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Simple Trisection Construction

Simple Trisection Construction
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2016-10-30 10:52
AngleTransformationGeometric Construction
This example demonstrates that if parallel lines cut off congruent segments on one transversal, they cut off congruent segments on every transversal. Key concepts include parallel lines, congruent segments, and geometric construction. It uses GeoGebra commands such as Point, Line, Rotate, Intersect, Segment, etc. Suitable for middle school and above students to learn geometric construction and related theorems.

GeoGebra Commands & Parameters

  1. Point(y = 0) → A
  2. Line(A, B) → f
  3. Rotate(B, α, A) → B'
  4. Angle(B, A, B') → β
  5. Line(A, B') → g
  6. Rotate(B', γ, A) → B''
  7. Angle(B', A, B'') → δ
  8. Ray(A, B'') → h
  9. Circle(A, a) → c
  10. OrthogonalLine(A, g) → i
  11. Intersect(c, i, 2) → C
  12. Circle(C, a) → d
  13. Intersect(d, i, 2) → D
  14. Circle(D, a) → e
  15. Intersect(e, i, 2) → E
  16. Line(C, g) → j
  17. Line(D, g) → k
  18. Line(E, g) → l
  19. Intersect(h, j) → F
  20. Intersect(k, h) → G
  21. Intersect(l, h) → H
  22. Segment(A, F) → m
  23. Circle(A, (0.3 * m)) → p
  24. Point(l) → I
  25. Point(g) → J
  26. Segment(I, J) → n
  27. Intersect(k, n) → K
  28. Intersect(j, n) → L
  29. Intersect(p, g, 2) → M
  30. Intersect(p, h, 1) → N
  31. CircleSector(A, M, N) → q
  32. Dilate(A, If[0 < b ≤ 1, 1 - b, 0], F) → A'
  33. Segment(A, A') → r
  34. Dilate(F, If[1 < b ≤ 2, 2 - b, 0], G) → F'
  35. Segment(F, F') → s
  36. Dilate(G, If[2 < b ≤ 3, 3 - b, 0], H) → G'
  37. Segment(G, G') → t
  38. Dilate(A, If[3 < b ≤ 4, 4 - b, 0], F) → A'_1
  39. Vector(A, A'_1) → u
  40. Translate(q, u) → q'
  41. Dilate(F, If[4 < b ≤ 5, 5 - b, 0], G) → F'_1
  42. Vector(F, F'_1) → v
  43. Translate(q', v) → q''
  44. Dilate(G, If[5 < b ≤ 6, 6 - b, 0], H) → G'_1
  45. Vector(G, G'_1) → w
  46. Translate(q'', w) → q'''
  47. Rotate(q''', If[6 < b ≤ 7, ((-(b - 6)) * π), (-π)], H) → c_1
  48. Rotate(A, If[7 < b ≤ 8, ((b - 7) * π), π], F) → A'_2
  49. Rotate(A', If[7 < b ≤ 8, ((b - 7) * π), π], F) → A''
  50. Segment(A'_2, A'') → r'
  51. Rotate(A'_2, If[8 < b ≤ 9, ((b - 8) * π), π], G) → O
  52. Rotate(A'', If[8 < b ≤ 9, ((b - 8) * π), π], G) → A'''
  53. Segment(O, A''') → r''
  54. Dilate(I, If[11.5 < b ≤ 12.5, 12.5 - b, 0], K) → I'
  55. Segment(I, I') → f_1
  56. Dilate(K, If[12.5 < b ≤ 13.5, 13.5 - b, 0], L) → K'
  57. Segment(K, K') → g_1
  58. Dilate(L, If[13.5 < b ≤ 14.5, 14.5 - b, 0], J) → L'
  59. Segment(L, L') → h_1
  60. Rotate(I, If[14.5 < b ≤ 15.5, ((-(b - 15.5)) * π), (-π)], K) → I'_1
  61. Rotate(I', If[14.5 < b ≤ 15.5, ((-(b - 15.5)) * π), (-π)], K) → I''
  62. Rotate(I, If[14.5 < b ≤ 15.5, ((-(b - 14.5)) * π), (-π)], K) → I'_2
  63. Rotate(I', If[14.5 < b ≤ 15.5, ((-(b - 14.5)) * π), (-π)], K) → I''_1
  64. Segment(I'_2, I''_1) → i_1
  65. Rotate(I'_2, If[15.5 < b ≤ 16.5, ((-(b - 15.5)) * π), (-π)], L) → P
  66. Rotate(I''_1, If[15.5 < b ≤ 16.5, ((-(b - 15.5)) * π), (-π)], L) → Q
  67. Segment(P, Q) → j_1

AI Drawing Prompt Reference

Draw a simple trisection construction. Start by drawing a horizontal line segment AB, then rotate B to get B'. Draw two new line segments g and h, and use intersection points C, D, E to create circles c, d, e. Draw a ray h and construct an arc q. Use scaling and translation operations around points A, F, G, H to perform multiple transformations, ultimately forming a complex geometric figure.
The prompt above was compiled from the original drawing commands and is for reference only.