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Congruent Chords and Circle Center Distance

Congruent Chords and Circle Center Distance
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2016-07-08 03:19
CircleTrigonometryGeometric Construction
This example demonstrates that if two congruent chords are drawn inside a circle, they are equidistant from the circle's center. It involves core math concepts like circle, congruence, distance, and geometric construction. The construction uses commands/objects such as Circle, Point, Segment, Intersect, Midpoint, Angle, CircleSector, Dilate, Line, Vector, Translate, and Rotate in GeoGebra. Suitable for high school mathematics teaching to explore circle properties and geometric transformations.

GeoGebra Commands & Parameters

  1. Circle(A, B) → c
  2. Point(c) → C
  3. Segment(A, C) → a
  4. Point(c) → D
  5. Point(c) → E
  6. Circle(D, Chord) → e
  7. Circle(E, Chord) → f
  8. Intersect(c, e, 1) → F
  9. Intersect(c, f, 1) → G
  10. Segment(G, E) → g
  11. Segment(D, F) → h
  12. Midpoint(g) → H
  13. Midpoint(h) → I
  14. Segment(A, H) → b
  15. Segment(A, I) → d
  16. Angle(A, I, D) → α
  17. Angle(G, H, A) → β
  18. Circle(I, (0.1 * g)) → k
  19. Circle(H, (0.1 * g)) → p
  20. Intersect(k, c, 1) → J
  21. Intersect(k, c, 1) → K
  22. Intersect(c, h, 2) → L
  23. Intersect(k, c, 1) → M
  24. Intersect(k, h) → N, O
  25. Intersect(p, g) → P, Q
  26. Intersect(k, d, 1) → R
  27. Intersect(p, b, 1) → S
  28. CircleSector(I, O, R) → q
  29. CircleSector(H, P, S) → r
  30. Dilate(E, If[0 < i ≤ 1, 1 - i, 0], G) → E'
  31. Dilate(D, If[0 < i ≤ 1, 1 - i, 0], L) → D'
  32. Segment(E, E') → j
  33. Segment(D, D') → l
  34. Line(D, j) → m
  35. Point(m) → T
  36. Angle(F, D, T) → γ
  37. Dilate(G, If[1 < i ≤ 2, 2 - i, 0], D) → G'
  38. Vector(G, G') → u
  39. Translate(E, u) → E'_1
  40. Translate(E', u) → E''
  41. Segment(E'_1, E'') → j'
  42. Rotate(E'_1, If[2 < i < 3, ((-(i - 2)) * γ), (-γ)], D) → U
  43. Rotate(E'', If[2 < i < 3, ((-(i - 2)) * γ), (-γ)], D) → E'''
  44. Segment(U, E''') → j''
  45. Dilate(A, If[3 < i ≤ 4, 4 - i, 0], H) → A'
  46. Dilate(A, If[3 < i ≤ 4, 4 - i, 0], I) → A'_1
  47. Segment(A, A') → n
  48. Segment(A, A'_1) → s
  49. Rotate(r, If[4.5 < i < 5.5, ((i - 4.5) * π / 2), π / 2], H) → r'
  50. Rotate(q, If[5.5 < i < 6.5, ((i - 5.5) * π / 2), π / 2], I) → q'
  51. Angle(H, A, I) → δ
  52. Rotate(A, If[7 < i < 8, ((i - 7) * δ), δ], A) → A'_2
  53. Rotate(A', If[7 < i < 8, ((-(i - 7)) * ((2 * π) - δ)), δ - (2 * π)], A) → A''
  54. Segment(A'_2, A'') → n'