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Geometric Construction of the Thompson Cubic

Geometric Construction of the Thompson Cubic
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2021-01-02 19:13
TriangleCircleFunctionComplex NumberGeometric Construction
This GeoGebra example demonstrates the geometric construction and key point relationships behind the “Thompson Cubic (Thompson cubic curve)”. It combines classic triangle center machinery with higher-order curve definition. The main mathematical ideas include cubic curve construction (cubic curve geometry), triangle-derived special points (incircle center/centers, centroid, midpoints), angle bisectors and reflection (mirror) constraints, and the use of conics with focus properties to encode the structure. Construction highlights include creating lines and polygons, building a TriangleCurve, using Circle and Incircle, then employing Center, Midpoint, Centroid, AngularBisector, Intersect, and OrthogonalLine to locate critical points; finally, Conic and Focus are used to assemble the curve’s defining elements. Suitable for advanced high-school problem solving, contest geometry modules, and undergraduate explorations of geometric constructions of higher-degree curves.

GeoGebra Commands & Parameters

  1. Line(A, B) → f
  2. Line(B, C) → g
  3. Line(C, A) → h
  4. Polygon(A, B, C) → t1, c, a, b
  5. TriangleCurve(A, B, C, (A * ((c^(2) * B^(2)) - (b^(2) * C^(2)))) + (B * ((a^(2) * C^(2)) - (c^(2) * A^(2)))) + (C * ((b^(2) * A^(2)) - (a^(2) * B^(2)))) = 0) → eq2
  6. Circle(B, A, C) → d
  7. Center(Incircle[A, B, C]) → I
  8. Center(d) → O
  9. Incircle(A, B, C) → e
  10. Midpoint(B, A) → M_c
  11. Midpoint(A, C) → M_b
  12. Midpoint(C, B) → M_a
  13. Centroid(t1) → G
  14. AngularBisector(f, h) → i, j
  15. AngularBisector(g, h) → k, l
  16. Intersect(i, k) → I_B
  17. Intersect(k, j) → I_A
  18. Intersect(l, i) → I_C
  19. Intersect(eq2, d, 3) → Q_1
  20. Intersect(eq2, d, 6) → Q_2
  21. Intersect(eq2, d, 2) → Q_3
  22. OrthogonalLine(I_C, g) → m
  23. OrthogonalLine(I_A, f) → n
  24. OrthogonalLine(I_B, g) → p
  25. Intersect(g, p) → D
  26. Circle(I_B, D) → q
  27. Intersect(g, m) → F
  28. Circle(I_C, F) → s
  29. Intersect(f, n) → E
  30. Circle(I_A, E) → r
  31. OrthogonalLine(A, g) → t
  32. OrthogonalLine(B, h) → f_1
  33. OrthogonalLine(C, f) → g_1
  34. Intersect(f_1, t) → H
  35. Mirror(M_a, G) → J
  36. Mirror(M_b, G) → K
  37. Conic(M_a, K, M_c, J, M_b) → c_1
  38. Focus(c_1) → F_1, F_2

AI Drawing Prompt Reference

Draw a triangle and construct three line segments based on its sides and vertices. Create a triangle using the vertices and draw a cubic curve defined by a specific mathematical formula. Utilize the incircle and center of the triangle, as well as the midpoints and centroid of the vertices, to draw angle bisectors and find key intersection points. Construct special points using circles and lines, and locate new points through reflection operations. Finally, build a conic with focus properties using these points and mark its foci.
The prompt above was compiled from the original drawing commands and is for reference only.