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Dynamic Geometry Illustration: Centroid and Orthocenter of a Triangle

Dynamic Geometry Illustration: Centroid and Orthocenter of a Triangle
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2018-04-30 10:14
TriangleVectorGeometric ConstructionLocus
This example dynamically demonstrates the geometric relationship between the centroid and orthocenter of a triangle. It involves core concepts such as centroid, orthocenter, vector, transformation, and geometric construction. The construction uses commands and objects like Circle, PointIn, Semicircle, Intersect, Line, Angle, Dilate, Midpoint, Segment, Rotate, Ray, CircleSector, Vector, and Translate. Suitable for high school students and math enthusiasts studying geometric properties and dynamic demonstrations.

GeoGebra Commands & Parameters

  1. Circle(A, B) → c
  2. Distance(A, B) → a
  3. PointIn(b) → C_1
  4. Semicircle(A, C_1) → d
  5. Semicircle(C_1, A) → e
  6. Intersect(c, d, 1) → D_1
  7. Intersect(c, e, 1) → C
  8. Line(C_1, C) → f
  9. Line(C_1, D_1) → g
  10. Line(C, g) → h
  11. Intersect(c, h) → F, D
  12. Line(C_1, D) → i
  13. Intersect(c, i, 2) → E
  14. Line(C, E) → j
  15. Intersect(g, j) → G
  16. Angle(A, D_1, G) → α
  17. Angle(C_1, C, A) → β
  18. Dilate(C_1, If[0 < k ≤ 1, 1 - k, 0], G) → H
  19. Midpoint(C, C_1) → I
  20. Dilate(C_1, If[0 < k ≤ 1, 1 - k, 0], I) → J
  21. Segment(C_1, J) → l
  22. Segment(C_1, H) → m
  23. Dilate(G, If[0 < k ≤ 1, 1 - k, 0], G) → G'
  24. Dilate(A, If[1 < k ≤ 2, 2 - k, 0], D_1) → A'
  25. Dilate(A, If[1 < k ≤ 2, 2 - k, 0], C) → A'_1
  26. Segment(A, A') → n
  27. Segment(A, A'_1) → p
  28. Dilate(G, If[1 < k ≤ 2, 2 - k, 0], D_1) → G'_1
  29. Dilate(I, If[1 < k ≤ 2, 2 - k, 0], C) → I'
  30. Segment(G, G'_1) → q
  31. Segment(I, I') → r
  32. Rotate(A, 90°, C) → A'_2
  33. Angle(A, C, A'_2) → γ
  34. Ray(C, A'_2) → s
  35. Circle(C_1, (0.4 * Distance[C_1, G])) → t
  36. Angle(G, C_1, C) → δ
  37. Intersect(i, h) → K
  38. Dilate(C, If[3 < k ≤ 4, 4 - k, 0], K) → C'
  39. Segment(C, C') → f_1
  40. Intersect(t, m) → L,
  41. Rotate(L, If[4 < k ≤ 5, ((k - 4) * δ), δ], C_1) → L'
  42. CircleSector(C_1, L, L') → c_1
  43. Dilate(C_1, If[5.5 < k ≤ 6.5, 6.5 - k, 0], C) → M
  44. Vector(C_1, M) → u
  45. Translate(c_1, u) → d_1
  46. Dilate(K, If[6 < k ≤ 7, 7 - k, 0], E) → K'
  47. Segment(K, K') → g_1
  48. Dilate(E, If[7 < k ≤ 8, 8 - k, 0], C_1) → E'
  49. Segment(E, E') → h_1
  50. Dilate(C, If[8.5 < k ≤ 9.5, 9.5 - k, 0], E) → C'_1
  51. Segment(C, C'_1) → i_1
  52. Dilate(E, If[9.5 < k ≤ 10.5, 10.5 - k, 0], G) → E'_1
  53. Segment(E, E'_1) → j_1
  54. Segment(C_1, G) → k_1
  55. Segment(G, D_1) → l_1
  56. Rotate(C_1, If[11.5 < k ≤ 12.5, ((-(k - 11.5)) * π), (-π)], G) → N
  57. Rotate(G, If[11.5 < k ≤ 12.5, ((-(k - 11.5)) * π), (-π)], G) → G'_2
  58. Segment(N, G'_2) → m_1

AI Drawing Prompt Reference

Draw a triangle and dynamically demonstrate the relationship between its centroid and orthocenter. First, construct the triangle and find its vertices, then use circles and semicircles to locate the midpoints of the sides and the feet of the altitudes. Next, draw the altitudes and angle bisectors, identifying the centroid and orthocenter. Finally, apply scaling and rotation to show the dynamic relationship between the centroid and orthocenter, and use vectors to represent the positions involved.
The prompt above was compiled from the original drawing commands and is for reference only.