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Similarity with Billiard: Dynamic Geometry Exploration

Similarity with Billiard: Dynamic Geometry Exploration
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2020-12-07 00:06
TriangleAngleProbability & StatisticsTransformationGeometric Construction
① This example simulates billiard ball reflections on a triangular table, demonstrating the connection between similarity transformations and special points such as the orthocenter. ② Core mathematical concepts include similarity transformation, orthocenter, special points, polygons, random experiments, angles, vectors, and reflections. ③ Construction uses GeoGebra commands: Segment, Rotate, Point, Line, Intersect, Mirror, Ray, Angle, and Vector, creating the triangle, reflection paths, orthocenter, and auxiliary lines, with random variables for stochastic simulation. ④ Suitable for high school or college geometry courses, ideal for exploring similarity in dynamic contexts and for inquiry-based learning or mathematical experimentation.

GeoGebra Commands & Parameters

  1. Segment(C_{1}, D) → f
  2. Rotate(D, 33°, C_{1}) → D'
  3. Segment(D, C) → g
  4. Point(f) → G
  5. Point(f) → H
  6. Segment(C_{1}, R) → j
  7. Line(C_{1}, R) → k
  8. Point(Circle[C, 4]) → F
  9. Line(F, C) → h
  10. Intersect(h, f) → W
  11. Segment(F, W) → i
  12. Point(i) → J
  13. Point(i) → K
  14. Segment(F, C) → l
  15. Mirror(C, eq1) → C'
  16. Ray(W, C') → m
  17. Intersect(m, eq2) → L
  18. Segment(W, L) → n
  19. Segment(D, W) → p
  20. Segment(W, C_{1}) → q
  21. Angle(F, W, D) → α
  22. Angle(C_{1}, W, L) → β
  23. Segment(L, R) → r
  24. Angle(W, D, C) → γ
  25. Angle(I, C_{1}, W) → δ
  26. Vector(C, W) → u

AI Drawing Prompt Reference

Construct a triangular table using three points and draw a dynamic billiard-style reflection path. Start by taking a segment on a reference side, rotate it around a triangle vertex by a fixed angle to get a new point, then draw connecting lines and form their intersection points. Use symmetry/mirroring about a chosen axis to define reflected points, and from one intersection point draw a ray that intersects another line to determine reflection endpoints. Include a circle centered at one triangle point with a given radius, place a point on the circle, and connect it to another triangle vertex to form key segments. Add several angle measures at the main intersection points and optionally draw a vector to represent the direction relationship related to similarity.
The prompt above was compiled from the original drawing commands and is for reference only.