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Alternate Segment Trisection

Alternate Segment Trisection
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2016-07-19 02:37
TrigonometryCoordinate GeometryCalculusGeometric Construction
This example demonstrates an unconventional method to trisect a line segment using geometric constructions. It involves core mathematical concepts like rational numbers, angles, transformations, and calculus elements such as definite integrals and differential equations. The construction is achieved through commands like Segment, Dilate, Circle, Intersect, Rotate, and Line in GeoGebra. Suitable for high school and advanced learners exploring complex geometric constructions and higher mathematics applications.

GeoGebra Commands & Parameters

  1. Segment(A, B) → f
  2. Dilate(A, If[0 < a ≤ 1, 1 - a, 0], B) → A'
  3. Circle(A, A') → c
  4. Dilate(B, If[1 < a ≤ 2, 2 - a, 0], A) → B'
  5. Circle(B, B') → d
  6. Intersect(c, d, 1) → C
  7. Intersect(c, d, 2) → D
  8. Rotate(B, If[2 < a ≤ 3, ((a - 2) * π / 3), π / 3], A) → B'_1
  9. Rotate(B, If[2 < a ≤ 3, ((-(a - 2)) * π / 3), ((-π)) / 3], A) → B'_2
  10. Dilate(B'_2, If[3 < a ≤ 4, 4 - a, 0], A) → E
  11. Circle(B'_2, E) → e
  12. Line(B, B'_2) → g
  13. Intersect(e, g, 2) → F
  14. Dilate(B, If[4 < a ≤ 5, 5 - a, 0], F) → B'_3
  15. Segment(B, B'_3) → h
  16. Line(A, B'_2) → i
  17. Intersect(e, i, 2) → G
  18. Dilate(A, If[5 < a ≤ 6, 6 - a, 0], G) → A'_1
  19. Segment(A, A'_1) → j
  20. Line(B'_1, F) → k
  21. Line(B'_1, G) → l
  22. Intersect(k, f) → H
  23. Intersect(l, f) → I
  24. Segment(A, H) → n
  25. Dilate(B'_1, If[6 < a ≤ 7, 7 - a, 0], H) → J
  26. Segment(B'_1, J) → p
  27. Dilate(B'_1, If[6 < a ≤ 7, 7 - a, 0], I) → K
  28. Segment(B'_1, K) → q
  29. Dilate(H, If[7 < a ≤ 8, 8 - a, 0], F) → H'
  30. Segment(H, H') → r
  31. Dilate(I, If[7 < a ≤ 8, 8 - a, 0], G) → I'
  32. Segment(I, I') → s
  33. Segment(H, I) → m
  34. Segment(I, B) → t
  35. Rotate(A, If[8 < a ≤ 9, ((a - 8) * π), π], H) → A'_2
  36. Rotate(H, If[8 < a ≤ 9, ((a - 8) * π), π], H) → H'_1
  37. Segment(A'_2, H'_1) → n'
  38. Rotate(A'_2, If[9 < a ≤ 10, ((a - 9) * π), π], I) → L
  39. Rotate(H'_1, If[9 < a ≤ 10, ((a - 9) * π), π], I) → M
  40. Segment(L, M) → n''