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Intersection of Two Parabolas by Tom Button

Intersection of Two Parabolas by Tom Button
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2014-12-24 19:35
VectorFunctionCoordinate GeometryGeometric Construction
This example demonstrates the dynamic behavior of the intersection points of two parabolas as their foci and directrices are moved using points A, B, C, and D. The position of the circle's center relative to these elements is also shown. Key math concepts include parabola definition, coordinate geometry, intersection calculations, and vector operations. Constructed using GeoGebra commands: OrthogonalLine, Parabola, Intersect, Circle, LineBisector, Vector, Translate. Suitable for high school and above students to explore parabolic properties and dynamic geometric relationships, as well as for calculus and geometric construction education.

GeoGebra Commands & Parameters

  1. OrthogonalLine(B, yAxis) → a
  2. OrthogonalLine(D, xAxis) → b
  3. Parabola(C, b) → c
  4. Parabola(A, a) → d
  5. Intersect(c, d) → E, F, G, H
  6. Circle(E, F, G) → e
  7. LineBisector(F, G) → f
  8. LineBisector(H, G) → g
  9. Intersect(f, g) → I
  10. OrthogonalLine(A, a) → h
  11. OrthogonalLine(C, b) → i
  12. Intersect(b, i) → J
  13. Intersect(a, h) → K
  14. OrthogonalLine(C, a) → j
  15. OrthogonalLine(A, b) → k
  16. Intersect(a, j) → L
  17. Intersect(b, k) → M
  18. Intersect(a, b) → N
  19. Vector(K, A) → u
  20. Vector(M, A) → v
  21. Vector(L, C) → w
  22. Vector(J, C) → l
  23. OrthogonalLine(I, b) → m
  24. OrthogonalLine(I, a) → n
  25. Intersect(b, m) → O
  26. Intersect(a, n) → P
  27. Translate(P, u) → P'
  28. Vector(P, P') → p
  29. Translate(P', w) → P''
  30. Vector(P', P'') → q
  31. Translate(O, l) → O'
  32. Vector(O, O') → r
  33. Translate(O', v) → O''
  34. Vector(O', O'') → s

AI Drawing Prompt Reference

Draw two parabolas defined by their foci and directrices, showing their intersection points. Encircle one of the intersection points with a circle. Find the angle bisectors of the lines formed by the directrices and their intersection. Extend lines from the intersection along the parabolas, highlighting their intersections and vector relationships. Finally, construct auxiliary points and vectors through translation and vector operations to demonstrate dynamic geometric relationships.
The prompt above was compiled from the original drawing commands and is for reference only.