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Parabola and Dandelin Sphere

Parabola and Dandelin Sphere
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2021-11-20 22:39
Coordinate GeometrySolid GeometryGeometric Construction
This example demonstrates the formation of a parabola and its relationship with Dandelin spheres. Key mathematical concepts include conic sections, parabola, sphere, isosceles triangles, intersection, continuity. Construction highlights the use of GeoGebra commands such as Intersect, Line, Mirror, Point, AngularBisector, OrthogonalPlane, OrthogonalLine, Circle, Sphere, Cone, Plane, IntersectPath, and Corner. Suitable for high school students and math enthusiasts studying the geometric properties of conic sections.

GeoGebra Commands & Parameters

  1. Intersect(yAxis, zAxis) → A
  2. Line(A, B) → f
  3. Mirror(f, yAxis) → f'
  4. Point(f') → C
  5. Line(C, f) → h
  6. AngularBisector(f', h) → i, j
  7. Intersect(yAxis, j) → D
  8. OrthogonalPlane(D, f') → p
  9. Intersect(f', p) → E
  10. OrthogonalLine(D, xOyPlane) → g
  11. Circle(g, E) → c
  12. OrthogonalLine(C, xOyPlane) → k
  13. Sphere(D, E) → a
  14. Intersect(h, c) → F, G
  15. Circle(yAxis, B) → d
  16. Cone(d, (-y(B))) → b, e, l
  17. Plane(h, k) → q
  18. Circle(yAxis, E) → r
  19. OrthogonalPlane(E, yAxis) → p_1
  20. IntersectPath(p_1, q) → m
  21. IntersectPath(q, b) → s
  22. Corner(s) → H, I
  23. Cone(Rotate[d, π, xAxis], (-y(B))) → n, t, o

AI Drawing Prompt Reference

Create a 3D geometric construction showing how a parabola arises from a cone and how it relates to Dandelin spheres. Start from two perpendicular coordinate axes to set key reference points, then draw a line and construct its mirror across one axis to obtain the corresponding symmetric boundary. Use intersections with the mirrored line to define further points and build an angle-bisector direction to locate an additional point. Construct a plane through that direction and obtain a circle by drawing the perpendicular circle in the appropriate plane that passes through the needed intersection point. Using that point, display a sphere and a cone that share the same geometric focal/tangency relationship, along with auxiliary perpendicular lines to the horizontal plane and cone side surfaces; the final scene should clearly combine the cone, the parabola-forming setup, and the Dandelin sphere arrangement.
The prompt above was compiled from the original drawing commands and is for reference only.