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Exeter Point as Focus of Inscribed Parabola

Exeter Point as Focus of Inscribed Parabola
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2015-02-17 09:36
Coordinate Geometry
In collections A Tour of Triangle Geometry
This example demonstrates the geometric properties when the Exeter point acts as the focus of an inscribed parabola, showing that its directrix is the Euler line and involving concepts like focus, directrix, perspector, equilateral triangles, ratios, and trigonometry. It is constructed using GeoGebra commands such as Line, Circle, Point, Mirror, Parabola, etc. Suitable for high school and advanced learners exploring parabolas and Euler lines.

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. Point(d) → F
  6. Mirror(F, c) → F'_c
  7. Mirror(F, a) → F'_a
  8. Mirror(F, b) → F'_b
  9. Line(F'_a, F'_b) → e
  10. Parabola(F, e) → f
  11. OrthogonalLine(F, e) → g
  12. OrthogonalLine(F'_c, e) → h
  13. OrthogonalLine(F'_a, e) → i
  14. OrthogonalLine(F'_b, e) → j
  15. Intersect(f, c, 1) → Z
  16. Intersect(f, a, 1) → X
  17. Intersect(b, j) → Y
  18. Polygon(A, B, C) → poly1, c_1, a_1, b_1
  19. Line(C, X) → k
  20. Line(A, Y) → l
  21. Line(B, Z) → m
  22. Intersect(k, l) → P
  23. Centroid(poly1) → G
  24. Mirror(C, G) → C'
  25. Mirror(A, G) → A'
  26. Mirror(B, G) → B'
  27. Conic(B', A, C', B, A') → p
  28. Line(F, P) → n
  29. Intersect(d, n) → D, S
  30. Center(d) → E_1
  31. Line(G, E_1) → q
  32. Mirror(q, b) → q'
  33. Mirror(q, c) → q'_1
  34. Intersect(q'_1, q') → E

AI Drawing Prompt Reference

Draw a triangle with its circumcircle. Pick a point on the circumcircle as the focus. Reflect this focus across each side of the triangle, obtaining three reflected points. Connect two of these reflected points to form a line, which will be the directrix of a parabola. Construct the parabola with the given focus and directrix, tangent to all three sides of the triangle. From each tangency point, draw a line to the opposite vertex; these three lines concur at a point. Then take the centroid of the triangle and reflect each vertex about it, creating three new points; use these to draw a conic. Connect the focus to the concurrence point, intersecting the circumcircle at two points. Finally, draw a line through the centroid and the circumcenter, reflect it across two sides, and the intersection of these two reflected lines is the desired point.
The prompt above was compiled from the original drawing commands and is for reference only.