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Incircle, Excircles, and Nagel Points

Incircle, Excircles, and Nagel Points
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2022-01-05 03:30
TriangleCircleGeometric Construction
This example demonstrates the geometric relationships and properties of incircles, excircles, and Nagel points in a triangle. Core concepts include triangles, circles, angle bisectors, intersections, perpendicular lines, and polygon constructions. It is constructed using commands like Line, AngularBisector, Intersect, OrthogonalLine, Circle, and Polygon, making it suitable for high school geometry education and construction learning.

GeoGebra Commands & Parameters

  1. Line(A, B) → f
  2. Line(B, C) → g
  3. Line(C, A) → h
  4. AngularBisector(f, h) → i, j
  5. AngularBisector(f, g) → k, l
  6. AngularBisector(h, g) → m, n
  7. Intersect(i, k) → E_C
  8. Intersect(j, l) → I
  9. Intersect(i, l) → E_B
  10. Intersect(j, k) → E_A
  11. OrthogonalLine(E_C, f) → p
  12. OrthogonalLine(I, f) → q
  13. OrthogonalLine(E_A, f) → r
  14. OrthogonalLine(E_B, f) → s
  15. Intersect(f, s) → H
  16. Intersect(f, p) → C'
  17. Intersect(f, q) → J
  18. Intersect(f, r) → K
  19. OrthogonalLine(E_C, h) → t
  20. OrthogonalLine(I, h) → a
  21. OrthogonalLine(E_B, h) → b
  22. OrthogonalLine(E_A, h) → c
  23. Intersect(h, t) → L
  24. Intersect(h, a) → M
  25. Intersect(h, b) → B'
  26. Intersect(h, c) → O
  27. OrthogonalLine(E_C, g) → d
  28. OrthogonalLine(I, g) → e
  29. OrthogonalLine(E_A, g) → f_1
  30. OrthogonalLine(E_B, g) → g_1
  31. Intersect(g, d) → P
  32. Intersect(g, e) → Q
  33. Intersect(g, f_1) → A'
  34. Intersect(g, g_1) → S
  35. Circle(E_C, L) → c_1
  36. Circle(E_B, B') → d_1
  37. Circle(I, M) → e_1
  38. Circle(E_A, O) → h_1
  39. Polygon(A, B, C) → t1, c_2, a_1, b_1
  40. Line(L, B) → i_1
  41. Line(P, A) → j_1
  42. Line(C, C') → k_1
  43. Line(A, S) → l_1
  44. Line(C, H) → m_1
  45. Line(B, B') → n_1
  46. Intersect(i_1, j_1) → N_C
  47. Intersect(l_1, m_1) → N_B
  48. Line(B, O) → p_1
  49. Line(C, K) → q_1
  50. Line(A, A') → r_1
  51. Intersect(p_1, q_1) → N_A
  52. Intersect(k_1, n_1) → N_a

AI Drawing Prompt Reference

Draw a triangle with its three sides and construct the angle bisectors of the triangle (the three pairs of bisectors are intersected pairwise to obtain the key intersection points). From each such bisector intersection, drop perpendicular lines to the corresponding triangle side, then take the intersections with the relevant sides/lines to create the corresponding foot-related points. Construct several circles: each circle is determined by a chosen center at one of the intersection-derived points together with a second point obtained from those perpendicular constructions, so the radius is fixed by two points. Connect the main constructed points to the appropriate triangle vertices and intersections, and include the polygon formed by the three original triangle vertices to complete the incircle–excircle–Nagel-point style configuration.
The prompt above was compiled from the original drawing commands and is for reference only.