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Hyperbola from Point on Circumcircle

Hyperbola from Point on Circumcircle
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2015-02-16 21:55
Coordinate GeometryLocus
In collections A Tour of Triangle Geometry
This example demonstrates the construction of a rectangular hyperbola C(G,P) where point P moves on the circumcircle of a triangle. The traces of P on the triangle's sides form the hyperbola. Key math concepts include circumcircle, locus, conic sections, and rectangular hyperbola. GeoGebra commands used are Circle, Line, OrthogonalLine, Intersect, Midpoint, Centroid, Point, Conic, and Center. Suitable for high school and above students to explore conic properties and geometric constructions.

GeoGebra Commands & Parameters

  1. Circle(A, B, C) → p
  2. Line(A, B) → a
  3. Line(B, C) → b
  4. Line(C, A) → c
  5. OrthogonalLine(Q_1, a) → g
  6. OrthogonalLine(Q_1, c) → h
  7. OrthogonalLine(Q_1, b) → i
  8. Intersect(b, i) → I
  9. Midpoint(A, C) → M
  10. Midpoint(C, B) → N
  11. Midpoint(B, A) → O_1
  12. Centroid(Polygon[A, B, C]) → G
  13. Line(B, G) → d
  14. Line(A, G) → e
  15. Line(C, G) → f
  16. Point(p) → P
  17. Line(B, P) → k
  18. Line(C, P) → s
  19. Line(A, P) → t
  20. Intersect(a, s) → D
  21. Intersect(a, f) → E
  22. Intersect(c, d) → F
  23. Intersect(c, k) → G_1
  24. Intersect(b, t) → H
  25. Intersect(b, e) → R
  26. Conic(D, F, G_1, H, R) → c_1
  27. Center(p) → O

AI Drawing Prompt Reference

Draw the circumcircle of triangle ABC, and let point P move on the circle. As P moves, its projections on the sides of the triangle form a hyperbola. First, draw the three sides of the triangle, then construct the projections of P on each side, connect these projections with P, and finally use these points to construct the hyperbola.
The prompt above was compiled from the original drawing commands and is for reference only.