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Smallest Ellipse Circumscribed in a Quadrilateral

Smallest Ellipse Circumscribed in a Quadrilateral
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2016-04-30 05:36
PolygonOptimization
This example demonstrates the construction of the smallest ellipse circumscribed within a quadrilateral. Key math concepts include ellipse, similarity transformation, quadratic equations, and vector operations. The construction utilizes GeoGebra commands such as midpoint, line, polygon, circle, intersection, segment, vector, angle, inversion, matrix application, and area calculation. Suitable for high school mathematics students and geometry enthusiasts to explore geometric relationships and optimization problems involving quadrilaterals and ellipses.

GeoGebra Commands & Parameters

  1. Midpoint(D, A) → E
  2. Midpoint(C, B) → F
  3. Line(F, E) → k
  4. Polygon(A, B, C, D) → poly1, a, b, c, d
  5. Circle(E, d) → q
  6. Intersect(q, k, 2) → H
  7. Segment(E, H) → r
  8. Vector(A, B) → u
  9. Angle(u) → α
  10. Vector(A, D) → v
  11. Angle(v) → β
  12. Invert(M) → N
  13. ApplyMatrix(N, C) → C'
  14. ApplyMatrix(N, A) → A'
  15. ApplyMatrix(N, B) → B'
  16. ApplyMatrix(N, D) → D'
  17. Polygon(A', B', C', D') → poly1', a', b', c', d'
  18. ApplyMatrix(M, h) → minima
  19. Area(h) → j
  20. PointIn(h) → Pointh
  21. Area(minima) → i
  22. PointIn(minima) → Pointh'
  23. Point(r) → G
  24. Conic(C, D, G, A, B) → p
  25. Area(p) → l
  26. PointIn(p) → Pointp