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Equilateral Triangle + Semicircle = ? (GoGeometry Action 88)

Equilateral Triangle + Semicircle = ? (GoGeometry Action 88)
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2017-08-16 05:13
TriangleCircleTransformationGeometric ConstructionSlider Animation
This dynamic GeoGebra example illustrates a geometric phenomenon that occurs when a semicircle is drawn on one side of an equilateral triangle. It explores the interplay between equilateral triangles, semicircles, circumcircles, and transformations such as rotation and dilation. The construction uses commands like Polygon, Segment, Midpoint, Semicircle, Rotate, Dilate, CircleSector, CircleArc, and Intersect to create a rich set of objects including rotated copies, intersection points, and auxiliary arcs. Suitable for middle and high school geometry classes, math clubs, or self-study, this interactive applet encourages hypothesis testing and visual reasoning about geometric relationships.

GeoGebra Commands & Parameters

  1. Polygon(A, B, 3) → poly1, f, g, h, C
  2. Segment(A, B) → i
  3. Segment(B, C) → j
  4. Segment(C, A) → k
  5. Midpoint(j) → D
  6. Semicircle(C, B) → c
  7. Rotate(B, π / 3, D) → B'
  8. Rotate(B, (2 * π / 3), D) → B'_1
  9. Rotate(B, If[0 < a ≤ 1, (a * π / 3), π / 3], B) → B'_2
  10. Rotate(C, If[0 < a ≤ 1, (a * π / 3), π / 3], B) → C'
  11. Segment(B'_2, C') → j'
  12. Rotate(B'_2, If[1 < a ≤ 2, ((a - 1) * π / 3), π / 3], A) → E
  13. Rotate(C', If[1 < a ≤ 2, ((a - 1) * π / 3), π / 3], A) → C''
  14. Segment(E, C'') → j''
  15. Dilate(C, If[2 < a ≤ 3, 3 - a, 0], D) → C'_1
  16. Segment(C, C'_1) → l
  17. Rotate(C, If[3 < a ≤ 4, ((a - 3) * π), π], D) → C'_2
  18. Rotate(C'_1, If[3 < a ≤ 4, ((a - 3) * π), π], D) → F
  19. Segment(C'_2, F) → l'
  20. Rotate(B, If[4 < a ≤ 5, ((a - 4) * π), π], D) → B'_3
  21. Segment(D, B'_3) → m
  22. CircleSector(D, B, B'_3) → d
  23. Rotate(B, If[5 < a ≤ 6, ((a - 5) * π / 3), π / 3], D) → B'_4
  24. CircleArc(D, B, B'_4) → e
  25. Rotate(e, If[7 < a ≤ 8, ((a - 7) * π / 3), π / 3], D) → e'
  26. Rotate(e', If[9 < a ≤ 10, ((a - 9) * π / 3), π / 3], D) → e''
  27. Segment(C, B) → n
  28. Segment(A, B'_1) → p
  29. Segment(A, B') → q
  30. Intersect(p, n) → G
  31. Intersect(d, n, 1) → I
  32. Intersect(q, n) → H
  33. Segment(C, G) → r
  34. Segment(G, H) → s
  35. Segment(H, B) → t
  36. Dilate(A, If[10 < a ≤ 11, 11 - a, 0], G) → A'
  37. Segment(A, A') → b
  38. Dilate(A, If[10 < a ≤ 11, 11 - a, 0], H) → A'_1
  39. Segment(A, A'_1) → f_1
  40. Dilate(G, If[11 < a ≤ 12, 12 - a, 0], B'_1) → G'
  41. Dilate(H, If[11 < a ≤ 12, 12 - a, 0], B') → H'
  42. Segment(G, G') → g_1
  43. Segment(H, H') → h_1
  44. Dilate(C, If[12 < a ≤ 13, 13 - a, 0], G) → C'_3
  45. Segment(C, C'_3) → i_1
  46. Rotate(C, If[13 < a ≤ 14, ((a - 13) * π), π], G) → C'_4
  47. Rotate(C'_3, If[13 < a ≤ 14, ((a - 13) * π), π], G) → J
  48. Segment(C'_4, J) → j_1
  49. Rotate(C'_4, If[14 < a ≤ 15, ((a - 14) * π), π], H) → K
  50. Rotate(J, If[14 < a ≤ 15, ((a - 14) * π), π], H) → J'
  51. Segment(K, J') → k_1
  52. Dilate(C, If[-1 < a < 0, (-a), 0], B) → C'_5
  53. Segment(C, C'_5) → l_1

AI Drawing Prompt Reference

Draw an equilateral triangle with a semicircle on one side. Use the midpoint of that side as a rotation center to rotate vertices and sides, creating multiple rotated copies. A slider controls the rotation angle, gradually revealing semicircles, sectors, arcs, and rotated segments. Inside the triangle, construct intersection points via rotation and scaling, connect them to form new segments, and apply scaling or rotation about certain points to generate dynamic geometric shapes. The overall composition shows the interplay of an equilateral triangle and semicircle under transformations.
The prompt above was compiled from the original drawing commands and is for reference only.