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Animation Demonstrating Incircle and 3D Vectors

Animation Demonstrating Incircle and 3D Vectors
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2017-05-03 02:18
CircleVectorCoordinate GeometryGeometric Construction
This example demonstrates the formation of a triangle's incircle and its relationship with 3D vectors through animation. It involves core mathematical concepts such as the incircle, 3D vectors, coordinate geometry, orthocenter, and fraction calculations. The construction utilizes commands like semicircle, point, segment, minimum, midpoint, orthogonal line, circle, intersection, circle sector, dilation, vector, polygon, angle, rotation, ray, line, and translation in GeoGebra. Suitable for high school and above, it illustrates the application of spatial geometry and algebra.

GeoGebra Commands & Parameters

  1. Semicircle(A, B) → d
  2. Point(d) → C
  3. Segment(C, B) → f
  4. Segment(A, C) → e
  5. Segment(A, B) → a
  6. Min(a, f) → m
  7. Min(a, e) → l
  8. Min(l, m) → n
  9. Midpoint(a) → C_1
  10. OrthogonalLine(C_1, a) → b
  11. Circle(C_1, A) → c
  12. OrthogonalLine(C, a) → g
  13. Intersect(g, a) → E
  14. Segment(C, E) → h
  15. Circle(A, (0.2 * n)) → k
  16. Circle(C, (0.2 * n)) → p
  17. Circle(B, (0.2 * n)) → q
  18. Intersect(p, h, 1) → I
  19. Intersect(p, f, 1) → J
  20. Intersect(q, a, 1) → K
  21. Intersect(q, f, 1) → L
  22. Intersect(k, a, 1) → M
  23. Intersect(k, e, 1) → N
  24. CircleSector(A, M, N) → r
  25. Dilate(C, If[-2 < i < -1, -1 - i, 0], E) → D'
  26. Dilate(A, If[-0.5 < i ≤ 0.5, 0.5 - i, 0], C) → A'
  27. Vector(A, A') → u
  28. Dilate(B, If[1.5 < i ≤ 2.5, 2.5 - i, 0], C) → B'
  29. Vector(B, B') → v
  30. Polygon(A, B, C) → poly3, d_3, a_2, b_2
  31. Angle(C, B, A) → δ
  32. Rotate(C, δ / 2, B) → C'
  33. Angle(C, B, C') → ε
  34. Ray(B, C') → h_1
  35. Intersect(e, h_1) → D
  36. Angle(B, A, C) → ζ
  37. Rotate(B, ζ / 2, A) → B'_1
  38. Angle(B, A, B'_1) → η
  39. Ray(A, B'_1) → i_1
  40. Intersect(f, i_1) → O
  41. Intersect(h_1, i_1) → P
  42. Intersect(q, h_1, 1) → Q
  43. Intersect(q, f, 1) → R
  44. Intersect(k, a, 1) → S
  45. Intersect(k, i_1, 1) → T
  46. CircleSector(A, S, T) → c_1
  47. CircleSector(B, R, Q) → k_1
  48. Polygon(A, B, C) → poly1, c_2, a_1, b_1
  49. OrthogonalLine(P, a) → j
  50. Intersect(a, j) → F
  51. Angle(P, F, A) → α
  52. Line(B, C) → j_1
  53. Circle(C, 1) → d_1
  54. Intersect(d_1, j_1, 2) → G
  55. Ray(C, G) → l_1
  56. Circle(C, (0.5 * Radius[k])) → e_1
  57. Intersect(e_1, e, 1) → H
  58. Intersect(e_1, l_1, 1) → U
  59. CircleSector(C, U, H) → p_1
  60. Rotate(p_1, If[0 < i ≤ 1, (i * π / 2), π / 2], C) → s
  61. Dilate(A, If[1 < i ≤ 2, 2 - i, 0], P) → A'_1
  62. Dilate(B, If[1 < i ≤ 2, 2 - i, 0], P) → B'_2
  63. Segment(A, T) → t
  64. Segment(B, B'_2) → f_1
  65. Segment(A, A'_1) → g_1
  66. OrthogonalLine(P, g_1) → m_1
  67. Line(C, m_1) → n_1
  68. Line(C, g_1) → q_1
  69. Rotate(k_1, If[2.5 < i ≤ 3.5, ((i - 2.5) * ε), ε], B) → r_1
  70. Rotate(c_1, If[3.5 < i ≤ 4.5, ((i - 3.5) * ζ / 2), ζ / 2], A) → s_1
  71. Intersect(m_1, a) → V
  72. Dilate(P, If[4.5 < i ≤ 5.5, 5.5 - i, 0], V) → P'
  73. Segment(P, P') → t_1
  74. Dilate(C, If[5.5 < i ≤ 6.5, 6.5 - i, 0], P) → C'_1
  75. Vector(C, C'_1) → u_1
  76. Translate(s, u_1) → s'
  77. Point(q_1) → W
  78. Angle(W, C, A) → β
  79. Rotate(s', If[6.5 < i ≤ 7.5, ((-(i - 6.5)) * β), (-β)], P) → s''
  80. Segment(C, B) → g_2
  81. OrthogonalLine(V, l_1) → h_2
  82. Line(C, B) → i_2
  83. Intersect(i_2, h_2) → Z
  84. Dilate(V, If[7.5 < i ≤ 8.5, 8.5 - i, 0], Z) → V'
  85. Segment(V, V') → j_2
  86. Dilate(C, If[8.5 < i ≤ 9.5, 9.5 - i, 0], Z) → C'_2
  87. Vector(C, C'_2) → v_1
  88. Translate(s, v_1) → s'_1
  89. Line(A, C) → k_2
  90. OrthogonalLine(P, k_2) → l_2
  91. Intersect(k_2, l_2) → A_1
  92. Dilate(P, If[9.5 < i ≤ 10.5, 10.5 - i, 0], A_1) → P'_1
  93. Segment(P, P'_1) → f_2
  94. Dilate(C, If[10.5 < i ≤ 11.5, 11.5 - i, 0], A_1) → C'_3
  95. Vector(C, C'_3) → w_1
  96. Translate(s, w_1) → s'_2
  97. Dilate(A_1, If[11.5 < i ≤ 12.5, 12.5 - i, 0], C) → B_1
  98. Vector(A_1, B_1) → u_2
  99. Translate(P, u_2) → P'_2
  100. Translate(P'_1, u_2) → D_1
  101. Segment(P'_2, D_1) → m_2
  102. Dilate(A_1, If[12.5 < i ≤ 13.5, 13.5 - i, 0], P'_2) → E_1
  103. Vector(A_1, E_1) → v_2
  104. Translate(P, v_2) → P'_3
  105. Translate(P'_1, v_2) → F_1
  106. Segment(P'_3, F_1) → n_2

AI Drawing Prompt Reference

Draw a triangle ABC with A, B, and C as vertices. Demonstrate the relationship between the incircle of the triangle and 3D vectors through animation. First, draw the three sides of the triangle, then find the incenter of the triangle and draw the incircle. Show the relationship between the incircle and the sides of the triangle. Finally, use rotation and scaling operations to demonstrate the interaction between vectors and the sides of the triangle.
The prompt above was compiled from the original drawing commands and is for reference only.