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Nine-Point Circle Construction in Triangle

Nine-Point Circle Construction in Triangle
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2021-09-27 21:49
TriangleCircleGeometric Construction
This example demonstrates the construction of the famous Nine-Point Circle (6-Punkte-Kreis) in a triangle. The Nine-Point Circle passes through nine significant points: the midpoints of the three sides, the feet of the three altitudes, and the midpoints of the segments from each vertex to the orthocenter. Construction: Use Polygon to create △ABC, Line and OrthogonalLine to construct altitudes and obtain foot points H_a, H_b, H_c, Segment to mark side midpoints M_a, M_b, M_c, and Circle with Center to draw the circle through these nine points. Objects include vertices A, B, C, orthocenter H, angles α, β, γ, etc. Ideal for secondary school geometry teaching, suitable for teaching special points in triangles and circle construction techniques.

GeoGebra Commands & Parameters

  1. Polygon(A, B, C) → d1, c, a, b
  2. Line(A, B) → f
  3. Line(B, C) → g
  4. Line(C, A) → h
  5. OrthogonalLine(A, g) → i
  6. OrthogonalLine(B, h) → j
  7. OrthogonalLine(C, f) → k
  8. Intersect(i, j) → H
  9. Intersect(k, f) → H_c
  10. Intersect(i, g) → H_a
  11. Intersect(j, h) → H_b
  12. Angle(B, H_c, H) → α
  13. Angle(A, H_b, H) → β
  14. Angle(C, H_a, H) → γ
  15. Segment(H, H_a) → l
  16. Segment(H, H_c) → m
  17. Segment(H, H_b) → n
  18. Segment(H, B) → p
  19. Segment(H, C) → q
  20. Segment(H, A) → r
  21. Circle(A, B, C) → d
  22. Center(d) → U
  23. OrthogonalLine(U, f) → s
  24. OrthogonalLine(U, g) → t
  25. OrthogonalLine(U, h) → e
  26. Intersect(e, h) → M_b
  27. Intersect(s, f) → M_c
  28. Intersect(t, g) → M_a
  29. Segment(U, M_c) → f_1
  30. Segment(U, M_a) → g_1
  31. Segment(U, M_b) → h_1
  32. Circle(M_c, H) → c_1
  33. Circle(M_a, H) → d_1
  34. Circle(M_b, H) → e_1
  35. Intersect(e_1, h) → T, S_{1}
  36. Intersect(d_1, g) → P, O
  37. Intersect(c_1, f) → R, Q
  38. Circle(S_{1}, R, Q) → k_1
  39. Segment(M_a, M_b) → i_1
  40. Segment(M_c, M_b) → j_1
  41. Segment(M_c, M_a) → l_1
  42. Circle(H_a, M_a, H_c) → p_1
  43. Center(p_1) → N
  44. Segment(N, U) → m_1
  45. Segment(N, H) → n_1
  46. Polygon(H_c, H_a, H_b) → d2, h_b, h_c, h_a
  47. Intersect(d_1, e_1, 2) → D
  48. Intersect(c_1, d_1, 2) → E
  49. Intersect(c_1, e_1, 2) → F
  50. Segment(A, M_a) → q_1
  51. Segment(M_b, B) → r_1
  52. Segment(C, M_c) → s_1
  53. Intersect(q_1, r_1) → S

AI Drawing Prompt Reference

Draw a triangle ABC first, then construct its nine-point circle. Start by creating triangle ABC with points A, B, and C; next, construct the altitudes from vertices A, B, and C to the opposite sides, finding the feet of the altitudes H_a, H_b, and H_c; then locate the orthocenter H of the triangle; after that, find the midpoints M_a, M_b, and M_c of the sides; subsequently, construct circles with H as the center and passing through H_a, H_b, and H_c, intersecting the sides to get nine significant points; finally, use these points to draw the nine-point circle.
The prompt above was compiled from the original drawing commands and is for reference only.