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Dynamic Demonstration of a Phenomenon When Two Medians of a Triangle Are Perpend

Dynamic Demonstration of a Phenomenon When Two Medians of a Triangle Are Perpend
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2018-01-10 22:23
TrianglePolygonAngleVectorGeometric Construction
This example dynamically illustrates the geometric phenomenon that occurs when any two medians of a triangle are perpendicular. It involves core mathematical concepts such as medians, perpendicularity, vectors, and polygon construction. The construction utilizes GeoGebra commands and objects like Segment, Circle, Intersect, OrthogonalLine, Ray, Distance, Min, Polygon, Dilate, Rotate, CircleSector, Line, Point, Angle, Vector, and Translate. It is suitable for high school and above students to explore special triangle properties and dynamic geometric transformations.

GeoGebra Commands & Parameters

  1. Segment(A, B) → f
  2. Circle(A, (2 * f / 3)) → c
  3. Intersect(c, f) → C,
  4. OrthogonalLine(C, f) → g
  5. Circle(C, 2) → d
  6. Intersect(d, g, 2) → D
  7. Segment(D, C) → h
  8. Circle(C, (2 * h)) → e
  9. Intersect(e, g, 1) → E
  10. Segment(C, E) → i
  11. Ray(A, D) → j
  12. Ray(E, B) → k
  13. Intersect(j, k) → F
  14. Distance(A, D) → distanceAD
  15. Distance(D, F) → distanceDF
  16. Distance(B, F) → distanceBF
  17. Distance(B, E) → distanceBE
  18. Distance(C, F) → distanceCF
  19. Distance(A, E) → distanceAE
  20. Segment(C, B) → l
  21. Min(h, l) → a
  22. Circle(C, (0.4 * a)) → p
  23. Segment(A, B) → m
  24. Intersect(p, m) → G, H
  25. Polygon(A, E, F) → t1, f_1, a_1, e_1
  26. Dilate(A, If[0 < b ≤ 1, 1 - b, 0], D) → A'
  27. Dilate(E, If[2 < b ≤ 3, 3 - b, 0], B) → E'
  28. Segment(A, A') → n
  29. Rotate(A, If[1 < b ≤ 2, ((b - 1) * π), π], D) → A'_1
  30. Rotate(A', If[1 < b ≤ 2, ((b - 1) * π), π], D) → A''
  31. Segment(A'_1, A'') → n'
  32. Segment(E, E') → q
  33. Rotate(E, If[3 < b ≤ 4, ((-(b - 3)) * π), (-π)], B) → E'_1
  34. Rotate(E', If[3 < b ≤ 4, ((-(b - 3)) * π), (-π)], B) → E''
  35. Segment(E'_1, E'') → q'
  36. Dilate(E, If[4 < b ≤ 5, 5 - b, 0], D) → E'_2
  37. Segment(E, E'_2) → r
  38. Dilate(A, If[5 < b ≤ 6, 6 - b, 0], C) → A'_2
  39. Segment(A, A'_2) → s
  40. Dilate(C, If[6 < b ≤ 7, 7 - b, 0], B) → C'
  41. Segment(C, C') → t
  42. Rotate(H, If[7 < b ≤ 8, ((b - 7) * π / 2), π / 2], C) → H'
  43. CircleSector(C, H, H') → c_1
  44. Rotate(c_1, If[9 < b ≤ 10, ((b - 9) * π / 2), π / 2], C) → d_1
  45. Dilate(F, If[10.5 < b ≤ 11.5, 11.5 - b, 0], C) → F'
  46. Segment(F, F') → g_1
  47. Line(A, g_1) → h_1
  48. Point(h_1) → I
  49. Angle(E, A, I) → α
  50. Dilate(C, If[12 < b ≤ 13, 13 - b, 0], A) → C'_1
  51. Vector(C, C'_1) → u
  52. Translate(F, u) → F'_1
  53. Translate(F', u) → F''
  54. Segment(F'_1, F'') → i_1
  55. Rotate(F'_1, If[13 < b ≤ 14, ((-(b - 13)) * α), (-α)], A) → J
  56. Rotate(F'', If[13 < b ≤ 14, ((-(b - 13)) * α), (-α)], A) → F'''
  57. Segment(J, F''') → j_1

AI Drawing Prompt Reference

Draw a triangle where two medians are perpendicular. Start with a base segment, construct a perpendicular line at a point on it, and use circle intersections to determine the third vertex. Connect vertices to midpoints of opposite sides to form medians, which intersect at the centroid. Animate the centroid moving along a median to show shape changes, label side and median lengths, and use circular sectors and rotating segments to illustrate the right angle relationship.
The prompt above was compiled from the original drawing commands and is for reference only.