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Loop of Coins: Tangent Circles in a Closed Geometric Chain

Loop of Coins: Tangent Circles in a Closed Geometric Chain
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2012-10-30 15:02
TriangleCirclePolygonTransformationGeometric Construction
This GeoGebra example demonstrates a closed loop of congruent circles (coins) arranged so that adjacent circles are tangent, forming a ring-like geometric chain. It highlights core concepts including circle tangency, equilateral triangles formed by connecting circle centers, circle sectors, rotational symmetry, and polygonal construction. The construction relies on GeoGebra commands such as Circle, Point, Intersect, PolyLine, Segment, Midpoint, and CircleSector, with objects A through H representing centers and intersection points, c through h representing circle objects, and auxiliary text and image elements (text1, pic1–pic7) for annotation. Suitable for high school geometry courses, it serves as a dynamic demonstration for lessons on circles, tangents, triangle area, geometric construction, and transformation, supporting both teacher-led instruction and student inquiry.

GeoGebra Commands & Parameters

  1. Circle(A, 2) → c
  2. Point(c) → B
  3. Circle(B, 2) → d
  4. Point(d) → C
  5. Circle(C, 2) → e
  6. Point(e) → D
  7. Circle(D, 2) → f
  8. Point(f) → E
  9. Circle(E, 2) → g
  10. Point(g) → F
  11. Circle(F, 2) → h
  12. Point(h) → G
  13. Circle(G, 2) → k
  14. Point(k) → H
  15. Circle(H, 2) → p
  16. Point(p) → I
  17. Circle(I, 2) → q
  18. Point(q) → J
  19. Circle(J, 2) → r
  20. Circle(A, 1) → s_1
  21. Circle(B, 1) → t_1
  22. Circle(C, 1) → c_1
  23. Circle(D, 1) → d_1
  24. Circle(E, 1) → e_1
  25. Circle(F, 1) → f_1
  26. Circle(G, 1) → g_1
  27. Circle(H, 1) → h_1
  28. Circle(I, 1) → k_1
  29. Circle(J, 1) → p_1
  30. Intersect(c, r, 2) → K
  31. Circle(K, 1) → q_1
  32. PolyLine(A, B, C, D, E, F, G, H, I, J) → linePoly
  33. Segment(K, A) → line2
  34. Segment(K, J) → line3
  35. Midpoint(A, B) → L
  36. Midpoint(B, C) → M
  37. Midpoint(C, D) → N
  38. Midpoint(D, E) → O
  39. Midpoint(E, F) → P
  40. Midpoint(F, G) → Q
  41. Midpoint(G, H) → R
  42. Midpoint(H, I) → S
  43. Midpoint(I, J) → T
  44. Midpoint(J, K) → U
  45. Midpoint(K, A) → V
  46. CircleSector(A, V, L) → s_2
  47. CircleSector(B, L, M) → t_2
  48. CircleSector(C, M, N) → r_2
  49. CircleSector(D, N, O) → c_2
  50. CircleSector(E, O, P) → d_2
  51. CircleSector(F, P, Q) → e_2
  52. CircleSector(G, Q, R) → f_2
  53. CircleSector(H, R, S) → g_2
  54. CircleSector(I, S, T) → h_2
  55. CircleSector(J, T, U) → k_2
  56. CircleSector(K, U, V) → p_2
  57. CircleSector(A, L, V) → q_3
  58. CircleSector(B, M, L) → r_3
  59. CircleSector(C, N, M) → s_3
  60. CircleSector(D, O, N) → t_3
  61. CircleSector(E, P, O) → c_3
  62. CircleSector(F, Q, P) → d_3
  63. CircleSector(G, R, Q) → e_3
  64. CircleSector(H, S, R) → f_3
  65. CircleSector(I, T, S) → g_3
  66. CircleSector(J, U, T) → h_3
  67. CircleSector(K, V, U) → k_3
  68. Defined(K) → loop

AI Drawing Prompt Reference

Draw a closed loop of 10 congruent circles with a radius of 2, where each circle is tangent to its neighbors. Start by drawing the first circle, then sequentially draw the remaining circles with their centers on the circumference of the previous circle. Next, draw smaller circles with a radius of 1, ensuring their centers are on the circumference of the larger circles. Finally, connect the centers of the circles to form a polygon, and draw line segments connecting the centers to the points of tangency, as well as some circular sectors.
The prompt above was compiled from the original drawing commands and is for reference only.