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Median, Angle Bisector, and Altitude Concurrency in a Triangle

Median, Angle Bisector, and Altitude Concurrency in a Triangle
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2017-04-29 00:50
TriangleVectorGeometric Construction
This example dynamically illustrates the concurrency of the median, angle bisector, and altitude in a triangle. Key mathematical concepts include medians, angle bisectors, and altitudes. The construction uses GeoGebra commands such as Segment, OrthogonalLine, Point, etc. Suitable for middle school and above students learning geometric construction.

GeoGebra Commands & Parameters

  1. Segment(A, B) → f
  2. OrthogonalLine(A, f) → g
  3. Point(g) → C
  4. Ray(A, C) → h
  5. Ray(A, C) → i
  6. Point(i) → D
  7. Segment(A, D) → j
  8. Segment(D, B) → k
  9. Angle(B, A, D) → α
  10. Circle(B, j) → c
  11. Intersect(c, k, 1) → E
  12. Segment(B, E) → l
  13. Segment(D, E) → m
  14. OrthogonalLine(E, j) → n
  15. Angle(A, D, E) → β
  16. Rotate(A, β / 2, D) → A'
  17. Angle(A, D, A') → γ
  18. Ray(D, A') → p
  19. Intersect(n, p) → F
  20. Midpoint(m) → G
  21. Intersect(n, j) → H
  22. Segment(A, E) → r
  23. Segment(G, F) → q
  24. Segment(F, A) → s
  25. Intersect(p, r) → I
  26. OrthogonalLine(B, f) → t
  27. Point(t) → J
  28. Angle(J, B, E) → δ
  29. Polygon(D, A, E) → poly1, e, d, a
  30. Polygon(A, B, E) → poly2, e_1, a_1, b
  31. Segment(D, H) → f_1
  32. Segment(D, G) → g_1
  33. Min(f_1, g_1) → o
  34. Circle(D, (0.6 * o)) → c_1
  35. Segment(D, E) → h_1
  36. Intersect(c_1, h_1, 1) → K
  37. Intersect(c_1, j, 1) → L
  38. Intersect(c_1, p, 1) → M
  39. CircleSector(D, L, M) → d_1
  40. Dilate(A, If[0 < u ≤ 1, 1 - u, 0], B) → A'_1
  41. Vector(A, A'_1) → v
  42. Translate(A, v) → A'_2
  43. Translate(D, v) → D'
  44. Segment(A'_2, D') → j'
  45. Rotate(A'_2, If[1 < u ≤ 2, ((u - 1) * δ), δ], B) → N
  46. Rotate(D', If[1 < u ≤ 2, ((u - 1) * δ), δ], B) → D''
  47. Segment(N, D'') → j''
  48. Dilate(E, If[2 < u ≤ 3, 3 - u, 0], G) → E'
  49. Segment(E, E') → i_1
  50. Rotate(E, If[3 < u ≤ 4, ((u - 3) * π), π], G) → E'_1
  51. Rotate(E', If[3 < u ≤ 4, ((u - 3) * π), π], G) → E''
  52. Segment(E'_1, E'') → j_1
  53. Dilate(A, If[4 < u ≤ 5, 5 - u, 0], E) → A'_3
  54. Segment(A, A'_3) → k_1
  55. Polygon(A, E, D) → poly3, d_2, a_2, e_2
  56. Segment(E, H) → l_1
  57. Angle(A, H, E) → ε
  58. Dilate(E, If[5 < u ≤ 6, 6 - u, 0], H) → E'_2
  59. Segment(E, E'_2) → m_1
  60. Dilate(D, If[6 < u ≤ 7, 7 - u, 0], F) → D'_1
  61. Segment(D, D'_1) → n_1
  62. Dilate(F, If[7 < u ≤ 8, 8 - u, 0], I) → F'
  63. Segment(F, F') → p_1
  64. Rotate(d_1, If[8 < u ≤ 9, ((u - 8) * γ), γ], D) → q_1
  65. Dilate(A, If[9 < u ≤ 10, 10 - u, 0], F) → A'_4
  66. Segment(A, A'_4) → r_1
  67. Dilate(F, If[10 < u ≤ 11, 11 - u, 0], G) → F'_1
  68. Segment(FF, F'_1) → s_1

AI Drawing Prompt Reference

Draw a triangle and construct its median, angle bisector, and altitude, highlighting their point of concurrency. Then, draw a circle centered at the intersection of the angle bisector, intersecting the altitude at two points. Connect these points to the triangle's vertices to form two smaller triangles. Finally, rotate and scale one of the smaller triangles to observe geometric transformations.
The prompt above was compiled from the original drawing commands and is for reference only.