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Pyramid Construction by JesusSD

Pyramid Construction by JesusSD
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2017-08-12 22:17
VectorSolid GeometryGeometric Construction
This example demonstrates the construction of a pyramid and its related geometric elements using GeoGebra. Key math concepts include vectors, spheres, cosine, line segments, limits, orthogonal lines, midpoints, angle bisectors, rotations, and more. The construction involves commands like Polygon, Line, Circle, Intersect, Segment, OrthogonalLine, Midpoint, AngularBisector, Angle, Rotate, and Ray, as detailed in the construction summary. It is suitable for high school and above, focusing on solid geometry and coordinate geometry exploration.

GeoGebra Commands & Parameters

  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