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Medians & Centroid Dance

Medians & Centroid Dance
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2016-11-14 22:38
TriangleVectorGeometric Construction
This example dynamically demonstrates the relationship between medians and centroids of a triangle, visually proving the centroid theorem. Key math concepts include medians, centroids, and geometric transformations. GeoGebra commands and objects used in construction are Midpoint, Segment, Intersect, Dilate, Rotate, Polygon. Suitable for high school mathematics teaching or exploration by geometry enthusiasts.

GeoGebra Commands & Parameters

  1. Midpoint(A, B) → D
  2. Midpoint(B, C) → E
  3. Midpoint(C, A) → F
  4. Segment(A, B) → a
  5. Segment(B, C) → b
  6. Segment(C, A) → c
  7. Segment(C, D) → d
  8. Segment(A, E) → e
  9. Segment(B, F) → f
  10. Intersect(d, e) → G
  11. Dilate(A, If[0 < g ≤ 1, 1 - g, 0], D) → A'
  12. Segment(A, A') → h
  13. Rotate(A, If[1 < g ≤ 2, ((g - 1) * π), 3.141592653589793], D) → A'_1
  14. Rotate(A', If[1 < g ≤ 2, ((g - 1) * π), 3.141592653589793], D) → A''
  15. Segment(A'_1, A'') → h'
  16. Dilate(B, If[2 < g ≤ 3, 3 - g, 0], E) → B'
  17. Segment(B, B') → i
  18. Rotate(B, If[3 < g ≤ 4, ((g - 3) * π), 3.141592653589793], E) → B'_1
  19. Rotate(B', If[3 < g ≤ 4, ((g - 3) * π), 3.141592653589793], E) → B''
  20. Segment(B'_1, B'') → i'
  21. Dilate(C, If[4 < g ≤ 5, 5 - g, 0], F) → C'
  22. Segment(C, C') → j
  23. Rotate(C, If[5 < g ≤ 6, ((g - 5) * π), 3.141592653589793], F) → C'_1
  24. Rotate(C', If[5 < g ≤ 6, ((g - 5) * π), 3.141592653589793], F) → C''
  25. Segment(C'_1, C'') → j'
  26. Midpoint(G, C) → H
  27. Midpoint(G, B) → I
  28. Midpoint(A, G) → J
  29. Dilate(C, If[6 < g ≤ 7, 7 - g, 0], H) → C'_2
  30. Segment(C, C'_2) → k
  31. Dilate(H, If[7 < g ≤ 8, 8 - g, 0], G) → H'
  32. Segment(H, H') → l
  33. Dilate(G, If[8 < g ≤ 9, 9 - g, 0], D) → G'
  34. Segment(G, G') → m
  35. Dilate(A, If[9 < g ≤ 10, 10 - g, 0], J) → A'_2
  36. Segment(A, A'_2) → n
  37. Dilate(J, If[10 < g ≤ 11, 11 - g, 0], G) → J'
  38. Segment(J, J') → p
  39. Dilate(G, If[11 < g ≤ 12, 12 - g, 0], E) → G'_1
  40. Segment(G, G'_1) → q
  41. Polygon(A, B, C) → poly1, c_1, a_1, b_1
  42. Dilate(F, If[12 < g ≤ 13, 13 - g, 0], G) → F'
  43. Segment(F, F') → r
  44. Rotate(F, If[14 < g ≤ 15, ((g - 14) * π), 3.141592653589793], G) → F'_1
  45. Rotate(F', If[14 < g ≤ 15, ((g - 14) * π), 3.141592653589793], G) → F''
  46. Segment(F'_1, F'') → r'
  47. Rotate(F'_1, If[15 < g ≤ 16, ((g - 15) * π), 3.141592653589793], I) → K
  48. Rotate(F'', If[15 < g ≤ 16, ((g - 15) * π), 3.141592653589793], I) → F'''
  49. Segment(K, F''') → r''
  50. Rotate(G, If[16 < g ≤ 17, ((g - 16) * π), 3.141592653589793], G) → G'_2
  51. Rotate(G'_1, If[16 < g ≤ 17, ((g - 16) * π), 3.141592653589793], G) → L
  52. Segment(G'_2, L) → q'
  53. Rotate(G'_2, If[17 < g ≤ 18, ((g - 17) * π), 3.141592653589793], J) → M
  54. Rotate(L, If[17 < g ≤ 18, ((g - 17) * π), 3.141592653589793], J) → L'
  55. Segment(M, L') → q''
  56. Rotate(G, If[18 < g ≤ 19, ((g - 18) * π), 3.141592653589793], G) → G'_3
  57. Rotate(G', If[18 < g ≤ 19, ((g - 18) * π), 3.141592653589793], G) → G''
  58. Segment(G'_3, G'') → m'
  59. Rotate(G'_3, If[19 < g ≤ 20, ((g - 19) * π), 3.141592653589793], H) → N
  60. Rotate(G'', If[19 < g ≤ 20, ((g - 19) * π), 3.141592653589793], H) → G'''
  61. Segment(N, G''') → m''

AI Drawing Prompt Reference

Draw a triangle ABC, then draw its three medians AD, BE, and CF, which intersect at point G, the centroid. Next, perform scaling and translation transformations around points D, E, and F on the vertices of the triangle to observe the changes in the medians. Finally, draw the midlines GH, GI, and GJ through point G and observe their relationships with the sides of the triangle.
The prompt above was compiled from the original drawing commands and is for reference only.