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Central Conic Cevian Complement of the Symmedian Point

Central Conic Cevian Complement of the Symmedian Point
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2015-02-17 02:41
TriangleCircleGeometric Construction
In collections A Tour of Triangle Geometry
This example demonstrates the geometric relationship between the symmetric center and symmetric complement of the symmedian point in a triangle. Key mathematical concepts include symmetric complement, centroid, and tritangent triangle. Construction involves GeoGebra commands such as Line, Incircle, AngularBisector, Centroid, and others. Suitable for high school students or geometry learners.

GeoGebra Commands & Parameters

  1. Line(A, B) → a
  2. Line(B, C) → b
  3. Line(C, A) → c
  4. Incircle(A, B, C) → d
  5. AngularBisector(a, c) → e, f
  6. AngularBisector(b, c) → g, h
  7. AngularBisector(b, a) → i, j
  8. Polygon(A, B, C) → poly1, c_1, a_1, b_1
  9. Centroid(Polygon[A, B, C]) → D
  10. Mirror(D, b_1) → D'
  11. Mirror(D, a_1) → D'_1
  12. Mirror(D, c_1) → D'_2
  13. LineBisector(D'_2, D') → k
  14. LineBisector(D', D'_1) → l
  15. Intersect(k, l) → K
  16. Circle(A, B, C) → p
  17. OrthogonalLine(A, a_1) → m
  18. OrthogonalLine(B, b_1) → n
  19. OrthogonalLine(C, c_1) → q
  20. Intersect(a, q) → E
  21. Intersect(c, n) → F
  22. Intersect(b, m) → G
  23. Polygon(E, F, G) → poly2, g_1, e_1, f_1
  24. Tangent(A, p) → r,
  25. Tangent(B, p) → s,
  26. Tangent(C, p) → t,
  27. Intersect(r, s) → H
  28. Intersect(r, t) → I
  29. Intersect(s, t) → J
  30. Polygon(H, I, J) → poly3, j_1, h_1, i_1
  31. Line(H, E) → d_1
  32. Line(I, F) → k_1
  33. Line(J, G) → l_1
  34. Intersect(d_1, k_1) → Q
  35. Line(A, Q) → m_1
  36. Line(Q, C) → n_1
  37. Line(Q, B) → p_1
  38. Line(K, A) → q_1
  39. Line(K, C) → r_1
  40. Line(K, B) → s_1
  41. Intersect(c, s_1) → L
  42. Intersect(b, q_1) → M
  43. Intersect(b, m_1) → N
  44. Intersect(c, p_1) → O
  45. Intersect(a, r_1) → P
  46. Intersect(a, n_1) → R
  47. Mirror(M, K) → M'
  48. Mirror(L, K) → L'
  49. Mirror(P, K) → P'
  50. Conic(M', O, L, P', M) → t_1

AI Drawing Prompt Reference

In a triangle, first draw the three sides and the incircle. Then construct the symmedian point and its symmetric complement. Next, draw the triangle formed by the complement and its relation to the symmedian point. Finally, draw three circles tangent to the incircle and connect their tangent points to form a triangle. Ensure the figure is clear, showing the geometric relationships between the symmedian point, complement, tangent triangle, and tangent circles.
The prompt above was compiled from the original drawing commands and is for reference only.