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Sphere Touching Three Cones

Sphere Touching Three Cones
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2017-04-10 18:11
CircleSolid GeometryGeometric Construction
In collections intersection: surface-surface, curve-curve intersection: surface-surface, curve-curve intersection: surface-surface, curve-curve intersection: surface-surface, curve-curve
This example demonstrates the geometric relationship formed by a sphere touching three cones. It involves core mathematical concepts such as sphere, cone, tangent, midpoint, perpendicular line, intersection point, etc. The construction uses commands and objects like Circle, Line, Intersect, Midpoint, OrthogonalLine, Point, Segment, Center, Cone, Sphere as per the summary. Suitable for high school solid geometry teaching and geometric construction practice.

GeoGebra Commands & Parameters

  1. Circle(A, B) → c
  2. Circle(B, A) → d
  3. Line(A, B) → f
  4. Intersect(d, f, 2) → D
  5. Circle(D, B) → e
  6. Circle(D, A) → g
  7. Circle(A, D) → h
  8. Intersect(g, h, 1) → C
  9. Midpoint(C, A) → E
  10. Circle(C, E) → k
  11. OrthogonalLine(A, xAxis) → i
  12. OrthogonalLine(D, xAxis) → j
  13. OrthogonalLine(C, xAxis) → l
  14. Intersect(e, k, 1) → F
  15. Point(i) → G
  16. OrthogonalLine(G, i) → m
  17. Intersect(l, m) → H
  18. Intersect(j, m) → I
  19. Intersect(i, xAxis) → J
  20. Intersect(l, xAxis) → K
  21. Intersect(j, xAxis) → L
  22. Circle(J, Segment[B, A]) → p
  23. Circle(K, Segment[B, A]) → q
  24. Circle(L, Segment[B, A]) → r_2
  25. Intersect(r_2, xAxis, 2) → M
  26. Intersect(q, xAxis, 2) → N
  27. Intersect(p, xAxis, 1) → O
  28. Intersect(q, xAxis, 1) → P
  29. Intersect(p, xAxis, 2) → Q
  30. Intersect(r_2, xAxis) → R, S
  31. Segment(O, G) → n
  32. Segment(G, Q) → s
  33. Segment(P, H) → t
  34. Segment(H, N) → a
  35. Segment(R, I) → b
  36. Segment(I, M) → f_1
  37. Segment(G, A) → g_1
  38. Segment(H, C) → h_1
  39. Segment(I, D) → i_1
  40. OrthogonalLine(M, xAxis) → j_1
  41. OrthogonalLine(R, xAxis) → k_1
  42. Intersect(e, k_1, 1) → T
  43. Intersect(e, j_1, 1) → U
  44. Segment(M, U) → l_1
  45. Segment(R, T) → m_1
  46. OrthogonalLine(N, xAxis) → n_1
  47. OrthogonalLine(P, xAxis) → p_1
  48. Intersect(k, p_1, 1) → V
  49. Intersect(k, n_1, 1) → W
  50. Segment(N, W) → q_1
  51. Segment(P, V) → r_1
  52. OrthogonalLine(O, xAxis) → s_1
  53. OrthogonalLine(Q, xAxis) → t_1
  54. Intersect(c, s_1, 1) → Z
  55. Intersect(c, t_1, 1) → A_1
  56. Segment(O, Z) → a_1
  57. Segment(Q, A_1) → b_1
  58. Circle(A, C, D) → c_1
  59. Center(c_1) → B_1
  60. Segment(F_1, C_1) → d_1
  61. Circle(d_1, G_1) → e_1
  62. Segment(A, B) → f_2
  63. Cone(C_1, F_1, f_2) → o, c_2, u
  64. Segment(H_1, E_1) → g_2
  65. Cone(E_1, H_1, f_2) → v, d_2, w
  66. Segment(I_1, D_1) → h_2
  67. Cone(D_1, I_1, f_2) → o_1, e_2, u_1
  68. OrthogonalLine(J_1, xOyPlane) → i_2
  69. Segment(B_1, D) → j_2
  70. Line(D, xAxis) → k_2
  71. Circle(D, B_1) → p_2
  72. Intersect(p_2, k_2, 2) → K_1
  73. OrthogonalLine(K_1, k_2) → l_2
  74. Intersect(l_2, xAxis) → L_1
  75. OrthogonalLine(M, f_1) → m_2
  76. Circle(M, r) → q_2
  77. Intersect(q_2, m_2, 2) → M_1
  78. OrthogonalLine(M_1, m_2) → n_2
  79. Intersect(l_2, n_2) → N_1
  80. Sphere(O_1, r) → v_1

AI Drawing Prompt Reference

Draw a sphere touching three cones to demonstrate their geometric relationship. Start by drawing two intersecting circles and find their intersection points. Use these points to construct additional circles and lines. Identify the intersection points of the circles with lines to determine the contact points of the sphere. Draw perpendicular lines to the x-axis to locate key points and connect them to form the tangent lines between the sphere and cones. Finally, connect the points with line segments to complete the geometric figure of the sphere touching three cones.
The prompt above was compiled from the original drawing commands and is for reference only.