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Fermat Point - Steiner Point

Fermat Point - Steiner Point
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2013-09-06 16:23
TriangleAngleGeometric Construction
In collections Besondere Punkte/Special Points
This example demonstrates the Fermat point, where point M satisfies ∠AMB=∠BMC=∠CMA=2π/3. Using commands like Segment, Rotate, Polygon, Line, Intersect, Angle, Circle, the construction of the Fermat point of triangle ABC is shown. Suitable for high school geometry, it helps in understanding the properties and construction of the Fermat point.

GeoGebra Commands & Parameters

  1. Segment(A, M) → a
  2. Segment(B, M) → b
  3. Segment(C, M) → c
  4. Rotate(C, -1.0471975511965976, B) → A'
  5. Rotate(C, 1.0471975511965976, A) → B'
  6. Rotate(B, -1.0471975511965976, A) → C'
  7. Polygon(A, B, C') → poly1, c', a_1, b_1
  8. Polygon(A', B, C) → poly2, c_1, a', b_2
  9. Polygon(A, B', C) → poly3, c_2, a_2, b'
  10. Line(A, A') → e
  11. Line(B, B') → f
  12. Line(C, C') → g
  13. Intersect(e, f) → M'
  14. Rotate(M, 1.0471975511965976, A) → M_B
  15. Rotate(M, -1.0471975511965976, A) → M_C
  16. Segment(M, M_B) → h_B
  17. Segment(B', M_B) → h'_B
  18. Rotate(M, -1.0471975511965976, B) → M_A
  19. Segment(M, M_A) → j
  20. Segment(A', M_A) → k
  21. Segment(C', M_C) → l
  22. Segment(M, M_C) → m
  23. Rotate(M, 1.0471975511965976, C) → M'_A
  24. Segment(M, M'_A) → q
  25. Segment(A', M'_A) → r
  26. Rotate(M, -1.0471975511965976, C) → M'_B
  27. Segment(B', M'_B) → s
  28. Segment(M'_B, M) → t
  29. Rotate(M, 1.0471975511965976, B) → M'_C
  30. Segment(C', M'_C) → d_1
  31. Segment(M, M'_C) → e_1
  32. Angle(C, M, A) → α
  33. Angle(A, M, B) → β
  34. Angle(B, M, C) → γ
  35. Segment(A, M_C) → f_1
  36. Segment(B, M'_C) → g_1
  37. Segment(B, M_A) → h_1
  38. Segment(C, M'_A) → i_1
  39. Segment(A, M_B) → j_1
  40. Segment(C, M'_B) → k_1
  41. Circle(A, B, C') → p_1
  42. Circle(A', B, C) → q_1
  43. Circle(A, B', C) → r_1

AI Drawing Prompt Reference

Draw a triangle ABC, then construct the Fermat point M such that ∠AMB=∠BMC=∠CMA=120°. First, use rotations and polygons to find point M, then draw line segments from M to each vertex of the triangle, and finally use rotations and intersections to locate the Fermat point M.
The prompt above was compiled from the original drawing commands and is for reference only.