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Dynamic Illustration of Two Internally Tangent Circles with an Inscribed Equilat

Dynamic Illustration of Two Internally Tangent Circles with an Inscribed Equilat
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2018-03-11 12:22
TriangleCircleTransformationGeometric ConstructionSlider Animation
This example demonstrates the dynamic relationship between two internally tangent circles and an inscribed equilateral triangle. Key mathematical concepts include circles, equilateral triangles, and geometric transformations. Using GeoGebra commands such as Circle, Rotate, Intersect, etc., the size and position of the circles can be dynamically adjusted to observe the changes in the inscribed equilateral triangle. Suitable for students at the junior high school level and above to learn about geometric transformations and properties of circles.

GeoGebra Commands & Parameters

  1. Circle(A, B) → c
  2. Point(c) → D
  3. Rotate(D, (2 * π / 3), A) → D'
  4. Rotate(D', (2 * π / 3), A) → D''
  5. CircleArc(A, D', D'') → s
  6. Point(s) → C
  7. Segment(A, C) → g
  8. Point(g) → F
  9. Semicircle(D'', F) → t
  10. Segment(F, C) → h
  11. Circle(F, h) → e
  12. Intersect(e, t, 1) → H
  13. Semicircle(D, F) → k
  14. Intersect(e, k, 1) → E
  15. Dilate(D'', If[6 < b ≤ 7, 7 - b, 0], H) → D'''
  16. Segment(D'', D''') → k_1
  17. Line(E, k_1) → p_1
  18. Point(p_1) → L
  19. Semicircle(D', F) → p
  20. Intersect(e, p, 1) → G
  21. Dilate(D', If[3 < b ≤ 4, 4 - b, 0], G) → D''_1
  22. Segment(D', D''_1) → f_1
  23. Line(D, f_1) → l_1
  24. Point(l_1) → S
  25. Polygon(D, D', D'') → t1, d'', d, d'
  26. Line(A, C) → f
  27. Segment(D, E) → i
  28. Segment(D', G) → j
  29. Semicircle(C, D'') → q
  30. Semicircle(D'', F) → r
  31. Segment(D'', H) → l
  32. Circle(D, j) → c_1
  33. Intersect(c_1, i) → I,
  34. Dilate(D, If[0 < b ≤ 1, 1 - b, 0], D') → D'_1
  35. Segment(D, D'_1) → m
  36. Rotate(D, If[1 < b ≤ 2, ((b - 1) * π / 3), π / 3], D) → D'_2
  37. Rotate(D'_1, If[1 < b ≤ 2, ((b - 1) * π / 3), π / 3], D) → J
  38. Segment(D'_2, J) → m'
  39. Rotate(D'_2, If[2 < b ≤ 3, ((b - 2) * π / 3), π / 3], D'') → K
  40. Rotate(J, If[2 < b ≤ 3, ((b - 2) * π / 3), π / 3], D'') → J'
  41. Segment(K, J') → m''
  42. Angle(D', G, F) → α
  43. Angle(D'', H, F) → β
  44. Angle(D, E, F) → γ
  45. Dilate(F, If[3 < b ≤ 4, 4 - b, 0], G) → F'
  46. Segment(F, F') → n
  47. Dilate(D, If[4 < b ≤ 5, 5 - b, 0], I) → D'_3
  48. Segment(D, D'_3) → g_1
  49. Dilate(I, If[5 < b ≤ 6, 6 - b, 0], E) → I'
  50. Segment(I, I') → h_1
  51. Dilate(F, If[5 < b ≤ 6, 6 - b, 0], E) → F'_1
  52. Segment(F, F'_1) → i_1
  53. Dilate(F, If[6 < b ≤ 7, 7 - b, 0], H) → F'_2
  54. Segment(F, F'_2) → j_1
  55. Angle(S, D, E) → δ
  56. Dilate(G, If[8 < b ≤ 9, 9 - b, 0], D) → G'
  57. Vector(G, G') → u
  58. Translate(D', u) → D''_2
  59. Translate(D''_1, u) → M
  60. Segment(D''_2, M) → m_1
  61. Rotate(D''_2, If[9 < b ≤ 10, ((b - 9) * δ), δ], D) → O
  62. Rotate(M, If[9 < b ≤ 10, ((b - 9) * δ), δ], D) → M'
  63. Segment(O, M') → n_1
  64. Angle(L, E, D) → ε
  65. Dilate(H, If[10 < b ≤ 11, 11 - b, 0], E) → H'
  66. Vector(H, H') → v
  67. Translate(D'', v) → D'''_1
  68. Translate(D''', v) → Q
  69. Segment(D'''_1, Q) → q_1
  70. Rotate(D'''_1, If[11 < b ≤ 12, ((b - 11) * ε), ε], E) → R
  71. Rotate(Q, If[11 < b ≤ 12, ((b - 11) * ε), ε], E) → Q'
  72. Segment(R, Q') → r_1