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Medians and Centroid Dance

Medians and Centroid Dance
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2019-11-13 18:59
TriangleVectorGeometric Construction
This example demonstrates the theorem of medians in a triangle, showing the properties of the centroid through a proof without words. It uses commands like Midpoint, Segment, Intersect, Dilate, Rotate, and Polygon, involving concepts such as triangles and vectors. Suitable for middle school and above students to learn geometric construction and theorem proofs.

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), (-π)], 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), (-π)], 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), (-π)], 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 with its three medians intersecting at the centroid. Animate six stages: first, shrink each segment from a vertex to the midpoint of the opposite side, then rotate that segment 180 degrees around the midpoint; next, shrink segments from the centroid to each vertex, to each midpoint, and from centroid to midpoints; finally, rotate the centroid-to-midpoint segment around the centroid and around another midpoint, creating a dynamic proof without words. Show the triangle, medians, centroid, and all moving segments.
The prompt above was compiled from the original drawing commands and is for reference only.