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Isogonal Conjugate of the Gergonne Point

Isogonal Conjugate of the Gergonne Point
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2015-02-14 19:07
TriangleCircleGeometric Construction
In collections A Tour of Triangle Geometry
This example demonstrates the isogonal conjugate of the Gergonne point, which is the internal center of similarity of the circumcircle and the incircle. Key mathematical concepts include isogonal conjugate, incenter, circumcircle, incircle. Construction involves commands like Line, AngularBisector, Intersect, OrthogonalLine, Circle, Segment, Polygon, Center, Mirror, Dilate, and objects A, B, a, C, b, c, d, e, f, I, g, D, h, E, C', F, B', H, A', i, j, k, G_e, poly1, g_1, j_1, k_1, p, O, G'_{e2}, G'_{e1}, G'_{e3}, q, G'_e, l, p', low. Suitable for high school and above students to explore the properties of the incenter and the construction of isogonal conjugates.

GeoGebra Commands & Parameters

  1. Line(A, B) → a
  2. Line(B, C) → b
  3. Line(C, A) → c
  4. AngularBisector(B, A, C) → d
  5. AngularBisector(C, B, A) → e
  6. AngularBisector(A, C, B) → f
  7. Intersect(d, e) → I
  8. OrthogonalLine(I, c) → g
  9. Intersect(c, g) → D
  10. Circle(I, D) → h
  11. Intersect(h, a) → E, C'
  12. Intersect(h, c) → F, B'
  13. Intersect(h, b) → H, A'
  14. Segment(E, C) → i
  15. Segment(A, H) → j
  16. Segment(B, F) → k
  17. Intersect(i, j) → G_e
  18. Polygon(A', C', B') → poly1, g_1, j_1, k_1
  19. Circle(B, A, C) → p
  20. Center(p) → O
  21. Mirror(G_e, c) → G'_{e2}
  22. Mirror(G_e, a) → G'_{e1}
  23. Mirror(G_e, b) → G'_{e3}
  24. Circle(G'_{e2}, G'_{e1}, G'_{e3}) → q
  25. Center(q) → G'_e
  26. Dilate(p, l, G'_e) → p'

AI Drawing Prompt Reference

Draw a triangle and find its incenter and circumcenter, which are the centers of similarity between the incircle and circumcircle. Then, construct the isogonal conjugate of the Gergonne point and show the lines connecting it to the triangle's vertices. Finally, create the locus of the isogonal conjugate using reflections and dilations, and draw the related circles and segments.
The prompt above was compiled from the original drawing commands and is for reference only.