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Conics Through Two Point Traces

Conics Through Two Point Traces
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2015-02-16 21:43
TriangleCoordinate GeometryGeometric Construction
In collections A Tour of Triangle Geometry
This example demonstrates constructing a unique conic through the traces of two points P and Q on the sides of triangle ABC. It involves core concepts like coordinate geometry, conic sections, midpoints, orthogonal lines, and intersections. Constructed using commands such as Line, OrthogonalLine, Intersect, Circle, Midpoint, and Conic. Ideal for high school students and educators exploring conics and geometric constructions.

GeoGebra Commands & Parameters

  1. Line(A, B) → a
  2. Line(B, C) → b
  3. Line(C, A) → c
  4. OrthogonalLine(Q_1, a) → g
  5. OrthogonalLine(Q_1, c) → h
  6. OrthogonalLine(Q_1, b) → i
  7. Intersect(b, i) → I
  8. OrthogonalLine(A, b) → j
  9. OrthogonalLine(C, a) → l
  10. OrthogonalLine(B, c) → m
  11. Intersect(a, l) → J
  12. Intersect(b, j) → K
  13. Intersect(c, m) → L
  14. Circle(L, J, K) → r
  15. Midpoint(A, C) → M
  16. Midpoint(C, B) → N
  17. Midpoint(B, A) → O
  18. Line(A, N) → n
  19. Line(C, O) → p
  20. Line(B, M) → q
  21. Line(B, Q) → d
  22. Line(A, Q) → e
  23. Line(C, Q) → f
  24. Line(B, P) → k
  25. Line(C, P) → s
  26. Line(A, P) → t
  27. Intersect(a, s) → D
  28. Intersect(a, f) → E
  29. Intersect(c, d) → F
  30. Intersect(c, k) → G
  31. Intersect(b, t) → H
  32. Intersect(b, e) → R
  33. Conic(D, F, G, H, R) → c_1

AI Drawing Prompt Reference

Draw a triangle ABC, then select points P and Q on each side. Construct a circle through the traces of points P and Q. Next, construct lines perpendicular to each side through point Q, and find their intersections I, J, K, L. Use these points to construct a circle r. Then, find the midpoints M, N, O of points P and Q, and construct lines n, p, q using these midpoints and points P and Q. Finally, use these lines and points to construct a circle c_1 passing through points D, E, F, G, H, and R.
The prompt above was compiled from the original drawing commands and is for reference only.