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Collinear Points, Reflection Points, Circum-Hyperbola

Collinear Points, Reflection Points, Circum-Hyperbola
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2015-02-17 20:32
In collections A Tour of Triangle Geometry
This example demonstrates the relationship between reflected points Pa, Pb, Pc of an arbitrary point P (not on the circumcircle or orthocenter) across the sides of a triangle, and the circum-hyperbola formed by intersections of circles through these points. Key mathematical concepts include collinearity, reflection, circles, circum-hyperbola, and geometric construction. It is constructed using commands like Line, Circle, Mirror, Polygon, OrthogonalLine, Intersect, LineBisector, Conic. Suitable for high school and above students to explore geometric transformations and properties of conic sections.

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. Mirror(P, c) → P_b
  6. Mirror(P, b) → P_a
  7. Mirror(P, a) → P_c
  8. Polygon(A, B, C) → poly1, c_1, a_1, b_1
  9. OrthogonalLine(A, b) → i
  10. OrthogonalLine(B, c) → j
  11. OrthogonalLine(C, a) → k
  12. Intersect(j, k) → H
  13. Circle(A, P_c, P_b) → e
  14. Circle(C, P_b, P_a) → f
  15. Circle(B, P_c, P_a) → g
  16. Intersect(e, f, 2) → T
  17. Circle(A, B, C) → h
  18. LineBisector(P_c, P_b) → l
  19. LineBisector(P_a, P_b) → m
  20. Circle(A, B, P_c) → p
  21. Circle(C, A, P_b) → q
  22. Circle(B, C, P_a) → r
  23. Intersect(p, q, 2) → E
  24. Conic(E, A, B, P, C) → s
  25. Intersect(h, s, 3) → Q
  26. Line(Q, T) → n

AI Drawing Prompt Reference

In triangle ABC, construct a point P. Reflect P across the sides of the triangle to get points Pa, Pb, and Pc. Then draw the intersection point T of circles through these reflected points. Also draw a conic through points E, A, B, P, and C. Finally, find the intersection point Q of circle h with the conic s, and connect Q and T. This shows the relationship between reflected points and conic sections.
The prompt above was compiled from the original drawing commands and is for reference only.