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Point and its Conjugate on Rectangular Hyperbola

Point and its Conjugate on Rectangular Hyperbola
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2015-02-17 20:07
Coordinate GeometryGeometric ConstructionLocus
In collections A Tour of Triangle Geometry
This example demonstrates that any point P (not the orthocenter) and its reflection conjugate r(P) lie on the same rectangular hyperbola. Key math concepts include rectangular hyperbola, point reflection, conjugate points, and analytical geometry. Construction involves commands like Line, Circle, Mirror, Intersect, Conic, and Polygon in GeoGebra. Suitable for high school and above students studying hyperbola properties and geometric transformations.

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. Circle(P_a, B, C) → e
  9. Circle(P_b, A, C) → f
  10. Circle(P_c, A, B) → g
  11. Circle(P_b, P_a, P_c) → h
  12. Intersect(e, f, 2) → D
  13. Mirror(D, b) → D'
  14. Mirror(D, c) → D'_2
  15. Conic(D, A, P, C, B) → r
  16. Polygon(A, B, C) → poly1, c_1, a_1, b_1