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The Central Cevian Complement of the Incenter and the Excentral Triangle

The Central Cevian Complement of the Incenter and the Excentral Triangle
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2015-02-17 02:31
TriangleCircleGeometric Construction
In collections A Tour of Triangle Geometry
This example demonstrates the relationship between the incenter P and the homothetic center Q of the excentral triangle and intouch triangle. Key concepts include the incircle, excentral triangle, intouch triangle, and homothetic center. Using GeoGebra commands such as Line, Incircle, Center, AngularBisector, Intersect, etc., these geometric elements can be constructed. Suitable for high school geometry learners and math enthusiasts.

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. Center(d) → P
  6. AngularBisector(a, c) → e, f
  7. AngularBisector(b, c) → g, h
  8. AngularBisector(b, a) → i, j
  9. Intersect(e, h) → D
  10. Intersect(e, g) → E
  11. Intersect(f, g) → F
  12. Intersect(d, a) → G, J
  13. Intersect(d, c) → H, L
  14. Intersect(d, b) → I, K
  15. Polygon(A, B, C) → poly1, c_1, a_1, b_1
  16. Line(D, G) → k
  17. Line(H, E) → l
  18. Line(F, I) → m
  19. Intersect(k, l) → Q
  20. Polygon(D, E, F) → poly2, f_1, d_1, e_1
  21. Polygon(G, H, I) → poly3, i_1, g_1, h_1
  22. Line(A, Q) → n
  23. Line(B, Q) → p
  24. Line(C, Q) → q
  25. Intersect(q, a) → M
  26. Intersect(p, c) → N
  27. Intersect(b, n) → O
  28. Line(A, P) → r
  29. Line(B, P) → s
  30. Line(C, P) → t
  31. Intersect(t, a) → R
  32. Intersect(r, b) → S
  33. Intersect(s, c) → T
  34. Mirror(T, P) → T'
  35. Mirror(S, P) → S'
  36. Mirror(R, P) → R'
  37. Conic(R', S, T', R, S') → k_1

AI Drawing Prompt Reference

Draw a triangle ABC, then construct its incircle and find its center P. Next, draw the angle bisectors to locate the homothetic center Q of the excentral triangle and intouch triangle. Finally, draw lines from Q to the vertices of the triangle, and the mirror points of P as R', S', T'. Use these points to draw a conic section.
The prompt above was compiled from the original drawing commands and is for reference only.