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Stammler Hyperbola

Stammler Hyperbola
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2015-02-16 20:30
TriangleCircleCoordinate Geometry
In collections A Tour of Triangle Geometry
The Stammler Hyperbola passes through the circumcenter, incenter, three excenters, and the vertices of the tangential triangle. This example uses GeoGebra commands such as Line, Circle, Tangent, Intersect, Polygon, Center, AngularBisector, OrthogonalLine, Conic, Mirror to demonstrate this geometric property. Suitable for high school geometry education.

GeoGebra Commands & Parameters

  1. Line(A, B) → a
  2. Line(B, C) → b
  3. Line(C, A) → c
  4. Circle(A, B, C) → d
  5. Tangent(B, d) → e,
  6. Tangent(A, d) → f,
  7. Tangent(C, d) → g,
  8. Intersect(e, f) → C'
  9. Intersect(f, g) → B'
  10. Intersect(e, g) → A'
  11. Polygon(A, B, C) → poly1, c_1, a_1, b_1
  12. Polygon(C', B', A') → poly2, f_1, d_1, e_1
  13. Center(d) → O
  14. AngularBisector(c, a) → h, i
  15. AngularBisector(b, c) → j, k
  16. AngularBisector(a, b) → l, m
  17. Intersect(i, k) → I
  18. Intersect(h, j) → E_1
  19. Intersect(i, j) → E_2
  20. Intersect(h, k) → E_3
  21. OrthogonalLine(E_1, b) → n
  22. OrthogonalLine(E_2, c) → p
  23. OrthogonalLine(E_3, c) → q
  24. Intersect(c, q) → D
  25. Circle(E_3, D) → r
  26. Intersect(p, c) → E
  27. Circle(E_2, E) → s
  28. Intersect(n, b) → F
  29. Circle(E_1, F) → t
  30. Conic(E_3, I, O, E_1, E_2) → g_1
  31. OrthogonalLine(A, b) → h_1
  32. OrthogonalLine(C, a) → i_1
  33. Intersect(h_1, i_1) → H
  34. Line(H, O) → j_1
  35. Mirror(j_1, b) → k_1
  36. Mirror(j_1, c) → l_1
  37. Intersect(l_1, k_1) → S_c
  38. Intersect(d, j_1) → G, J
  39. Line(S_c, J) → m_1
  40. Line(G, S_c) → n_1

AI Drawing Prompt Reference

Draw a geometric figure that includes a triangle, a circle, and coordinate geometry elements. The figure is constructed using lines, circles, tangents, intersections, triangles, and centers. Key objects include triangle vertices, circle center, tangent intersection points, triangle center, angle bisector intersection points, perpendicular feet, and circles. The construction order starts with the triangle, then adds the circle and tangents, followed by calculating intersections, then constructing triangles and centers, and finally adding angle bisectors and perpendicular feet, and finally drawing circles and conic sections.
The prompt above was compiled from the original drawing commands and is for reference only.