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Animation 111: Parabola, Secants, and Orthocenter in Dynamic Geometry

Animation 111: Parabola, Secants, and Orthocenter in Dynamic Geometry
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2016-09-08 09:20
TransformationGeometric ConstructionSlider AnimationLocus
This GeoGebra example dynamically illustrates the interplay between a parabola, secant lines, the orthocenter of a triangle, and related geometric elements. The core mathematical concepts include properties of parabolas, secant intersections, triangle orthocenters, congruence and similarity, as well as transformations such as rotation and dilation. The construction employs commands like PointIn, OrthogonalLine, Segment, Intersect, Dilate, Plane, IntersectPath, Point, Angle, and Rotate, involving multiple points, lines, planes, and an interactive button. It is suitable for high school or early college students studying conic sections and triangle centers, and serves as a rich resource for teachers creating dynamic geometry lessons. The animation uses sliders and buttons to show how geometric quantities change with parameters, enhancing spatial reasoning and understanding of dynamic geometric relationships.

GeoGebra Commands & Parameters

  1. PointIn(b) → P
  2. OrthogonalLine(P, c) → f
  3. Segment(F, P) → g
  4. Intersect(f, c) → C
  5. Segment(P, C) → h
  6. Dilate(F, If[0 < d ≤ 1, 1 - d, 0], P) → B'
  7. Segment(F, B') → j
  8. Dilate(P, If[1 < d ≤ 2, 2 - d, 0], C) → A'
  9. Segment(P, A') → k
  10. Plane(P, C, F) → l
  11. IntersectPath(l, c) → m
  12. Point(m) → D
  13. Angle(P, C, D) → α
  14. Angle(F, P, C) → β
  15. Rotate(F, If[2 < d ≤ 3, ((-(d - 2)) * β), (-β)], P, l) → B'_1
  16. Rotate(B', If[2 < d ≤ 3, ((-(d - 2)) * β), (-β)], P, l) → B''
  17. Segment(B'_1, B'') → j'
  18. IntersectPath(l, b) → p

AI Drawing Prompt Reference

Draw a dynamic geometry scene with a parabola as the base curve. A movable point lies on the parabola, and a perpendicular line from this point meets the parabola at another point. Connect the fixed focus to the movable point with a segment. Use scaling to create two auxiliary points and draw segments from the focus and the movable point to them. Define a plane through the movable point, the perpendicular foot, and the focus. This plane intersects the parabola in a curve, on which place a point. Mark two angles involving the focus, the movable point, and the intersection point. Then, depending on a parameter, rotate the focus and one auxiliary point around the movable point within the plane, producing a new segment. The final figure includes the parabola, multiple segments, a plane, an intersection curve, angle labels, and dynamic rotation.
The prompt above was compiled from the original drawing commands and is for reference only.