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The Nagel Point and the Extouch Triangle

The Nagel Point and the Extouch Triangle
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2015-02-14 21:18
TriangleCircleGeometric Construction
In collections A Tour of Triangle Geometry
This example demonstrates the perspective relationship between a triangle and its extouch triangle centered at the Nagel point. Key math concepts include incircles, excircles, angle bisectors, and intersection points. Construction involves commands like Line, AngularBisector, Intersect, OrthogonalLine, Circle, Center, Polygon, Segment, Mirror, and Dilate in GeoGebra. Suitable for high school and advanced learners exploring advanced triangle geometry.

GeoGebra Commands & Parameters

  1. Line(A, B) → a
  2. Line(B, C) → b
  3. Line(C, A) → c
  4. AngularBisector(B, A, C) → d
  5. AngularBisector(C, B, A) → e
  6. AngularBisector(A, C, B) → f
  7. Intersect(d, e) → I
  8. OrthogonalLine(I, c) → g
  9. Intersect(c, g) → D
  10. Circle(I, D) → h
  11. Circle(B, A, C) → p
  12. Center(p) → O
  13. OrthogonalLine(A, d) → i
  14. OrthogonalLine(B, e) → j
  15. Intersect(f, i) → E
  16. Intersect(e, i) → F
  17. Intersect(d, j) → G
  18. OrthogonalLine(E, c) → k
  19. Intersect(c, k) → H
  20. Circle(E, H) → q
  21. OrthogonalLine(F, b) → m
  22. Intersect(b, m) → J
  23. Circle(F, J) → r
  24. OrthogonalLine(G, a) → n
  25. Intersect(a, n) → K
  26. Circle(G, K) → s
  27. Intersect(r, c, 1) → B'
  28. Intersect(q, a, 1) → C'
  29. Intersect(s, b, 1) → A'
  30. Polygon(A', C', B') → poly1, l_1, n_1, m_1
  31. Segment(B, B') → t
  32. Segment(A, A') → a_1
  33. Segment(C', C) → b_1
  34. Intersect(t, a_1) → N_a
  35. Mirror(N_a, c) → N_{a1}
  36. Mirror(N_a, b) → N_{a2}
  37. Mirror(N_a, a) → N_{a3}
  38. Circle(N_{a3}, N_{a1}, N_{a2}) → c_1
  39. Center(c_1) → N'_a
  40. Dilate(p, l, N'_a) → p'

AI Drawing Prompt Reference

Draw a triangle and its three angle bisectors. Find the intersection points of the angle bisectors with the opposite sides. Draw circles centered at these points. Locate the centers of these circles and draw their circumcircles. Then, for each vertex, draw a circle that intersects the adjacent sides perpendicularly, getting intersection points. Connect these points to the corresponding vertices to form a new triangle. Finally, perform mirror operations on the new triangle relative to the original triangle's sides to complete the construction.
The prompt above was compiled from the original drawing commands and is for reference only.