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Dynamic Demonstration of Triangle's Orthocenter & Midpoint

Dynamic Demonstration of Triangle's Orthocenter & Midpoint
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2017-05-19 20:44
TriangleCircleVectorGeometric Construction
This example dynamically illustrates the geometric relationship between a triangle's orthocenter and the midpoint of one side. It involves core mathematical concepts such as orthocenter, midpoint, vectors, rotation, and geometric transformations. The construction utilizes commands like Rotate, Angle, Segment, Ray, Point, Circle, Intersect, CircleSector, Line, OrthogonalLine, Polygon, Midpoint, and Dilate from GeoGebra. It is suitable for high school and above students to understand triangle geometry and dynamic geometric 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. Angle(A, H, C) → δ
  25. Line(C, B) → l
  26. OrthogonalLine(A, l) → q
  27. Intersect(q, l) → F
  28. Angle(B, F, A) → γ
  29. Intersect(q, r) → G
  30. Segment(C, H) → s
  31. Line(C, H) → t
  32. Circle(C, 2) → d_1
  33. Circle(H, 2) → e_1
  34. Intersect(d_1, t, 1) → L
  35. Intersect(e_1, t) → M, N
  36. Ray(C, L) → f_1
  37. Ray(H, N) → g_1
  38. Segment(C, H) → h_1
  39. Midpoint(C, G) → O
  40. Circle(B, 1) → k_1
  41. Circle(A, 1) → p_1
  42. Intersect(k_1, k, 2) → P
  43. Intersect(p_1, k, 1) → Q
  44. Ray(B, P) → i_1
  45. Ray(A, Q) → j_1
  46. Dilate(B, If[0 < o ≤ 1, 1 - o, 0], I) → B'
  47. Segment(B, B') → l_1
  48. Rotate(B, If[1 < o ≤ 2, ((-(o - 1)) * π), (-π)], I) → B'_1
  49. Rotate(B', If[1 < o ≤ 2, ((-(o - 1)) * π), (-π)], I) → B''
  50. Segment(B'_1, B'') → m_1
  51. Intersect(k_1, l, 2) → K
  52. Circle(C, 2) → q_1
  53. Intersect(q_1, l, 1) → R
  54. Segment(B, C) → n_1
  55. Ray(C, R) → r_1
  56. Ray(B, K) → s_1
  57. Dilate(A, If[2 < o ≤ 3, 3 - o, 0], F) → A'_1
  58. Segment(A, A'_1) → t_1
  59. Dilate(C, If[3 < o ≤ 4, 4 - o, 0], O) → C'
  60. Segment(C, C') → f_2
  61. Dilate(O, If[4 < o ≤ 5, 5 - o, 0], G) → O'
  62. Segment(O, O') → g_2
  63. Dilate(G, If[5 < o ≤ 6, 6 - o, 0], H) → G'
  64. Segment(G, G') → h_2
  65. Rotate(C, If[6 < o ≤ 7, ((-(o - 6)) * π), (-π)], O) → C'_1
  66. Rotate(C', If[6 < o ≤ 7, ((-(o - 6)) * π), (-π)], O) → C''
  67. Segment(C'_1, C'') → i_2
  68. Angle(I, F, O) → ε
  69. Dilate(I, If[8 < o ≤ 9, 9 - o, 0], F) → I'
  70. Dilate(O, If[8 < o ≤ 9, 9 - o, 0], F) → O'_1
  71. Segment(I, I') → j_2
  72. Segment(O, O'_1) → k_2
  73. Line(A, F) → l_2
  74. Line(C, B) → m_2