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Cut-The-Knot Action 19: Dynamic Exploration of Quadrilaterals and Quadratic Equa

Cut-The-Knot Action 19: Dynamic Exploration of Quadrilaterals and Quadratic Equa
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2018-02-17 12:08
PolygonAngleGeometric ConstructionSlider Animation
① This example is a dynamic and modifiable geometric illustration of a problem posted by Alexander Bogomolny on Twitter, demonstrating hidden relationships within a general quadrilateral (including kites) through interactive manipulation. ② Core mathematical concepts include general quadrilaterals, kites, quadratic equations, standard deviation, secant lines, and constraints among angles and segments, highlighting the interplay between algebra and geometry. ③ Construction highlights: Based on the construction summary, the GeoGebra commands used include Segment, Rotate, Ray, Point, Line, Angle, Intersect, OrthogonalLine, Polygon, Min, Circle, CircleSector, Dilate, Vector, and Translate, building multiple points (e.g., A, B, C, D, E, F, G), segments, circles, sectors, vectors, and translated objects, with sliders or dynamic points enabling real-time changes. ④ Suitable for high school and above students or math enthusiasts, this example serves to explore quadrilateral properties, quadratic equations, and geometric constraints, and can be used for classroom demonstrations or independent student inquiry.

GeoGebra Commands & Parameters

  1. Segment(A, B) → f
  2. Rotate(A, (a * π / 180), B) → A'
  3. Ray(B, A') → g
  4. Point(g) → C
  5. Line(C, A) → n_2
  6. Segment(C, A) → i
  7. Angle(B, C, A) → γ
  8. Rotate(B, γ / 2, C) → B'
  9. Ray(C, B') → k
  10. Intersect(f, k) → I
  11. Angle(C, A, B) → β
  12. Rotate(C, β / 2, A) → C'
  13. Ray(A, C') → j
  14. Segment(B, C) → h
  15. Intersect(h, j) → J
  16. Segment(J, I) → l
  17. Point(l) → K
  18. OrthogonalLine(K, i) → h_1
  19. Line(C, B) → m_2
  20. OrthogonalLine(K, h) → g_2
  21. Line(B, A) → l_2
  22. OrthogonalLine(K, f) → n
  23. Angle(A, B, A') → α
  24. Polygon(B, A, C) → t1, c, b, a_1
  25. Angle(C, A, C') → δ
  26. Angle(B, C, B') → ε
  27. Intersect(j, k) → D
  28. Min({f, i, h}) → d
  29. Circle(B, (0.15 * d)) → e
  30. Circle(A, (0.15 * d)) → p
  31. Circle(C, (0.15 * d)) → q
  32. Intersect(q, h, 1) → E
  33. Intersect(p, i) → F,
  34. Intersect(e, f) → G,
  35. Intersect(e, h, 1) → H
  36. CircleSector(B, G, H) → r
  37. OrthogonalLine(K, l) → m
  38. OrthogonalLine(K, h) → s
  39. Intersect(s, h) → N
  40. Intersect(n, l_2) → M
  41. Segment(K, M) → t
  42. Segment(K, N) → f_1
  43. Angle(K, N, C) → ζ
  44. Angle(A, M, K) → η
  45. OrthogonalLine(D, i) → g_1
  46. Circle(K, f_1) → c_1
  47. Intersect(n_2, h_1) → L
  48. Angle(K, L, A) → θ
  49. Ray(K, L) → i_1
  50. Circle(K, f_1) → d_1
  51. Intersect(d_1, i_1, 1) → O
  52. Dilate(A, If[0 < u ≤ 1, 1 - u, 0], D) → A'_1
  53. Dilate(C, If[0 < u ≤ 1, 1 - u, 0], D) → C'_1
  54. Segment(A, A'_1) → j_1
  55. Segment(C, C'_1) → k_1
  56. Dilate(D, If[1 < u ≤ 2, 2 - u, 0], J) → D'
  57. Dilate(D, If[1 < u ≤ 2, 2 - u, 0], I) → D'_1
  58. Segment(D, D') → l_1
  59. Segment(D, D'_1) → m_1
  60. Rotate(F, If[1 < u ≤ 2, ((u - 1) * δ), δ], A) → F'
  61. Rotate(E, If[1 < u ≤ 2, ((u - 1) * ε), ε], C) → E'
  62. CircleSector(A, F, F') → e_1
  63. CircleSector(C, E, E') → p_1
  64. Rotate(p_1, If[3 < u ≤ 4, ((u - 3) * ε), ε], C) → q_1
  65. Rotate(e_1, If[3 < u ≤ 4, ((u - 3) * δ), δ], A) → r_1
  66. Dilate(J, If[4 < u ≤ 5, 5 - u, 0], I) → J'
  67. Segment(J, J') → n_1
  68. Dilate(K, If[7 < u ≤ 8, 8 - u, 0], M) → K'
  69. Dilate(K, If[7 < u ≤ 8, 8 - u, 0], N) → K'_1
  70. Segment(K, K') → s_1
  71. Segment(K, K'_1) → t_1
  72. Dilate(K, If[7 < u ≤ 8, 8 - u, 0], O) → K'_2
  73. Segment(K, K'_2) → b_1
  74. Dilate(O, If[8 < u ≤ 9, 9 - u, 0], L) → O'
  75. Segment(O, O') → f_2
  76. Intersect(g_2, m_2) → P
  77. Angle(L, K, P) → κ
  78. Rotate(K, If[10 < u ≤ 11, ((-(u - 10)) * κ), (-κ)], K) → K'_3
  79. Rotate(K'_1, If[10 < u ≤ 11, ((-(u - 10)) * κ), (-κ)], K) → Q
  80. Segment(K'_3, Q) → h_2
  81. Line(O, s_1) → i_2
  82. Point(i_2) → R
  83. Angle(R, O, L) → λ
  84. Dilate(K, If[11 < u ≤ 12, 12 - u, 0], O) → K'_4
  85. Vector(K, K'_4) → v
  86. Translate(K, v) → K'_5
  87. Translate(K', v) → K''
  88. Segment(K'_5, K'') → j_2
  89. Rotate(K'_5, If[12 < u ≤ 13, ((u - 12) * λ), λ], O) → S
  90. Rotate(K'', If[12 < u ≤ 13, ((u - 12) * λ), λ], O) → K'''
  91. Segment(S, K''') → k_2