  111. Intersect(LBisector2_A, LBisector2_B) → Incenter
  112. Intersect(LBisector2_A, a_2) → pBisector2_a
  113. Intersect(LBisector2_B, b_2) → pBisector2_b
  114. Intersect(LBisector2_C, c_2) → pBisector2_c
  115. Segment(C_2, pBisector2_c) → Bisector2_C
  116. Segment(B_2, pBisector2_b) → Bisector2_B
  117. Segment(A_2, pBisector2_a) → Bisector2_A
  118. OrthogonalLine(Incenter, L2_a) → LIncenter2_a
  119. Intersect(L2_a, LIncenter2_a) → pIncenter2_a
  120. Segment(Incenter, pIncenter2_a) → RadiusCI_2
  121. Circle(Incenter, pIncenter2_a) → Incircle_2
  122. OrthogonalLine(Incenter, L2_b) → LIncenter2_b
  123. Intersect(L2_b, LIncenter2_b) → pIncenter2_b
  124. OrthogonalLine(Incenter, L2_c) → LIncenter2_c
  125. OrthogonalLine(pMidpoint2_a, L2_a) → LBisector2_a
  126. OrthogonalLine(pMidpoint2_b, L2_b) → LBisector2_b
  127. OrthogonalLine(pMidpoint2_c, L2_c) → LBisector2_c
  128. Intersect(LBisector2_c, LBisector2_b) → Circumcenter
  129. Segment(pMidpoint2_a, Circumcenter) → Bisector2_a
  130. Segment(pMidpoint2_b, Circumcenter) → Bisector2_b
  131. Segment(pMidpoint2_c, Circumcenter) → Bisector2_c
  132. Angle(C_2, A_2, B_2) → α_2
  133. Angle(A_2, B_2, C_2) → β_2
  134. Angle(B_2, C_2, A_2) → γ_2
  135. Circle(Circumcenter, C_2) → Circumcircle_2
  136. Midpoint(A_2, Orthocenter) → pMidpointO2_A
  137. Segment(ph2_a, Orthocenter) → Oh2_a
  138. Segment(ph2_b, Orthocenter) → Oh2_b
  139. Segment(ph2_c, Orthocenter) → Oh2_c
  140. Circle(ph2_a, ph2_b, ph2_c) → C9Points_2
  141. Line(Barycenter, Orthocenter) → LEuler_2
  142. Intersect(C9Points_2, LEuler_2) → p9PEuler_2, p9PEuler'_2
  143. Midpoint(p9PEuler_2, p9PEuler'_2) → Center9P
  144. Segment(Center9P, p9PEuler_2) → Radius9P_2
  145. Angle(C_2, ph2_a, A_2) → ha2_A
  146. Angle(C_2, ph2_c, B_2) → hc2_C
  147. Angle(B_2, ph2_b, C_2) → hb2_B
  148. Polygon(pMidpoint2_a, pMidpoint2_b, pMidpoint2_c) → TInner_2, TInner2_c, TInner2_a, TInner2_b
  149. Rotate(C_2, -30°, B_2) → aux2_B
  150. Angle(aux2_B, B_2, C_2) → aux2_β
  151. Ray(B_2, aux2_B) → aux2_b
  152. Rotate(B_2, 120°, C_2) → aux2_C
  153. Angle(B_2, C_2, aux2_C) → aux2_γ
  154. Ray(C_2, aux2_C) → aux2_c
  155. Intersect(aux2_b, aux2_c) → aux2_A
  156. Segment(pBarycenter, A) → RadiusCC
  157. Segment(Barycenter, A_2) → RadiusCC_2
  158. Midpoint(pOrthocenter, B) → pMidpointO_B
  159. Midpoint(Orthocenter, B_2) → pMidpointO2_B
  160. Midpoint(C, pOrthocenter) → pMidpointO_C
  161. Midpoint(Orthocenter, C_2) → pMidpointO2_C
  162. Intersect(L_c, LIncenter_c) → pIncircle_c
  163. Intersect(L2_c, LIncenter2_c) → pIncenter2_c
  164. Midpoint(pPerimeter_B, pPerimeter_C) → pMidpointPer
  165. Line(A, pMidpoint2_a) → LMedian_a
  166. Line(B, pMidpoint2_b) → LMedian_b
  167. Line(C, pMidpoint_c) → LMedian_c
  168. OrthogonalLine(ph_a, b) → f
  169. OrthogonalLine(ph_a, f) → g
  170. Intersect(f, b) → D
  171. Intersect(b, hb) → E
  172. Intersect(hb, g) → F
  173. Intersect(f, g) → G
  174. Segment(D, G) → radio
  175. Segment(F, G) → h
  176. Angle(D, E, F) → δ
  177. Angle(E, F, G) → ε
  178. Angle(F, G, D) → ζ
  179. Angle(G, D, E) → η
  180. Circle(G, C) → Eath
  181. Intersect(Eath, ha) → H,
  182. OrthogonalLine(H, ha) → i
  183. OrthogonalLine(B, a) → j
  184. OrthogonalLine(C, a) → k
  185. Intersect(k, i) → GreatPyramid_1
  186. Intersect(i, j) → J
  187. Segment(C, GreatPyramid_1) → l
  188. Segment(GreatPyramid_1, J) → m
  189. Segment(J, B) → n
  190. Angle(C, GreatPyramid_1, H) → θ
  191. Circle(A, H) → Moon
  192. Circle(H, A) → d
  193. Intersect(d, m) → I, K
  194. Intersect(Moon, Lh_a) → L, M
  195. OrthogonalLine(L, Lh_a) → GreatPyramid
  196. OrthogonalLine(I, GreatPyramid) → p
  197. OrthogonalLine(K, GreatPyramid) → q
  198. Intersect(GreatPyramid, p) → N
  199. Intersect(q, GreatPyramid) → O
  200. Segment(N, I) → r
  201. Segment(N, O) → s
  202. Segment(K, O) → t
  203. Intersect(Moon, r) → P, Q
  204. Intersect(Moon, t) → R, S
  205. Polygon(O, K, J) → poly1, Ratio3, Ratio4, Ratio5
  206. Angle(J, K, O) → ι
  207. Angle(L, O, K) → κ
  208. Segment(L, O) → e
  209. Angle(I, J, B) → λ
  210. Intersect(m, Ratio4) → T
  211. Intersect(m, Ratio4) → U
  212. Intersect(Eath, Moon, 1) → V
  213. Segment(V, J) → f_1
  214. Angle(O, K, O) → μ