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Generalization of Fermat-Dao-Nhi Equilateral Triangle GeoGebra Demo

Generalization of Fermat-Dao-Nhi Equilateral Triangle GeoGebra Demo
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2022-07-16 17:31
TriangleCoordinate GeometryGeometric Construction
This example demonstrates the relationship between an outer-Fermat triangle with centroid G and an arbitrary point P in the plane of triangle ABC. Core concepts include outer-Fermat triangle, centroid, mirror, midpoint, segment, line, and intersection. Constructed using commands like Polygon, Centroid, Mirror, Midpoint, Segment, Line, Intersect. Suitable for high school mathematics education or geometric research.

GeoGebra Commands & Parameters

  1. Polygon(A, B, 3) → poly1, f, g, h, C
  2. Polygon(B, D, 3) → poly2, i, j, k, E
  3. Polygon(D, A, 3) → poly3, l, m, n, B_f
  4. Centroid(poly1) → I
  5. Centroid(poly3) → J
  6. Centroid(poly2) → K
  7. Polygon(K, I, J) → t2, j_1, k_1, i_1
  8. Centroid(t2) → M
  9. Mirror(B, F) → B'
  10. Mirror(A, F) → A'
  11. Mirror(D, F) → D'
  12. Midpoint(B', B_f) → G
  13. Midpoint(A', E) → H
  14. Midpoint(D', C) → L
  15. Polygon(H, L, G) → t1, g_1, h_1, l_1
  16. Segment(B, F) → p
  17. Segment(F, B') → q
  18. Segment(B', G) → r
  19. Segment(G, B_f) → s
  20. Segment(A, F) → t
  21. Segment(F, A') → a
  22. Segment(A', H) → b
  23. Segment(H, E) → c
  24. Segment(D, F) → d
  25. Segment(F, D') → e
  26. Segment(D', L) → f_1
  27. Segment(L, C) → m_1
  28. Line(I, L) → n_1
  29. Line(K, H) → p_1
  30. Line(G, J) → q_1
  31. Intersect(n_1, p_1) → N
  32. Segment(F, M) → r_1
  33. Segment(M, N) → s_1

AI Drawing Prompt Reference

Construct an equilateral triangle with vertices A, B, and C, and its corresponding outer Fermat triangle determined by points A, B, C and an additional point D. Find the centroids of the three outer triangles: obtain I from the triangle through A and B, J from the one through D and A, and K from the one through B and D. Create the internal triangle by connecting K, I, and J, then take the centroid M of that triangle. Reflect B, A, and D across a shared point F to get mirrored points, and use midpoints between these mirrors and the indicated originals to define G, H, and L. Draw the lines through I and L, K and H, and G and J, take the intersection point N of the first two lines and use it together with M to form segments linking F to M and M to N, yielding a centroid–reflection–midpoint geometric construction.
The prompt above was compiled from the original drawing commands and is for reference only.