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Dynamic & Modifiable Illustration of a Theorem Involving 2 Intersecting Circles

Dynamic & Modifiable Illustration of a Theorem Involving 2 Intersecting Circles
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2017-03-14 02:03
TriangleCircleTransformationGeometric ConstructionSlider Animation
In collections Circles (Theorems)
This GeoGebra example dynamically illustrates a geometric theorem involving two intersecting circles, inspired by a tweet from Antonio Gutierrez. It allows users to drag points and adjust sliders to explore relationships among intersection points, segments, and angles. The core mathematical concepts include circles, intersecting chords, similar triangles, and distance measurement. The construction uses GeoGebra commands such as Circle, Segment, PointIn, Distance, Intersect, CircleArc, Point, Line, Dilate, and Midpoint, creating multiple circles, segments, intersection points, arcs, and midpoints, with dilations and midpoints used to generate auxiliary elements. This resource is ideal for middle and high school geometry classes, teacher demonstrations, or student self-study, fostering understanding of circle-triangle relationships and dynamic geometric reasoning.

GeoGebra Commands & Parameters

  1. Circle(A, B) → c
  2. Segment(A, B) → f
  3. PointIn(a) → C
  4. Distance(C, c) → b
  5. PointIn(d) → D
  6. Circle(C, D) → e
  7. Intersect(c, e, 1) → E
  8. Intersect(c, e, 2) → F
  9. CircleArc(C, F, E) → g
  10. CircleArc(C, E, F) → h
  11. Point(g) → G
  12. Line(G, C) → i
  13. Intersect(g, i, 2) → H
  14. Segment(G, H) → j
  15. CircleArc(A, E, F) → k
  16. CircleArc(A, F, E) → p
  17. Point(k) → I
  18. Line(I, A) → l
  19. Intersect(k, l, 2) → J
  20. Segment(I, J) → m
  21. Segment(J, G) → n
  22. Segment(H, I) → q
  23. Intersect(p, q, 1) → K
  24. Intersect(p, n, 1) → L
  25. Intersect(h, q, 1) → M
  26. Intersect(h, n, 1) → N
  27. Segment(H, N) → r
  28. Segment(J, K) → s
  29. Segment(G, M) → t
  30. Segment(I, L) → f_1
  31. Intersect(t, f_1) → O
  32. Intersect(n, q) → P
  33. Intersect(r, s) → Q
  34. Dilate(I, If[0 < o ≤ 1, 1 - o, 0], M) → I'
  35. Segment(I, I') → g_1
  36. Dilate(M, If[1 < o ≤ 2, 2 - o, 0], K) → M'
  37. Segment(M, M') → h_1
  38. Dilate(K, If[2 < o ≤ 3, 3 - o, 0], H) → K'
  39. Segment(K, K') → i_1
  40. Dilate(G, If[3 < o ≤ 4, 4 - o, 0], L) → G'
  41. Segment(G, G') → j_1
  42. Dilate(L, If[4 < o ≤ 5, 5 - o, 0], P) → L'
  43. Segment(L, L') → k_1
  44. Dilate(P, If[5 < o ≤ 6, 6 - o, 0], N) → P'
  45. Segment(P, P') → l_1
  46. Dilate(N, If[6 < o ≤ 7, 7 - o, 0], J) → N'
  47. Segment(N, N') → m_1
  48. Dilate(I, If[7 < o ≤ 8, 8 - o, 0], L) → I'_1
  49. Segment(I, I'_1) → n_1
  50. Dilate(G, If[8 < o ≤ 9, 9 - o, 0], M) → G'_1
  51. Segment(G, G'_1) → p_1
  52. Dilate(J, If[9 < o ≤ 10, 10 - o, 0], K) → J'
  53. Segment(J, J') → q_1
  54. Dilate(H, If[10 < o ≤ 11, 11 - o, 0], N) → H'
  55. Segment(H, H') → r_1
  56. Midpoint(Q, O) → R
  57. Dilate(R, If[12 < o ≤ 13, 13 - o, 0], Q) → R'
  58. Dilate(R, If[12 < o ≤ 13, 13 - o, 0], O) → R'_1
  59. Segment(R, R'_1) → s_1
  60. Segment(R, R') → t_1
  61. Dilate(O, If[13 < o ≤ 14, 14 - o, 0], E) → O'
  62. Dilate(Q, If[13 < o ≤ 14, 14 - o, 0], F) → Q'
  63. Segment(O, O') → a_1
  64. Segment(Q, Q') → b_1