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The Reflection Triangle in Perspective and the Nueberg Cubic

The Reflection Triangle in Perspective and the Nueberg Cubic
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2015-02-17 22:17
TriangleCircleFunction
In collections A Tour of Triangle Geometry
This example demonstrates the locus of a point P such that its reflections Pa, Pb, and Pc in the sides of a triangle form a specific triangle. Key mathematical concepts include reflection, hyperbolas, and cubic curves. The construction uses GeoGebra commands such as Line, Circle, OrthogonalLine, Intersect, Cubic, Mirror, and Polygon. Suitable for high school geometry and coordinate geometry courses.

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. OrthogonalLine(A, b) → e
  6. OrthogonalLine(B, c) → f
  7. Intersect(e, f) → H
  8. Center(d) → O
  9. Line(H, O) → g
  10. Cubic(A, B, C, 1) → l
  11. Point(l) → P
  12. Mirror(P, b) → P_a
  13. Mirror(P, c) → P_b
  14. Mirror(P, a) → P_c
  15. Polygon(P_c, P_b, P_a) → poly1, p_a, p_c, p_b
  16. Line(P_c, C) → h
  17. Line(P_b, B) → i
  18. Line(A, P_a) → j
  19. Intersect(h, i) → D
  20. Polygon(A, B, C) → poly2, c_1, a_1, b_1