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Dynamic Geometry: Mean Inequality and Hyperbola

Dynamic Geometry: Mean Inequality and Hyperbola
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2018-04-26 03:03
FunctionTrigonometryCoordinate GeometryGeometric ConstructionSlider Animation
This example dynamically illustrates the mean inequality in relation to the incircle of a triangle and a hyperbola. It involves core concepts like mean inequality, hyperbola, incircle of a triangle, and coordinate geometry. Construction uses commands such as Ray, Point, Polygon, Midpoint, Segment, Dilate, Rotate, Circle, Intersect, CircleSector, Vector, Translate, Line, and Angle. Suitable for high school students and math enthusiasts exploring the connection between geometric transformations and algebraic relationships.

GeoGebra Commands & Parameters

  1. Ray((0, 0), (1, 0)) → h_1
  2. Point(h_1) → B
  3. Polygon(A, B, 4) → poly1, f, g, h, i, C, D
  4. Midpoint(A, C) → E
  5. Midpoint(A, B) → F
  6. Segment(A, C) → j
  7. Midpoint(A, E) → G
  8. Midpoint(E, C) → H
  9. Midpoint(h) → I
  10. Dilate(A, If[0 < a ≤ 1, 1 - a, 0], B) → A'
  11. Segment(A, A') → k
  12. Rotate(A, If[1 < a ≤ 2, ((-(a - 1)) * π / 2), ((-π)) / 2], B) → A'_1
  13. Rotate(A', If[1 < a ≤ 2, ((-(a - 1)) * π / 2), ((-π)) / 2], B) → A''
  14. Segment(A'_1, A'') → k'
  15. Rotate(A'_1, If[2 < a ≤ 3, ((-(a - 2)) * π / 2), ((-π)) / 2], C) → J
  16. Rotate(A'', If[2 < a ≤ 3, ((-(a - 2)) * π / 2), ((-π)) / 2], C) → A'''
  17. Segment(J, A''') → k''
  18. Rotate(J, If[3 < a ≤ 4, ((-(a - 3)) * π / 2), ((-π)) / 2], D) → J'
  19. Rotate(A''', If[3 < a ≤ 4, ((-(a - 3)) * π / 2), ((-π)) / 2], D) → K
  20. Segment(J', K) → k'''
  21. Circle(B, (0.15 * Distance[A, B])) → c
  22. Segment(A, B) → l
  23. Intersect(c, l) → L,
  24. Rotate(L, If[4 < a ≤ 5, ((-(a - 4)) * π / 2), ((-π)) / 2], B) → L'
  25. CircleSector(B, L', L) → d
  26. Dilate(B, If[6 < a ≤ 7, 7 - a, 0], C) → B'
  27. Vector(B, B') → u
  28. Translate(d, u) → d'
  29. Rotate(d', If[6 < a ≤ 7, ((a - 6) * π / 2), π / 2], B') → d''
  30. Dilate(C, If[7.5 < a ≤ 8.5, 8.5 - a, 0], D) → C'
  31. Vector(C, C') → v
  32. Translate(d'', v) → d'''
  33. Rotate(d''', If[7.5 < a ≤ 8.5, ((a - 7.5) * π / 2), π / 2], C') → e
  34. Dilate(D, If[9 < a ≤ 10, 10 - a, 0], A) → D'
  35. Vector(D, D') → w
  36. Translate(e, w) → e'
  37. Rotate(e', If[9 < a ≤ 10, ((a - 9) * π / 2), π / 2], D') → e''
  38. Dilate(A, If[10.5 < a ≤ 11.5, 11.5 - a, 0], B) → A'_2
  39. Vector(A, A'_2) → b
  40. Translate(e'', b) → e'''
  41. Rotate(e''', If[10.5 < a ≤ 11.5, ((a - 10.5) * π / 2), π / 2], A'_2) → p
  42. Dilate(A, If[12 < a ≤ 13, 13 - a, 0], G) → A'_3
  43. Segment(A, A'_3) → m
  44. Dilate(G, If[13 < a ≤ 14, 14 - a, 0], E) → G'
  45. Segment(G, G') → n
  46. Dilate(E, If[14 < a ≤ 15, 15 - a, 0], H) → E'
  47. Segment(E, E') → q
  48. Dilate(H, If[15 < a ≤ 16, 16 - a, 0], C) → H'
  49. Segment(H, H') → r
  50. Dilate(C, If[16 < a ≤ 17, 17 - a, 0], I) → C'_1
  51. Segment(C, C'_1) → s
  52. Rotate(C, If[17 < a < 18, ((-(a - 17)) * π), (-π)], I) → C'_2
  53. Rotate(C'_1, If[17 < a < 18, ((-(a - 17)) * π), (-π)], I) → M
  54. Segment(C'_2, M) → s'
  55. Rotate(A, If[18 < a < 19, ((a - 18) * π), π], G) → A'_4
  56. Rotate(A'_3, If[18 < a < 19, ((a - 18) * π), π], G) → N
  57. Segment(A'_4, N) → m'
  58. Rotate(A'_4, If[19 < a < 20, ((a - 19) * π), π], E) → O
  59. Rotate(N, If[19 < a < 20, ((a - 19) * π), π], E) → N'
  60. Segment(O, N') → m''
  61. Rotate(O, If[20 < a < 21, ((a - 20) * π), π], H) → O'
  62. Rotate(N', If[20 < a < 21, ((a - 20) * π), π], H) → N''
  63. Segment(O', N'') → m'''
  64. Dilate(B, If[22 < a ≤ 23, 23 - a, 0], G) → B'_1
  65. Dilate(I, If[22 < a ≤ 23, 23 - a, 0], G) → I'
  66. Segment(I, I') → t
  67. Segment(B, B'_1) → f_1
  68. Line(B, t) → g_1
  69. Point(g_1) → P
  70. Angle(P, B, C) → α
  71. Dilate(B, If[24 < a ≤ 25, 25 - a, 0], G) → B'_2
  72. Vector(B, B'_2) → u_1
  73. Translate(p, u_1) → p'
  74. Rotate(p', If[24 < a ≤ 25, ((-(a - 24)) * (α + π / 2)), (-(α + π / 2))], B'_2) → p''

AI Drawing Prompt Reference

Create a dynamic geometry illustration showing the relationship between a triangle's incircle and a hyperbola, controlled by a slider. Start with a horizontal ray and a point on it to form a quadrilateral. Find midpoints of key points and perform rotations and dilations to generate several points and segments. Draw the incircle and rotate a part of it, then translate and combine transformations to form a shape related to the hyperbola, dynamically illustrating the mean inequality.
The prompt above was compiled from the original drawing commands and is for reference only.