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Viviani's Theorem Exploration

Viviani's Theorem Exploration
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2019-05-07 23:29
VectorSolid GeometryCalculusDerivativeGeometric Construction
This example demonstrates Viviani's Theorem, showing the sum of distances from a point inside a cube to its three coordinate planes equals the cube's altitude. It involves core concepts in solid geometry, vectors, derivatives, and geometric constructions. The construction uses commands like Polygon, Point, Circle, Intersect, Line, etc., from GeoGebra. Ideal for high school and above students exploring geometric theorems.

GeoGebra Commands & Parameters

  1. Polygon(B, C, 3) → poly1, a, b, c, A
  2. Point(a) → U
  3. Circle(U, 0.01) → f_1
  4. Intersect(f_1, b, 1) → Z
  5. Line(Z, a) → h_2
  6. Intersect(h_2, c) → A_1
  7. Intersect(f_1, a, 1) → V
  8. Line(V, b) → g_2
  9. Intersect(g_2, c) → W
  10. Polygon(W, A_1, V) → t1, i_2, k_2, n_2
  11. PointIn(t1) → M
  12. OrthogonalLine(M, b) → d
  13. OrthogonalLine(M, a) → e
  14. OrthogonalLine(M, c) → f
  15. Intersect(d, b) → P
  16. Intersect(e, a) → N
  17. Intersect(f, c) → Q
  18. Segment(M, P) → g
  19. Segment(M, Q) → h
  20. Segment(M, N) → i
  21. Segment(A, Q) → j
  22. Segment(B, N) → k
  23. Segment(C, P) → l
  24. Angle(M, Q, A) → α
  25. Angle(M, P, C) → β
  26. Angle(M, N, B) → γ
  27. Segment(A, M) → p
  28. Segment(B, M) → q
  29. Segment(M, C) → r
  30. Rotate(P, 30°, M) → P'
  31. Angle(P, M, P') → δ
  32. Ray(M, P') → s
  33. Intersect(b, s) → D
  34. Intersect(i, poly1) → E, E_1
  35. Polygon(D, M, 3) → poly2, t, a_1, b_1, F
  36. Segment(M, F) → c_1
  37. Segment(M, D) → d_1
  38. OrthogonalLine(D, a_1) → e_1
  39. Intersect(a_1, c_1) → G
  40. Intersect(a_1, c_1) → H
  41. Intersect(e_1, a_1) → I
  42. Rotate(N, 30°, M) → N'
  43. Angle(N, M, N') → ε
  44. Ray(M, N') → g_1
  45. Intersect(a, g_1) → J
  46. Polygon(J, M, 3) → poly3, h_1, i_1, j_1, K
  47. Line(M, a) → k_1
  48. Intersect(c, k_1) → L
  49. Polygon(L, M, 3) → poly4, l_1, m_1, n_1, O
  50. OrthogonalLine(O, k_1) → p_1
  51. Intersect(k_1, p_1) → R
  52. Segment(O, R) → q_1
  53. Segment(M, K) → r_1
  54. Segment(M, J) → s_1
  55. Segment(D, A) → t_1
  56. Point(t_1) → S
  57. Vector(D, S) → u
  58. Translate(poly2, u) → poly2'
  59. Translate(D, u) → D'
  60. Translate(I, u) → I'
  61. Segment(D', I') → a_2
  62. Translate(M, u) → M'
  63. Translate(F, u) → F'
  64. Segment(M', F') → b_2
  65. Mirror(J, N) → J'
  66. Polygon(B, L, M, J') → q1, b_3, l_2, m_2, j'
  67. Polygon(A, O, M', S) → q2, a_3, o_1, m', s_2
  68. Point(b) → T
  69. Polygon(M, F, T, J) → q3, m_3, f_2, t_2, j_2
  70. Translate(M, u) → M'_1
  71. Translate(P, u) → P'_1
  72. Segment(M'_1, P'_1) → g'

AI Drawing Prompt Reference

Draw a point inside a cube and show that the sum of its distances to the three coordinate planes equals the cube's height. Start by constructing the cube, then draw perpendicular segments from the point to each coordinate plane. Calculate the lengths of these segments and intersect them with the cube's edges to form new shapes. Finally, use rotations and intersections to illustrate the relationships between these shapes, demonstrating Viviani's theorem.
The prompt above was compiled from the original drawing commands and is for reference only.