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Reflections of Lines Through the Orthocenter: Concurrency on the Circumcircle

Reflections of Lines Through the Orthocenter: Concurrency on the Circumcircle
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2014-07-29 02:16
TriangleCircleGeometric Construction
This example demonstrates the reflections of lines passing through the orthocenter H of ΔABC on the three sidelines, which concur on the circumcircle of the triangle. Key concepts include orthocenter, reflection, and circle. GeoGebra commands used include Polygon, OrthogonalLine, Intersect, Circle, Point, Line, Mirror, Locus, and VerticalText. Suitable for high school geometry education.

GeoGebra Commands & Parameters

  1. Polygon(A, B, C) → poli1, c, a, b
  2. OrthogonalLine(B, b) → d
  3. OrthogonalLine(C, c) → e
  4. Intersect(e, d) → H
  5. Circle(H, D) → f
  6. Point(f) → E
  7. Line(H, E) → g
  8. Mirror(g, c) → g'
  9. Mirror(g, b) → g'_1
  10. Intersect(g'_1, g') → F
  11. Locus(F, E) → luogo1
  12. OrthogonalLine(A, a) → h
  13. Mirror(g, b) → g'_2
  14. Mirror(g, b) → g'_3
  15. Mirror(g, a) → g'_4
  16. VerticalText(testo3) → testo4

AI Drawing Prompt Reference

Draw the circumcircle of triangle ABC and find its orthocenter H. Construct the perpendiculars from H to sides AB and AC, then reflect these perpendiculars over H to get lines g' and g'_1. The intersection point F of g' and g'_1 lies on the circumcircle. Next, construct the perpendicular from H to side AC, and reflect g' and g'_1 over H to get lines g'_2 and g'_3, which also intersect the circumcircle.
The prompt above was compiled from the original drawing commands and is for reference only.