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Ellipse and the Circle Through Vertices of Its Inscribed Rectangles

Ellipse and the Circle Through Vertices of Its Inscribed Rectangles
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2017-08-23 01:39
CirclePolygonCoordinate GeometryGeometric Construction
This example demonstrates the geometric relationship between an ellipse and the circle that passes through the vertices of rectangles inscribed in the ellipse. Core mathematical concepts include the standard equation of an ellipse, properties of inscribed rectangles, circle definition, and point symmetry about the center. Construction highlights: using GeoGebra commands such as Point, Mirror, Midpoint to construct the ellipse and its foci/vertices, Ellipse, FirstAxisLength, SecondAxisLength to define the ellipse parameters, Circle, Tangent, Intersect to build related circles and tangents, and Polygon, Angle, PathParameter to analyze the relationship between rectangle vertices and the circle. Suitable for high school or college analytic geometry courses, this example helps visualize the circular locus of vertices of inscribed rectangles and explore how parameter changes affect the configuration.

GeoGebra Commands & Parameters

  1. Point(xAxis) → A
  2. Mirror(A, J) → B_1
  3. Midpoint(A, B_1) → D
  4. Point(yAxis) → B
  5. Ellipse(A, B_1, B) → c
  6. SecondAxisLength(c) → b
  7. FirstAxisLength(c) → a
  8. Circle(D, f) → g
  9. Point(g, Mod[e, 1]) → E
  10. Tangent(E, c) → h, i
  11. Intersect(g, h, 2) → F
  12. Intersect(g, i, 2) → G
  13. Mirror(E, D) → E'
  14. Polygon(F, E, G, E') → poly1, f_1, e_1, g_1, e'
  15. Circle(D, E) → k
  16. Angle(G, E', F) → α
  17. Angle(E', F, E) → β
  18. Angle(F, E, G) → γ
  19. Angle(E, G, E') → δ
  20. Circle(C, K) → p
  21. Point(p) → L
  22. PathParameter(L) → j