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Inscribed Parabola

Inscribed Parabola
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2015-02-17 09:17
CircleGeometric Construction
In collections A Tour of Triangle Geometry
This example demonstrates that when a point F is on the circumcircle, its isogonal conjugate is at infinity, and the inscribed conic is a parabola. The construction uses GeoGebra commands such as Line, Circle, Point, Mirror, Parabola, OrthogonalLine, Intersect, and Polygon. Suitable for high school geometry education.

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'_3
  7. Mirror(F, a) → F'_1
  8. Mirror(F, b) → F'_2
  9. Line(F'_1, F'_2) → e
  10. Parabola(F, e) → f
  11. OrthogonalLine(F, e) → g
  12. OrthogonalLine(F'_3, e) → h
  13. OrthogonalLine(F'_1, e) → i
  14. OrthogonalLine(F'_2, e) → j
  15. Intersect(f, c, 1) → E
  16. Intersect(f, a, 1) → G
  17. Intersect(b, j) → D
  18. Polygon(A, B, C) → poly1, c_1, a_1, b_1

AI Drawing Prompt Reference

Draw a triangle and take a point on its circumcircle. Create three mirror lines through this point, intersecting at another point. Use this point and one mirror line as the focus and directrix to draw a parabola. Then, draw three orthogonal lines intersecting the mirror lines and the parabola. Finally, complete the triangle.
The prompt above was compiled from the original drawing commands and is for reference only.