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Parabolic Segment: Archimedes Insight

Parabolic Segment: Archimedes Insight
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2019-05-20 21:40
FunctionCalculusGeometric Construction
This example dynamically illustrates the formation of a parabolic segment and its relationship with tangent lines and angles. It involves core mathematical concepts such as parabola, tangent, angle, straight line, and vector. Construction highlights include commands like Line, Segment, Slope, Intersect, Polygon, OrthogonalLine, Circle, Dilate, Vector, Translate, Point, Angle, Rotate, and Midpoint from GeoGebra. Suitable for high school students and math enthusiasts to understand parabolic properties and geometric constructions.

GeoGebra Commands & Parameters

  1. Line(A, B) → g
  2. Segment(A, B) → j
  3. Slope(g) → d
  4. Intersect(f', h, 1) → C
  5. Polygon(A, D, B) → t1, b_1, a_1, d_1
  6. OrthogonalLine(D, g) → l
  7. Intersect(g, l) → E
  8. Slope(b_1) → m
  9. Slope(a_1) → m_1
  10. Intersect(f', n) → F
  11. Intersect(f', p) → H
  12. Polygon(A, I, D) → t2, d_2, a_2, i_1
  13. Polygon(D, G, B) → t3, b_2, d_3, g_1
  14. Segment(A, D) → q
  15. Segment(D, B) → r
  16. Segment(B, G) → s
  17. Segment(G, D) → t
  18. Segment(A, I) → e
  19. Segment(I, D) → f_1
  20. Line(A, B) → h_1
  21. Circle(A, t) → c_1
  22. Intersect(c_1, j_1, 2) → J
  23. Circle(A, r) → e_1
  24. Circle(J, s) → p_1
  25. Intersect(e_1, p_1, 2) → K
  26. Dilate(I, If[0 < o ≤ 1, 1 - o, 0], J) → I'
  27. Polygon(A, I', D) → t4, d_4, a_3, i'
  28. Segment(A, I') → l_1
  29. Segment(I', D) → n_1
  30. Line(D, l_1) → q_1
  31. Dilate(D, If[3 < o ≤ 4, 4 - o, 0], K) → D'
  32. Polygon(A, I', D') → t5, d', a_4, i'_1
  33. Segment(I', D') → r_1
  34. Segment(D', A) → s_1
  35. Dilate(I', If[5 < o ≤ 6, 6 - o, 0], G) → I''
  36. Vector(I', I'') → u
  37. Translate(A, u) → A'
  38. Translate(I', u) → I''_1
  39. Translate(D', u) → D''
  40. Polygon(A', I''_1, D'') → t5', d'', a', i''_1
  41. Line(I', t) → t_1
  42. Point(t_1) → L
  43. Angle(L, I', A) → α
  44. Rotate(A', If[5 < o ≤ 6, ((-(o - 5)) * α), (-α)], I''_1) → A''
  45. Rotate(I''_1, If[5 < o ≤ 6, ((-(o - 5)) * α), (-α)], I''_1) → M
  46. Rotate(D'', If[5 < o ≤ 6, ((-(o - 5)) * α), (-α)], I''_1) → D'''
  47. Polygon(A'', M, D''') → t5'', d''', a'', m_2
  48. Midpoint(A, B) → N
  49. Midpoint(N, B) → O
  50. Midpoint(A, N) → P
  51. Midpoint(A, P) → Q
  52. Midpoint(P, N) → R
  53. Midpoint(N, O) → S
  54. Midpoint(O, B) → T
  55. Dilate(A, If[7 < o ≤ 8, 8 - o, 0], Q) → A'_1
  56. Segment(A, A'_1) → f_2
  57. Rotate(A, If[8 < o ≤ 9, ((o - 8) * π), π], Q) → A'_2
  58. Rotate(A'_1, If[8 < o ≤ 9, ((o - 8) * π), π], Q) → U
  59. Segment(A'_2, U) → g_2
  60. Rotate(A'_2, If[9 < o ≤ 10, ((o - 9) * π), π], A'_2) → V
  61. Rotate(U, If[9 < o ≤ 10, ((o - 9) * π), π], A'_2) → U'
  62. Segment(V, U') → h_2
  63. Rotate(V, If[10 < o ≤ 11, ((o - 10) * π), π], U') → V'
  64. Rotate(U', If[10 < o ≤ 11, ((o - 10) * π), π], U') → U''
  65. Segment(V', U'') → i_2
  66. Rotate(V', If[11 < o ≤ 12, ((o - 11) * π), π], V') → V''
  67. Rotate(U'', If[11 < o ≤ 12, ((o - 11) * π), π], V') → U'''
  68. Segment(V'', U''') → j_2
  69. Rotate(V'', If[12 < o ≤ 13, ((o - 12) * π), π], U''') → V'''
  70. Rotate(U''', If[12 < o ≤ 13, ((o - 12) * π), π], U''') → W
  71. Segment(V''', W) → k_2
  72. Rotate(V''', If[13 < o ≤ 14, ((o - 13) * π), π], V''') → Z
  73. Rotate(W, If[13 < o ≤ 14, ((o - 13) * π), π], V''') → W'
  74. Segment(Z, W') → l_2
  75. Rotate(Z, If[14 < o ≤ 15, ((o - 14) * π), π], W') → Z'
  76. Rotate(W', If[14 < o ≤ 15, ((o - 14) * π), π], W') → W''
  77. Segment(Z', W'') → n_2
  78. Circle(D, n_2) → c_2
  79. Intersect(c_2, k_1) → A_1, B_1
  80. Dilate(M, If[16 < o ≤ 17, 17 - o, 0], B_1) → M'
  81. Polygon(D, M', B) → t6, b_3, d_5, m'
  82. Segment(D, M') → p_2
  83. Segment(M', B) → q_2
  84. Midpoint(M', W') → C_1
  85. Line(A, B) → r_2
  86. Circle(A, (7 / 8 * Distance[A, B])) → e_2
  87. Intersect(e_2, r_2) → D_1, E_1
  88. Polygon(A, E_1, D) → t7, d_6, a_5, e_3
  89. Polygon(E_1, B, D) → t8, d_7, e_4, b_4
  90. Midpoint(M', E_1) → F_1
  91. Dilate(D, If[17 < o ≤ 18, 18 - o, 0], E_1) → D'_1
  92. Segment(D, D'_1) → s_2
  93. Rotate(D, If[19 < o ≤ 20, ((-(o - 19)) * π), (-π)], F_1) → D'_2
  94. Rotate(M', If[19 < o ≤ 20, ((-(o - 19)) * π), (-π)], F_1) → M''
  95. Rotate(B, If[19 < o ≤ 20, ((-(o - 19)) * π), (-π)], F_1) → B'
  96. Polygon(D'_2, M'', B') → t6', b', d'_2, m''