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Secants: Proof Hint

Secants: Proof Hint
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2016-06-02 20:01
TriangleCircleAngleGeometric Construction
This example demonstrates the proof hint for the Secant Segment Lengths Theorem using similar triangles. Key math concepts include secants, circles, similar triangles, and angles. It is constructed using GeoGebra commands such as Circle, Segment, PointIn, Tangent, Intersect, CircleArc, Point, Dilate, CircleSector, Line, Angle, Vector, Translate, Rotate, Polygon, and OrthogonalLine. Suitable for high school geometry students to understand the proof process and application of similar triangles.

GeoGebra Commands & Parameters

  1. Circle(A_1, B_1) → c
  2. Segment(A_1, B_1) → f
  3. PointIn(a_2) → C
  4. Tangent(C, c) → g, h
  5. Intersect(c, g) → D_1
  6. Intersect(c, h) → E_1
  7. CircleArc(A_1, D_1, E_1) → d
  8. Point(d) → A
  9. CircleArc(A_1, D_1, A) → e
  10. CircleArc(A_1, A, E_1) → k
  11. Point(k) → D
  12. CircleArc(A_1, E_1, D_1) → p
  13. Segment(C, A) → i
  14. Segment(C, D) → j
  15. Intersect(p, i, 1) → E
  16. Intersect(p, j, 1) → B
  17. Segment(D, E) → l
  18. Segment(B, A) → m
  19. CircleArc(A_1, B, E) → q
  20. CircleArc(A_1, E, D_1) → r
  21. CircleArc(A_1, E_1, B) → s
  22. Circle(D, (0.2 * f)) → t
  23. Circle(A, (0.2 * f)) → c_1
  24. Intersect(t, j, 1) → J
  25. Intersect(t, l, 1) → K
  26. Dilate(D, If[0 < b_2 ≤ 1, 1 - b_2, 0], E) → G'
  27. Segment(D, G') → n
  28. Dilate(B, If[1 < b_2 ≤ 2, 2 - b_2, 0], A) → I'
  29. Segment(B, I') → f_1
  30. CircleSector(D, J, K) → d_1
  31. Line(A, n) → g_1
  32. Intersect(c_1, g_1, 2) → L
  33. Intersect(c_1, i, 1) → M
  34. Angle(M, A, L) → α
  35. Dilate(D, If[2 < b_2 ≤ 3, 3 - b_2, 0], A) → G'_1
  36. Vector(D, G'_1) → u
  37. Translate(d_1, u) → e_1
  38. Rotate(e_1, If[3 < b_2 ≤ 4, ((-(b_2 - 3)) * α), (-α)], A) → h_1
  39. Circle(C, (0.2 * f)) → k_1
  40. Intersect(k_1, j, 1) → N
  41. Intersect(k_1, i, 1) → O
  42. CircleSector(C, O, N) → p_1
  43. Segment(A, E) → i_1
  44. Segment(E, C) → j_1
  45. Segment(D, B) → l_1
  46. Segment(B, C) → m_1
  47. Polygon(D, C, E) → poly1, h_2, g_2, c_2
  48. OrthogonalLine(E, i_1) → n_1
  49. OrthogonalLine(B, l_1) → q_1
  50. Circle(E, (0.1 * f)) → r_1
  51. Circle(B, (0.1 * f)) → s_1
  52. Intersect(s_1, q_1, 1) → P
  53. Intersect(r_1, n_1, 2) → Q
  54. Line(Q, i_1) → t_1
  55. Line(P, l_1) → a_1
  56. OrthogonalLine(A, t_1) → b_1
  57. OrthogonalLine(C, t_1) → f_2
  58. OrthogonalLine(D, a_1) → i_2
  59. OrthogonalLine(C, a_1) → j_2
  60. Intersect(t_1, b_1) → R
  61. Intersect(t_1, f_2) → S
  62. Intersect(a_1, j_2) → T
  63. Intersect(a_1, i_2) → U
  64. Dilate(R, If[4 < b_2 ≤ 5, 5 - b_2, 0], S) → R'
  65. Segment(R, R') → a
  66. Polygon(A, C, B) → poly2, i_3, f_3, c_3
  67. Dilate(U, If[5 < b_2 ≤ 6, 6 - b_2, 0], T) → U'
  68. Segment(U, U') → b
  69. Segment(C, E) → k_2
  70. Segment(C, B) → l_2
  71. CircleArc(A_1, B, E) → d_2