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Reflection Triangle for Points on Euler Line

Reflection Triangle for Points on Euler Line
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2015-02-18 01:11
TriangleCoordinate Geometry
In collections A Tour of Triangle Geometry
This example demonstrates the triangle formed by the reflections of a point on the Euler line in the perpendicular bisectors of the sides of a triangle. Key concepts include reflection, the Euler line, and perpendicular bisectors. GeoGebra commands used include Centroid, Circle, LineBisector, Mirror, and Polygon. Suitable for high school geometry.

GeoGebra Commands & Parameters

  1. Centroid(Polygon[A, B, C]) → G
  2. Circle(A, C, B) → g
  3. Center(g) → O
  4. Line(G, O) → k
  5. Line(A, B) → a
  6. Line(B, C) → b
  7. Line(C, A) → c
  8. LineBisector(B, C) → d
  9. LineBisector(A, C) → e
  10. LineBisector(B, A) → f
  11. Point(k) → P
  12. Mirror(P, e) → P_b
  13. Mirror(P, d) → P_a
  14. Mirror(P, f) → P_c
  15. Polygon(P_a, P_b, P_c) → poly1, p_c, p_a, p_b
  16. Line(A, P_a) → h
  17. Line(B, P_b) → i
  18. Line(P_c, C) → j
  19. Intersect(h, i) → F
  20. Locus(F, P) → loc1
  21. Polygon(A, B, C) → poly2, c_1, a_1, b_1
  22. OrthogonalLine(A, b) → l
  23. OrthogonalLine(B, b_1) → m
  24. Intersect(l, m) → H

AI Drawing Prompt Reference

Draw a triangle ABC, then choose a point P on the Euler line. Reflect P over the perpendicular bisectors of BC, AC, and AB to get points P_a, P_b, and P_c respectively. Connect P_a, P_b, and P_c to form triangle P_aP_bP_c. Then, draw lines from A to P_a, B to P_b, and C to P_c, and find their intersection point F. Finally, use F and P to create locus loc1.
The prompt above was compiled from the original drawing commands and is for reference only.