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Archimedes' Semicircle Exploration

Archimedes' Semicircle Exploration
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2018-06-10 08:09
CircleTransformationGeometric Construction
This dynamic and modifiable geometric problem illustrates Archimedes' semicircle properties, involving circles, angles, and geometric transformations. Using GeoGebra commands such as point, midpoint, semicircle, line, circle arc, intersection, dilation, segment, angle, circle, rotation, etc., the example provides a clear understanding of geometric transformations and circle properties. Suitable for high school geometry learners and educators.

GeoGebra Commands & Parameters

  1. Point(Ray[A, A + (1, 0)]) → B
  2. Midpoint(A, B) → C
  3. Semicircle(A, B) → c
  4. Line(A, B) → f
  5. PointIn(p) → D
  6. CircleArc(C, B, A) → d
  7. Semicircle(D, C) → g
  8. Semicircle(C, D) → h
  9. Midpoint(D, C) → E
  10. Intersect(h, c) → F, , ,
  11. Intersect(g, c) → G, , ,
  12. Line(D, F) → i
  13. Line(D, G) → j
  14. Line(A, G) → k
  15. Line(B, F) → l
  16. Intersect(k, l) → H
  17. Line(D, H) → m
  18. Dilate(D, If[0 < q ≤ 1, 1 - q, 0], F) → D'
  19. Dilate(D, If[0 < q ≤ 1, 1 - q, 0], G) → D'_1
  20. Segment(D, D') → r
  21. Segment(D, D'_1) → s
  22. Angle(D, G, C) → α
  23. Angle(C, F, D) → β
  24. Dilate(C, If[0 < q ≤ 1, 1 - q, 0], G) → C'
  25. Dilate(C, If[0 < q ≤ 1, 1 - q, 0], F) → C'_1
  26. Segment(C, C') → t
  27. Segment(C, C'_1) → f_1
  28. Dilate(A, If[2 < q ≤ 3, 3 - q, 0], G) → A'
  29. Segment(A, A') → g_1
  30. Dilate(B, If[3 < q ≤ 4, 4 - q, 0], H) → B'
  31. Segment(B, B') → h_1
  32. Dilate(H, If[4 < q ≤ 5, 5 - q, 0], F) → H'
  33. Segment(H, H') → i_1
  34. Intersect(f, m) → I
  35. Min({Distance[I, H], Distance[I, B], Distance[I, A]}) → u
  36. Dilate(D, If[5 < q ≤ 6, 6 - q, 0], I) → D'_2
  37. Segment(D, D'_2) → j_1
  38. Circle(I, (0.45 * u)) → c_1
  39. Intersect(c, f, 2) → L
  40. Intersect(c_1, f) → M, N
  41. Rotate(N, If[6.5 < q ≤ 7.5, ((q - 6.5) * π / 2), π / 2], I) → N'
  42. CircleSector(I, N, N') → d_1
  43. Rotate(d_1, If[8.25 < q ≤ 9.25, ((q - 8.25) * π / 2), π / 2], I) → e_1