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Orthocenter and Circumcenter are Isogonal

Orthocenter and Circumcenter are Isogonal
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2015-02-12 05:54
TriangleAngleGeometric Construction
In collections A Tour of Triangle Geometry
This example demonstrates the isogonal conjugacy between the orthocenter (H) and circumcenter (O) of a triangle. Key math concepts include isogonal conjugacy, triangle centers, angle bisectors, perpendiculars, intersections, and angles. Construction involves commands like Segment, LineBisector, Intersect, OrthogonalLine, Angle, and Line in GeoGebra. Suitable for high school geometry education or exploration by enthusiasts.

GeoGebra Commands & Parameters

  1. Segment(A, B) → a
  2. Segment(B, C) → b
  3. Segment(C, A) → c
  4. LineBisector(c) → d
  5. LineBisector(b) → e
  6. LineBisector(a) → f
  7. Intersect(d, e) → O
  8. OrthogonalLine(A, b) → g
  9. OrthogonalLine(B, c) → h
  10. OrthogonalLine(C, a) → i
  11. Intersect(g, h) → H
  12. Angle(B, A, H) → α
  13. Angle(O, A, C) → β
  14. Line(A, O) → j

AI Drawing Prompt Reference

Draw a triangle and construct the angle bisectors of its three sides, which intersect at point O. Then, draw perpendiculars from each vertex to the opposite side, meeting at point H. Connect each vertex to O and observe the symmetric relationship between H and O. Label the angles at vertex BAC and OAC to demonstrate the isogonal conjugacy between H and O.
The prompt above was compiled from the original drawing commands and is for reference only.