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Four Mutually Tangent Spheres and the Radius of the Fifth Tangent Sphere

Four Mutually Tangent Spheres and the Radius of the Fifth Tangent Sphere
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2014-10-25 23:28
CircleSolid GeometryTransformationGeometric Construction
This GeoGebra example demonstrates a classic solid geometry problem: four spheres of equal radius R are mutually tangent to each other, forming a symmetric tetrahedral arrangement, and the task is to find the radius of a fifth sphere that is tangent to all four. The core mathematical concepts involve sphere tangency, spatial distance calculations, and exploiting symmetry to simplify the problem. The construction uses GeoGebra commands such as Sphere to create the spheres, Intersect and Distance to determine centers and tangency conditions, and Rotate, Dilate, and other transformations to build the symmetric configuration. The solution can be derived through geometric reasoning or algebraic equations. This example is suitable for high school or university-level solid geometry instruction, as well as for mathematical modeling or competition training, helping students understand tangency and distance relationships in 3D space.

GeoGebra Commands & Parameters

  1. Intersect(Circle[O3, 5.1961524227], LineBisector[O3, O4], 2) → A4
  2. Intersect(OrthogonalLine[A4, Line[O3, O4]], OrthogonalLine[O3, Line[A4, O4]]) → OO
  3. Angle(O3, OO, O4) → α
  4. Rotate(O_4, (-α), Line[O, yAxis]) → O_3
  5. Rotate(O_3, 120*°, O - (0, 0, ((0.25 * M) * h))) → O_1
  6. Rotate(O_3, -120*°, O - (0, 0, ((0.25 * M) * h))) → O_2
  7. Length(Segment[O_1, O_2]) → d
  8. Sphere(O_1, (0.5 * d)) → φ_1
  9. Sphere(O_2, (0.5 * d)) → φ_2
  10. Sphere(O_3, (0.5 * d)) → φ_3
  11. Sphere(O_4, (0.5 * d)) → φ_4
  12. Pyramid(O_1, O_2, O_3, O_4) → a, грань4, ребро1, грань3, грань2, грань1, ребро6, ребро5, ребро4, ребро2, ребро3
  13. Intersect(Line[O_4, O_1], φ_1, 1) → A_1
  14. Intersect(Line[O_4, O_2], φ_2, 1) → A_2
  15. Intersect(Line[O_4, O_3], φ_3, 1) → A_3
  16. Intersect(Line[O, O_1], φ_1, 2) → B_1
  17. Intersect(Line[O, O_2], φ_2, 2) → B_2
  18. Intersect(Line[O, O_3], φ_3, 2) → B_3
  19. Intersect(Line[O, O_4], φ_4, 2) → B_4
  20. Sphere(O, (d * ((0.25 * k6) + 0.5))) → Ω
  21. Sphere(O, (d * ((0.25 * k6) - 0.5))) → ω
  22. Intersect(Line[O_1, O_2], φ_1, 2) → A_4
  23. Intersect(Line[O_2, O_3], φ_2, 2) → A_5
  24. Intersect(Line[O_3, O_1], φ_3, 2) → A_6
  25. Distance(O, O_1) → OO1
  26. Distance(O_2, O_1) → O2O1
  27. Dilate(O, 0.8164965809277261, O_1) → O'
  28. Dilate(O, f, O_2) → O'_1
  29. Dilate(O, f, O_3) → O'_2
  30. Dilate(O, f, O_4) → O'_3
  31. Segment(O, O_1) → g
  32. Segment(O_1, B_1) → i
  33. Line(C, xAxis) → l
  34. Point(l) → D
  35. Sphere(O_1, (0.5 * d)) → p1
  36. Sphere(O_2, (0.5 * d)) → p2
  37. Sphere(O_3, (0.5 * d)) → p3
  38. Sphere(O_4, (0.5 * d)) → p4
  39. RotateText("\copyright T.C.\;\&\;uStas", 90*°) → надпись1

AI Drawing Prompt Reference

Draw four equal-radius spheres that are mutually tangent, forming a symmetric tetrahedral arrangement. Then find and depict the fifth sphere that is tangent to all four. Show the symmetric geometric structure, labeling the radii of all spheres and their tangency relationships to understand spatial distance and tangency in 3D space.
The prompt above was compiled from the original drawing commands and is for reference only.