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Feuerbach Circle (Nine-Point Circle)

Feuerbach Circle (Nine-Point Circle)
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2019-05-02 03:26
TriangleCircleGeometric Construction
This example demonstrates the Feuerbach Circle, also known as the Nine-Point Circle, which is the excircle tangent to both the incircle and the circumcircle of a triangle. Key math concepts include triangle geometry, the Nine-Point Circle, the Feuerbach Theorem, and properties of circles. Construction involves commands like Polygon, OrthogonalLine, Intersect, Segment, Midpoint, Circle, Angle, and AngularBisector. Suitable for high school and above students to explore the relationship between circles and triangles and geometric constructions.

GeoGebra Commands & Parameters

  1. Polygon(A, B, C) → poli1, c, a, b
  2. OrthogonalLine(A, a) → d
  3. OrthogonalLine(B, b) → e
  4. OrthogonalLine(C, c) → f
  5. Intersect(d, e) → Ortocentro
  6. Segment(A, Ortocentro) → g
  7. Segment(B, Ortocentro) → h
  8. Segment(C, Ortocentro) → i
  9. Midpoint(g) → E
  10. Midpoint(i) → F
  11. Midpoint(h) → G
  12. Midpoint(b) → H
  13. Midpoint(a) → I
  14. Midpoint(c) → J
  15. OrthogonalLine(A, a) → j
  16. OrthogonalLine(C, c) → k
  17. OrthogonalLine(B, b) → l
  18. Intersect(j, a) → K
  19. Intersect(l, b) → L
  20. Circle(H, F, I) → p
  21. OrthogonalLine(B, b) → m
  22. OrthogonalLine(A, a) → n
  23. OrthogonalLine(C, c) → q
  24. Intersect(q, c) → M
  25. Angle(Ortocentro, L, A) → α
  26. Angle(B, K, Ortocentro) → β
  27. Angle(A, M, Ortocentro) → γ
  28. Segment(C, F) → r
  29. Segment(F, Ortocentro) → s
  30. Segment(Ortocentro, G) → t
  31. Segment(G, B) → f_1
  32. Segment(Ortocentro, E) → g_1
  33. Segment(E, A) → h_1
  34. Segment(C, I) → i_1
  35. Segment(I, B) → j_1
  36. Segment(B, J) → k_1
  37. Segment(J, A) → l_1
  38. Segment(C, L) → m_1
  39. Segment(L, A) → n_1
  40. AngularBisector(C, B, A) → p_1
  41. AngularBisector(C, A, B) → q_1
  42. AngularBisector(B, C, A) → r_1
  43. Intersect(q_1, p_1) → Incentro
  44. OrthogonalLine(Incentro, c) → s_1
  45. Intersect(s_1, poli1) → D, N
  46. Circle(Incentro, N) → c_1
  47. Segment(C, M) → t_1
  48. Segment(B, L) → a_1
  49. Segment(A, K) → b_1

AI Drawing Prompt Reference

Draw a triangle ABC, then construct its orthocenter, midpoints, circumcenter, and incenter, and finally draw the nine-point circle, ensuring it passes through the triangle's feet of altitudes, midpoints, and the circumcenter.
The prompt above was compiled from the original drawing commands and is for reference only.