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Constructing the Feynman Triangle: Rotations and Ratio Geometry

Constructing the Feynman Triangle: Rotations and Ratio Geometry
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2025-02-02 16:52
TriangleCirclePolygonTransformationGeometric Construction
In collections Besondere Punkte/Special Points
This example demonstrates how to build the so-called “Feynman triangle” configuration and visualize the geometric relationships driven by rotation/transform ideas and resulting ratio structures. Key concepts include triangle geometry, transformation via rotations (conceptually), circle–line intersections, midpoint-based constructions, and assembling polygons to reveal the overall figure. Construction is carried out in GeoGebra using Segment to define base edges, OrthogonalLine for perpendicular lines, Circle to generate auxiliary circles, and Intersect to obtain intersection points. Points and Midpoint are then used to create dependent geometric elements and to form the main Polygon structure for inspection. To express and visualize constraints, the construction also employs ImplicitCurve (e.g., equation objects like eq3 and related curve objects) and Locus to generate the set of points satisfying the evolving conditions (tracking how the configuration changes under parameter/condition variations). It is intended for advanced secondary students, contest geometry learners, and introductory college-level courses focusing on geometric construction, transformation visualization, and constraint-based locus/implicit-curve rendering.

GeoGebra Commands & Parameters

  1. Segment(A_{4}, B_{4}) → a
  2. OrthogonalLine(A_{4}, a) → f_{3}
  3. Circle(A_{4}, h) → c_2
  4. Intersect(c_2, f_{3}, 2) → C_{3}
  5. OrthogonalLine(C_{3}, f_{3}) → g
  6. Point(g) → D_{3}
  7. Polygon(D_{3}, A_{4}, B_{4}) → t0, b, d, c
  8. Midpoint(D_{3}, A_{4}) → Q_{3}
  9. Midpoint(A_{4}, B_{4}) → T
  10. Midpoint(D_{3}, B_{4}) → W
  11. Circle(D_{3}, b / 5) → p_1
  12. Circle(D_{3}, c / 5) → q_1
  13. Circle(A_{4}, a / 5) → r_1
  14. Circle(D_{3}, b / 6) → d_2
  15. Circle(A_{4}, b / 6) → e_2
  16. Circle(D_{3}, c / 6) → p_2
  17. Circle(B_{4}, c / 6) → q_2
  18. Circle(A_{4}, a / 6) → r_2
  19. Circle(B_{4}, a / 6) → s_2
  20. Circle(D_{3}, b / 7) → d_3
  21. Intersect(a, b) → J_2
  22. Circle(A_{4}, b / 7) → e_3
  23. Circle(D_{3}, c / 7) → g_3
  24. Circle(B_{4}, c / 7) → p_3
  25. Circle(A_{4}, a / 7) → q_3
  26. Circle(B_{4}, a / 7) → r_3
  27. ImplicitCurve({M, N, O, P_{2}, Q}) → eq3
  28. Circle(D_{3}, b / 9) → c_4
  29. Circle(A_{4}, b / 9) → d_4
  30. Circle(A_{4}, a / 9) → e_4
  31. Circle(B_{4}, a / 9) → k_4
  32. Circle(B_{4}, c / 9) → c_5
  33. Circle(D_{3}, c / 9) → d_5
  34. Midpoint(b) → T_3
  35. Midpoint(T_3, D_{3}) → U_3
  36. Midpoint(T_3, A_{4}) → W_3
  37. Midpoint(a) → C_4
  38. Midpoint(A_{4}, C_4) → D_4
  39. Midpoint(C_4, B_{4}) → F_4
  40. Midpoint(c) → H_4
  41. Midpoint(H_4, B_{4}) → I_4
  42. Midpoint(H_4, D_{3}) → K_4
  43. Circle(A_{4}, a / n) → k_1
  44. Intersect(k_1, a, 1) → F
  45. Circle(B_{4}, c / n) → t_1
  46. Intersect(t_1, c, 1) → S_1
  47. Circle(D_{3}, b / n) → f_2
  48. Intersect(f_2, b, 1) → T_1
  49. Segment(A_{4}, S_1) → j_1
  50. Segment(D_{3}, F) → m_1
  51. Segment(B_{4}, T_1) → n_1
  52. Intersect(m_1, n_1) → C_2
  53. Intersect(j_1, m_1) → D_2
  54. Intersect(j_1, n_1) → E_2
  55. Polygon(D_2, C_2, E_2) → t1, a_1, b_1, g_2
  56. Locus(P, n) → l
  57. Locus(H_2, n) → lieu1
  58. Point(lieu1) → R_2
  59. Segment(F_2, R_2) → s
  60. Intersect(yAxis, xAxis) → T_2
  61. Segment(F_2, U_2) → h_1
  62. Intersect(p, q, (1, 1)) → B
  63. Polygon(D_2, E_2, C_2) → t2, i, j, k
  64. Segment(D_{3}, T_1) → m
  65. Polygon(D_2, E_2, C_2) → t3, r, t, f
  66. Polygon(D_2, E_2, C_2) → t4, f_1, g_1, i_1
  67. Polygon(D_2, E_2, C_2) → t5, l_1, s_1, c_1
  68. Polygon(C_2, D_2, E_2) → t6, d_1, e_1, h_2
  69. Polygon(E_2, D_2, C_2) → t7, e, i_2, j_2

AI Drawing Prompt Reference

Draw the Feynman triangle by first creating a base triangle, then constructing auxiliary lines and circles through rotations and geometric transformations, finding their intersections. Use these points to form new geometric shapes. Calculate midpoints of key segments and draw small circles around them. Finally, employ implicit curves and loci to visualize geometric constraints and dynamic relationships.
The prompt above was compiled from the original drawing commands and is for reference only.