MathPlotAI Logo MathPlotAI

Isogonal Conjugate of Line Through the Circumcenter

Isogonal Conjugate of Line Through the Circumcenter
Loading the math board and drawing, please wait…
2015-02-15 06:17
CircleAngleLocus
In collections A Tour of Triangle Geometry
This example demonstrates the locus of isogonal conjugates of a line passing through the circumcenter, intersecting the circumcircle at antipodal points D and D'. Key mathematical concepts include isogonal conjugation, antipodal points, and locus. GeoGebra commands used include Line, Circle, Point, Center, Mirror, OrthogonalLine, Intersect, Locus, etc. Suitable for high school geometry learners and educators.

GeoGebra Commands & Parameters

  1. Line(A, B) → a
  2. Line(B, C) → b
  3. Line(C, A) → c
  4. Circle(A, B, C) → d
  5. Point(d) → D
  6. Center(d) → O
  7. Mirror(D, O) → D'
  8. OrthogonalLine(D, c) → e
  9. OrthogonalLine(D, b) → f
  10. Intersect(c, e) → E
  11. Intersect(b, f) → F
  12. Line(E, F) → g
  13. OrthogonalLine(D', a) → h
  14. OrthogonalLine(D', b) → i
  15. Intersect(a, h) → G
  16. Intersect(b, i) → H_1
  17. Line(G, H_1) → j
  18. Line(D', O) → k
  19. Point(k) → I
  20. Mirror(I, b) → I'
  21. Mirror(I, c) → I'_1
  22. Mirror(I, a) → I'_2
  23. Circle(I'_2, I', I'_1) → p
  24. Center(p) → J
  25. Locus(J, I) → loc1
  26. OrthogonalLine(A, b) → l
  27. OrthogonalLine(B, c) → m
  28. Intersect(l, m) → H
  29. Polygon(A, B, C) → poly1, c_1, a_1, b_1

AI Drawing Prompt Reference

Draw a triangle ABC and construct perpendiculars from the circumcenter O to each side. Find the feet of the perpendiculars D and D'. Then, reflect D over each side to get D'1, D'2. Draw circles with D'1, D'2, and D as centers and D as radius. Observe the locus of the intersection points of these circles. Finally, draw a line through these intersection points and observe the locus of the intersection points of this line with the line connecting the circumcenter O.
The prompt above was compiled from the original drawing commands and is for reference only.