MathPlotAI Logo MathPlotAI

Dynamic Illustration of Two Circles Internally Tangent to a Larger Circle

Dynamic Illustration of Two Circles Internally Tangent to a Larger Circle
Loading the math board and drawing, please wait…
2017-03-12 12:28
CircleTransformationGeometric ConstructionSlider AnimationLocus
In collections Circles (Theorems)
This example demonstrates the relationship between two circles that are internally tangent to a larger circle. Key mathematical concepts include circle properties and geometric construction. Commands used include Point, Circle, Line, Intersect, Semicircle, Segment, CircleArc, LineBisector, Tangent, Dilate, Angle, Min, Rotate, CircleSector, Vector, Translate. Suitable for middle school and above students to learn about circle geometry and dynamic geometry.

GeoGebra Commands & Parameters

  1. Point(yAxis) → B
  2. Circle(A, B) → c
  3. Point(c) → C
  4. Line(C, A) → g
  5. Intersect(c, g, 2) → D
  6. Semicircle(D, C) → d
  7. Point(d) → E
  8. Segment(E, C) → i
  9. CircleArc(A, C, E) → e
  10. Point(e) → F
  11. Segment(F, D) → h
  12. Intersect(h, i) → G
  13. LineBisector(E, G) → l
  14. Line(E, A) → m
  15. Intersect(m, l) → H
  16. Intersect(c, m) → J, K
  17. Segment(H, A) → g_1
  18. Point(g_1) → M
  19. Segment(A, B) → f
  20. Tangent(F, c) → j,
  21. Tangent(E, c) → k,
  22. Line(F, A) → n
  23. LineBisector(G, F) → q
  24. Intersect(n, q) → I
  25. Circle(I, F) → r
  26. Circle(M, E) → p
  27. Dilate(F, If[0 < a ≤ 1, 1 - a, 0], G) → F'
  28. Dilate(G, If[1 < a ≤ 2, 2 - a, 0], D) → G'
  29. Dilate(E, If[2 < a ≤ 3, 3 - a, 0], G) → E'
  30. Dilate(G, If[3 < a ≤ 4, 4 - a, 0], C) → G'_1
  31. Dilate(D, If[6 < a ≤ 7, 7 - a, 0], A) → D'
  32. Dilate(A, If[7 < a ≤ 8, 8 - a, 0], C) → A'
  33. Intersect(p, r) → N, O, ,
  34. Line(F, O) → h_1
  35. Line(E, O) → i_1
  36. Intersect(c, i_1) → P, Q
  37. Intersect(c, h_1) → R, S
  38. Point(g_1) → T
  39. Segment(T, Q) → k_1
  40. Segment(N, M) → l_1
  41. Segment(N, I) → m_1
  42. Segment(T, S) → j_1
  43. Angle(M, N, I) → α
  44. Angle(S, T, Q) → β
  45. Min(m_1, l_1) → b
  46. Circle(N, (0.5 * b)) → s
  47. Intersect(s, m_1) → U,
  48. Intersect(s, l_1) → V,
  49. Dilate(N, If[0 < a ≤ 1, 1 - a, 0], M) → N'
  50. Segment(N, N') → t
  51. Dilate(N, If[1 < a ≤ 2, 2 - a, 0], I) → N'_1
  52. Segment(N, N'_1) → f_1
  53. Rotate(V, If[2 < a ≤ 3, ((a - 2) * α), α], N) → V'
  54. CircleSector(N, V, V') → c_1
  55. Dilate(E, If[3 < a ≤ 4, 4 - a, 0], O) → E'_1
  56. Segment(E, E'_1) → n_1
  57. Dilate(O, If[4 < a ≤ 5, 5 - a, 0], Q) → O'
  58. Segment(O, O') → p_1
  59. Dilate(F, If[5 < a ≤ 6, 6 - a, 0], O) → F'_1
  60. Segment(F, F'_1) → q_1
  61. Dilate(O, If[6 < a ≤ 7, 7 - a, 0], S) → O'_1
  62. Segment(O, O'_1) → r_1
  63. Dilate(A, If[7 < a ≤ 8, 8 - a, 0], S) → A'_1
  64. Dilate(A, If[7 < a ≤ 8, 8 - a, 0], Q) → A'_2
  65. Segment(A, A'_1) → s_1
  66. Segment(A, A'_2) → t_1
  67. Line(N, t_1) → a_1
  68. Point(a_1) → L
  69. Angle(U, N, L) → γ
  70. Dilate(N, If[8 < a ≤ 9, 9 - a, 0], A) → N'_2
  71. Vector(N, N'_2) → u
  72. Translate(c_1, u) → d_1
  73. Rotate(d_1, If[9 < a ≤ 10, ((a - 9) * γ), γ], A) → e_1

AI Drawing Prompt Reference

Draw a large circle with two smaller circles internally tangent to it. Start by taking a point B on the y-axis and drawing a circle c with centers A and B. Then, take a point C on circle c and draw a line g through C and A, intersecting circle c at point D. Draw a semicircle d with diameter endpoints D and C, and take a point E on semicircle d. Draw a line segment i from E to C, and draw a circular arc e with E as the center and C on the arc. Take a point F on circular arc e and draw a line segment h from F to D, intersecting line segment i at point G. Draw a line bisection l through E and G, and draw a line m from E to A, intersecting line bisection l at point I. Draw a line n from I to F, intersecting line bisection l at point Q, and draw a circle r with center I and radius IF, and a circle p with center E.
The prompt above was compiled from the original drawing commands and is for reference only.