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Dynamic Demonstration of Archimedes' Proposition 3

Dynamic Demonstration of Archimedes' Proposition 3
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2018-06-24 12:39
CircleTrigonometryGeometric Construction
This example dynamically demonstrates Proposition 3 from Archimedes' Book of Lemmas. ① It illustrates the geometric relationships in the proposition through dynamic illustrations. ② Core mathematical concepts involved include circles, right triangles, angles, rotations, and similarity. ③ Construction highlights the use of GeoGebra commands such as semicircle, line, midpoint, orthogonal line, intersect, circle arc, point, angle, rotate, circle, dilate, and segment. ④ Suitable for high school mathematics education, helping to understand geometric proofs and dynamic changes.

GeoGebra Commands & Parameters

  1. Semicircle(A, B) → c
  2. Line(A, B) → f
  3. Midpoint(A, B) → C
  4. OrthogonalLine(C, f) → g
  5. Intersect(c, g) → D,
  6. CircleArc(C, D, A) → d
  7. Point(d) → E
  8. OrthogonalLine(E, f) → h
  9. Intersect(h, f) → F
  10. Angle(E, C, A) → β
  11. Rotate(E, (-β), C) → E'
  12. Angle(E', C, E) → γ
  13. Circle(F, A) → e
  14. Intersect(e, f, 2) → G
  15. Semicircle(A, B) → k
  16. Angle(A, F, E) → α
  17. Dilate(A, If[0 < a ≤ 1, 1 - a, 0], F) → A'
  18. Segment(A, A') → i
  19. Dilate(E, If[0 < a ≤ 1, 1 - a, 0], F) → E'_1
  20. Segment(E, E'_1) → j
  21. Rotate(A, If[1 < a ≤ 2, ((-(a - 1)) * π), (-π)], F) → A'_1
  22. Rotate(A', If[1 < a ≤ 2, ((-(a - 1)) * π), (-π)], F) → A''
  23. Segment(A'_1, A'') → i'
  24. Rotate(A, If[2 < a ≤ 3, ((-(a - 2)) * β), (-β)], C) → A'_2
  25. CircleArc(C, A'_2, A) → p
  26. Rotate(p, If[4 < a ≤ 5, ((-(a - 4)) * β), (-β)], C) → p'
  27. Rotate(E, If[4 < a ≤ 5, ((-(a - 4)) * β), (-β)], C) → E'_2
  28. Angle(E', B, A) → δ
  29. Dilate(A'_1, If[6 < a ≤ 7, 7 - a, 0], B) → H
  30. Segment(A'_1, H) → l
  31. Rotate(A'_1, If[7.5 < a ≤ 8.5, ((-(a - 7.5)) * δ), (-δ)], B) → I
  32. Rotate(H, If[7.5 < a ≤ 8.5, ((-(a - 7.5)) * δ), (-δ)], B) → H'
  33. Segment(I, H') → l'