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Another Reflective Property of a Parabola

Another Reflective Property of a Parabola
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2016-04-08 21:32
Coordinate GeometryGeometric Construction
In collections Conic Sections Conic Sections- Maths Made Easy
This applet illustrates another reflective property of a parabola. Key mathematical concepts include the geometric properties of parabolas. Using GeoGebra commands such as Line, Parabola, Tangent, Intersect, etc., the construction of the parabola and related geometric elements is demonstrated. Suitable for high school mathematics education and learning.

GeoGebra Commands & Parameters

  1. Line(A, B) → a
  2. Slope(a) → b
  3. PointIn(d) → C
  4. Parabola(C, a) → e
  5. OrthogonalLine(C, a) → f
  6. Point(e) → D
  7. Tangent(D, e) → g,
  8. OrthogonalLine(C, f) → h
  9. Intersect(e, h, 1) → E
  10. Intersect(e, h, 2) → F
  11. Line(D, f) → i
  12. Circle(D, 1) → k
  13. Intersect(k, i, 2) → G
  14. Ray(D, G) → j
  15. OrthogonalLine(D, f) → l
  16. Intersect(f, l) → H
  17. Distance(C, D) → m
  18. Circle(D, (2 * m)) → p
  19. Intersect(p, j, 1) → I
  20. Dilate(C, If[0 < n ≤ 1, 1 - n, 0], D) → C'
  21. Vector(C, C') → u
  22. Dilate(D, If[1 < n ≤ 2, 2 - n, 0], I) → D'
  23. Vector(D, D') → v
  24. Distance(C, D) → o
  25. Circle(D, (0.3 * o)) → q
  26. Intersect(q, g, 2) → K
  27. Segment(C, D) → r
  28. Intersect(q, r, 1) → J
  29. CircleSector(D, J, K) → s
  30. Angle(J, D, K) → α
  31. Rotate(s, If[2 < s ≤ 3, ((-(n - 2)) * (180° - α)), ((-(n - 2)) * (180° - α))], D) → s'
  32. CircleArc(D, J, K) → t
  33. Circle(D, J) → c_1
  34. Intersect(c_1, g, 1) → L
  35. CircleSector(D, L, J) → d_1
  36. Rotate(d_1, If[2 < s ≤ 3, ((n - 2) * α), ((n - 2) * α)], D) → e_1
  37. CircleArc(D, L, J) → f_1
  38. Circle(D, C) → g_1
  39. Intersect(g_1, g, 1) → M
  40. Intersect(g_1, g) → N, O
  41. Vector(D, N) → w
  42. Vector(D, O) → u_1
  43. Intersect(a, f) → P
  44. Angle(B, P, C) → β

AI Drawing Prompt Reference

Draw a parabola with a focus on a given line, which serves as the directrix. Pick a point on the parabola, draw its tangent, and construct a perpendicular from the focus to the directrix. Using the point as center, create a circle intersecting the tangent and the line from focus to point, forming a sector. Then, with the focus as center, draw another circle intersecting the tangent to form another sector. Rotate these sectors to illustrate a reflective property of the parabola. Also, connect the focus to two points on the parabola and label the relevant angle.
The prompt above was compiled from the original drawing commands and is for reference only.