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Two Ellipses

Two Ellipses
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2020-12-18 16:49
Coordinate GeometryGeometric Construction
This example demonstrates the geometric relationships and transformations between two ellipses. Key math concepts include ellipses, straight lines, similarity transformations, inequalities, and diagram construction. It utilizes GeoGebra commands/objects like Line, Point, Mirror, Ellipse, Midpoint, Intersect, Distance, OrthogonalLine, Tangent, Segment, Polygon, Angle, and CurveCartesian as outlined in the construction summary. Suitable for high school mathematics education to understand ellipse properties and transformation applications.

GeoGebra Commands & Parameters

  1. Line(F', xAxis) → p
  2. Point(p) → F
  3. Mirror(F', F) → A'_{1}
  4. Ellipse(F', F, C) → d
  5. Midpoint(F', F) → E
  6. Line(F', F) → l
  7. Intersect(d, l) → H, I
  8. Distance(E, I) → R
  9. OrthogonalLine(E, l) → m
  10. Intersect(d, m) → J, K
  11. Distance(E, K) → r
  12. Tangent(D, d) → h, n
  13. Mirror(F', h) → A'
  14. Segment(F', F) → f
  15. Ellipse(F', F, C) → c
  16. Line(F', F) → g
  17. Polygon(F, D, F') → t1, a_1, b, d_1
  18. Mirror(F, h) → B'
  19. Mirror(D, h) → D'
  20. Polygon(B', D', A') → t1', a', b', d'
  21. Ellipse(B', A', D) → e
  22. Point(e) → F_{1}
  23. Angle(F, F', B') → α
  24. Angle(A'_{1}, F, A') → β
  25. CurveCartesian(x(F') + (Distance[F, A'] * cos(i)), y(F') + (Distance[F, A'] * sin(i)), i, 0, α) → a
  26. CurveCartesian(x(F) + (Distance[F, A'] * cos(i)), y(F) + (Distance[F, A'] * sin(i)), i, 0, β) → i
  27. Midpoint(F', A) → B
  28. CurveCartesian(x(F') + (Distance[F, B] * cos(i)), y(F') + (Distance[F, B] * sin(i)), i, 0, α) → k

AI Drawing Prompt Reference

Draw two ellipses, one with foci at points F' and F, and the other with foci at the mirrored point A'1 of F'. Construct a perpendicular line to line F'F through point E, intersecting ellipse d at points J and K, and calculate the distance r from E to K. Construct the tangent line h at point F', mirror point F' to get point A', and connect F' and F to form segment f. Construct an ellipse c with foci at F' and F. Construct a perpendicular line to line F'F through point F, intersecting ellipse d at points B' and D', and form a polygon t1' with vertices B', D', and A'. Construct an ellipse e with foci at B', A', and D, find point F1 on ellipse e, calculate angles α and β, and draw curves a and i, and finally locate point B.
The prompt above was compiled from the original drawing commands and is for reference only.