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Medians and Centroid Theorem

Medians and Centroid Theorem
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2019-06-25 04:44
TriangleSlider Animation
This example demonstrates the medians and centroid theorem in a triangle. By constructing the medians, the properties of the centroid are observed. Commands used include Midpoint, Segment, Intersect, Dilate, Rotate, and Polygon. Suitable for students at middle school level and above for geometric construction.

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 ABC, then find the midpoints D, E, and F of sides AB, BC, and CA respectively. Draw the medians AD, BE, and CF, which intersect at point G, which is the centroid of the triangle.
The prompt above was compiled from the original drawing commands and is for reference only.