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Butterfly Theorem Dynamic Illustration

Butterfly Theorem Dynamic Illustration
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2016-10-30 07:59
CircleTransformationGeometric Construction
This example demonstrates the Butterfly Theorem, which states that in a circle, if two chords intersect at a point, the line segment joining the midpoints of the two chords is equal to the distance from the midpoint to the circle's center. Key mathematical concepts include circles, geometric construction, and transformations. It uses commands such as Circle, Point, CircleArc, Segment, Midpoint, Ray, Intersect, Distance, Dilate, Rotate, and Polygon in GeoGebra. Suitable for middle school and above students to learn geometric construction and theorem verification.

GeoGebra Commands & Parameters

  1. Circle(O, R) → c
  2. Point(c) → Q
  3. Point(c) → P
  4. CircleArc(O, Q, P) → d
  5. Point(d) → A
  6. CircleArc(O, Q, A) → c_1
  7. Segment(P, Q) → f
  8. CircleArc(O, P, Q) → e
  9. Midpoint(f) → C_1
  10. Ray(A, C_1) → g
  11. Intersect(e, g, 1) → B
  12. Point(c_1) → C
  13. Ray(C, C_1) → h
  14. Intersect(e, h, 1) → D
  15. Segment(A, D) → i
  16. Segment(C, B) → j
  17. Intersect(f, i) → E
  18. Intersect(f, j) → F
  19. Distance(E, C_1) → distanceEC1
  20. Distance(C_1, F) → distanceC1F
  21. Dilate(P, If[0 < a ≤ 1, 1 - a, 0], E) → P'
  22. Segment(P, P') → k
  23. Dilate(E, If[1 < a ≤ 2, 2 - a, 0], C_1) → E'
  24. Segment(E, E') → l
  25. Dilate(C_1, If[2 < a ≤ 3, 3 - a, 0], F) → G
  26. Segment(C_1, G) → m
  27. Dilate(F, If[3 < a ≤ 4, 4 - a, 0], Q) → F'
  28. Segment(F, F') → n
  29. Rotate(E, If[4 < a ≤ 5, ((a - 4) * π), π], C_1) → E'_1
  30. Rotate(E', If[4 < a ≤ 5, ((a - 4) * π), π], C_1) → E''
  31. Segment(E'_1, E'') → l'
  32. Rotate(P, If[4 < a ≤ 5, ((a - 4) * π), π], C_1) → P'_1
  33. Rotate(P', If[4 < a ≤ 5, ((a - 4) * π), π], C_1) → P''
  34. Segment(P'_1, P'') → k'
  35. Dilate(A, If[9.5 < a ≤ 10.5, 10.5 - a, 0], E) → A'
  36. Segment(A, A') → p
  37. Dilate(E, If[10.5 < a ≤ 11.5, 11.5 - a, 0], D) → E'_2
  38. Segment(E, E'_2) → q
  39. Dilate(C, If[9.5 < a ≤ 10.5, 10.5 - a, 0], F) → C'
  40. Segment(C, C') → r
  41. Dilate(F, If[10.5 < a ≤ 11.5, 11.5 - a, 0], B) → F'_1
  42. Segment(F, F'_1) → s
  43. Dilate(E, If[14.5 < a ≤ 15.5, 15.5 - a, 0], C_1) → E'_3
  44. Segment(E, E'_3) → t
  45. Dilate(C_1, If[15.5 < a ≤ 16.5, 16.5 - a, 0], F) → H
  46. Segment(C_1, H) → b
  47. Rotate(E, If[16.5 < a ≤ 17.5, ((a - 16.5) * π), π], C_1) → E'_4
  48. Rotate(E'_3, If[16.5 < a ≤ 17.5, ((a - 16.5) * π), π], C_1) → I
  49. Segment(E'_4, I) → t'
  50. CircleArc(O, A, P) → d_1
  51. Dilate(A, If[7.5 < a ≤ 8.5, 8.5 - a, 0], C_1) → A'_1
  52. Segment(A, A'_1) → f_1
  53. Dilate(C_1, If[8.5 < a ≤ 9.5, 9.5 - a, 0], B) → J
  54. Segment(C_1, J) → g_1
  55. Dilate(C, If[7.5 < a ≤ 8.5, 8.5 - a, 0], C_1) → C'_1
  56. Segment(C, C'_1) → h_1
  57. Dilate(C_1, If[8.5 < a ≤ 9.5, 9.5 - a, 0], D) → K
  58. Segment(C_1, K) → i_1
  59. Polygon(D, C_1, A) → poly1, a_1, d_2, c_2
  60. Polygon(B, C_1, C) → poly2, c_3, b_1, c_4

AI Drawing Prompt Reference

Draw a circle with two intersecting chords inside it. Find the midpoints of these chords and connect them. Demonstrate that the distance from the midpoint connection to the circle's center equals the length of the segment joining the midpoints. Use dynamic illustrations to show circles, geometric constructions, and transformations including arcs, segments, rotations, and dilations.
The prompt above was compiled from the original drawing commands and is for reference only.