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Dynamic Orthocenter and Midpoint Demonstration in Triangles

Dynamic Orthocenter and Midpoint Demonstration in Triangles
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2017-10-09 08:24
TriangleGeometric Construction
This example dynamically illustrates a geometric phenomenon involving a triangle's orthocenter and the midpoint of one of its sides. It covers core concepts like orthocenter, median, rotation, angle, and geometric transformations. The construction utilizes commands such as Rotate, Angle, Segment, Ray, Point, Circle, Intersect, CircleSector, Line, OrthogonalLine, Polygon, Midpoint, Dilate. Suitable for high school and above students to explore triangle geometry and dynamic transformations.

GeoGebra Commands & Parameters

  1. Rotate(A, (a * π / 180), B) → A'
  2. Angle(A, B, A') → α
  3. Segment(B, A) → f
  4. Ray(B, A') → g
  5. Point(g) → C
  6. Segment(B, C) → h
  7. Circle(B, b) → c
  8. Intersect(c, f, 1) → D
  9. Intersect(c, h, 1) → E
  10. CircleSector(B, D, E) → d
  11. Segment(C, A) → i
  12. Line(C, A) → j
  13. Line(A, B) → k
  14. OrthogonalLine(C, k) → m
  15. Intersect(m, k) → H
  16. Circle(C, 2) → e
  17. Circle(H, 2) → p
  18. Intersect(e, j, 1) → J
  19. Ray(C, J) → n
  20. Line(C, H) → r
  21. Polygon(B, A, C) → poly1, c_1, b_1, a_1
  22. Angle(C, A, B) → β
  23. Midpoint(f) → I
  24. Line(C, B) → l
  25. OrthogonalLine(A, l) → q
  26. Intersect(q, l) → F
  27. Angle(B, F, A) → γ
  28. Intersect(q, r) → G
  29. Segment(C, H) → s
  30. Line(C, H) → t
  31. Circle(C, 2) → d_1
  32. Circle(H, 2) → e_1
  33. Intersect(d_1, t, 1) → L
  34. Intersect(e_1, t) → M, N
  35. Segment(C, H) → h_1
  36. Midpoint(C, G) → O
  37. Circle(B, 1) → k_1
  38. Circle(A, 1) → p_1
  39. Intersect(k_1, k, 2) → P
  40. Intersect(p_1, k, 1) → Q
  41. Dilate(B, If[0 < o ≤ 1, 1 - o, 0], I) → B'
  42. Segment(B, B') → l_1
  43. Rotate(B, If[1 < o ≤ 2, ((-(o - 1)) * π), (-π)], I) → B'_1
  44. Rotate(B', If[1 < o ≤ 2, ((-(o - 1)) * π), (-π)], I) → B''
  45. Segment(B'_1, B'') → m_1
  46. Intersect(k_1, l, 2) → K
  47. Circle(C, 2) → q_1
  48. Intersect(q_1, l, 1) → R
  49. Segment(B, C) → n_1
  50. Ray(C, R) → r_1
  51. Ray(B, K) → s_1
  52. Dilate(A, If[2 < o ≤ 3, 3 - o, 0], F) → A'_1
  53. Segment(A, A'_1) → t_1
  54. Dilate(C, If[3 < o ≤ 4, 4 - o, 0], O) → C'
  55. Segment(C, C') → f_2
  56. Dilate(O, If[4 < o ≤ 5, 5 - o, 0], G) → O'
  57. Segment(O, O') → g_2
  58. Dilate(G, If[5 < o ≤ 6, 6 - o, 0], H) → G'
  59. Rotate(C, If[5 < o ≤ 6, ((-(o - 5)) * π), (-π)], O) → C'_1
  60. Rotate(C', If[6 < o ≤ 7, ((-(o - 6)) * π), (-π)], O) → C''
  61. Segment(C'_1, C'') → i_2
  62. Dilate(I, If[8 < o ≤ 9, 9 - o, 0], F) → I'
  63. Dilate(O, If[8 < o ≤ 9, 9 - o, 0], F) → O'_1
  64. Line(A, F) → l_2
  65. Line(C, B) → m_2
  66. Line(C, A) → n_2
  67. OrthogonalLine(B, n_2) → p_2
  68. Intersect(n_2, p_2) → S
  69. Dilate(B, If[2 < o ≤ 3, 3 - o, 0], S) → B'_2
  70. Segment(B, B'_2) → q_2
  71. Angle(C, S, B) → ζ
  72. Segment(I, O) → j_2
  73. Line(F, S) → k_2
  74. Intersect(j_2, k_2) → T
  75. Segment(T, O) → g_1
  76. Segment(T, S) → h_2
  77. Segment(T, F) → r_2
  78. Segment(T, I) → s_2
  79. Min({s_2, r_2, h_2, g_1}) → v
  80. Circle(T, (0.5 * v)) → c_2
  81. Intersect(c_2, k_2) → U, V
  82. Dilate(I, If[6 < o ≤ 7, 7 - o, 0], O) → I'_1
  83. Segment(I, I'_1) → t_2
  84. Dilate(F, If[7 < o ≤ 8, 8 - o, 0], T) → F'
  85. Segment(F, F') → a_2
  86. Dilate(T, If[8 < o ≤ 9, 9 - o, 0], S) → T'
  87. Segment(T, T') → b_2
  88. Rotate(U, If[9 < o ≤ 10, ((o - 9) * π / 2), π / 2], T) → U'
  89. CircleSector(T, U, U') → d_2
  90. Rotate(d_2, If[11 < o ≤ 12, ((o - 11) * π / 2), π / 2], T) → e_2