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Relationship Between a Chord and a Tangent Circle

Relationship Between a Chord and a Tangent Circle
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2017-04-07 10:12
CircleAngleGeometric Construction
In collections Circles (Theorems)
This example illustrates the relationship between a chord of a circle and another circle that is tangent to this chord and the original circle. By dynamically adjusting, one can observe how these circles and chords change under different conditions. Key mathematical concepts include the properties of circles, tangents, and chords. The construction uses GeoGebra commands such as Circle, Point, Segment, Tangent, Angle, Rotate, etc. Suitable for middle school and above students or teachers for teaching purposes.

GeoGebra Commands & Parameters

  1. Circle(A, B) → c
  2. Point(c) → C
  3. Point(c) → D
  4. Segment(D, C) → f
  5. CircleArc(A, C, D) → d
  6. Point(d) → E
  7. Tangent(E, c) → g,
  8. Angle(D, A, C) → α
  9. Rotate(D, α / 2, A) → D'
  10. Angle(D, A, D') → β
  11. Segment(D', E) → h
  12. Intersect(f, h) → F
  13. OrthogonalLine(E, g) → i
  14. LineBisector(F, E) → j
  15. Intersect(i, j) → G
  16. Circle(G, F) → e
  17. Circle(D', C) → k
  18. Intersect(e, k, 2) → H
  19. Intersect(e, k, 1) → I
  20. Segment(D', I) → l
  21. Segment(D', H) → m
  22. Angle(G, I, D') → γ
  23. Angle(D', H, G) → δ
  24. Dilate(D, If[0 < a ≤ 1, 1 - a, 0], C) → D'_1
  25. Segment(D, D'_1) → n
  26. Point(g) → J
  27. Angle(J, E, G) → ε
  28. Angle(C, A, D) → ζ
  29. Rotate(C, If[1 < a ≤ 2, ((a - 1) * ζ), ζ], A) → C'
  30. CircleArc(A, C, C') → p
  31. Angle(D, A, D') → θ
  32. Rotate(D, If[2 < a ≤ 3, ((a - 2) * θ), θ], A) → D'_2
  33. CircleArc(A, D, D'_2) → q
  34. Dilate(D, If[3 < a ≤ 4, 4 - a, 0], D') → D'_3
  35. Segment(D, D'_3) → r
  36. Rotate(q, If[4.5 < a ≤ 5.5, ((a - 4.5) * θ), θ], A) → q'
  37. Rotate(D, If[4.5 < a ≤ 5.5, ((a - 4.5) * θ), θ], A) → D'_4
  38. Rotate(D'_3, If[4.5 < a ≤ 5.5, ((a - 4.5) * θ), θ], A) → K
  39. Segment(D'_4, K) → r'
  40. Dilate(G, If[7 < a ≤ 8, 8 - a, 0], E) → G'
  41. Segment(G, G') → s
  42. Rotate(E, If[8 < a ≤ 10, ((a - 8) * π), (2 * π)], G) → E'
  43. CircleArc(G, E, E') → t
  44. Dilate(G, If[7 < a ≤ 8, 8 - a, 0], F) → G'_1
  45. Segment(G, G'_1) → b
  46. Angle(G, F, D) → η
  47. Dilate(D', If[10 < a ≤ 11, 11 - a, 0], I) → D''
  48. Segment(D', D'') → f_1
  49. Dilate(G, If[10 < a ≤ 11, 11 - a, 0], I) → G'_2
  50. Segment(G, G'_2) → g_1
  51. Angle(C, D', I) → ρ
  52. Dilate(DD, If[12.5 < a ≤ 13.5, 13.5 - a, 0], F) → DD'
  53. Dilate(E, If[12 < a ≤ 13, 13 - a, 0], F) → E'_1
  54. Segment(E, E'_1) → h_1
  55. Dilate(F, If[13 < a ≤ 14, 14 - a, 0], D') → F'
  56. Segment(FF, F') → i_1
  57. Rotate(D'_4, If[14 < a ≤ 15, ((a - 14) * ρ), ρ], D') → L
  58. Rotate(K, If[14 < a ≤ 15, ((a - 14) * ρ), ρ], D') → K'
  59. Segment(L, K') → r''