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Bertrand Paradox - Third Method Demo

Bertrand Paradox - Third Method Demo
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2017-01-23 03:24
TrigonometryCoordinate GeometrySolid GeometryProbability & Statistics
In collections probability
This example dynamically demonstrates the third method of Bertrand's Paradox, showing the distribution differences of the ratio of randomly selected line segments to the diameter under various geometric conditions. It involves core concepts in Probability & Statistics, Coordinate Geometry, Geometric Construction, Rotation, and Intersection Calculations. The construction utilizes GeoGebra commands and objects like Circle, Point, Tangent, Rotate, Angle, Line, Intersect, Polygon, Segment, OrthogonalLine, PointIn, highlighting ratio distribution analysis. Suitable for high school and above learners to understand the connection between probability paradoxes and geometric constructions.

GeoGebra Commands & Parameters

  1. Circle(O, 3) → c
  2. Point(c) → A
  3. Tangent(A, c) → tangentLine,
  4. Point(tangentLine) → B_1
  5. Rotate(B_1, α, A) → B'
  6. Angle(B_1, A, B') → β
  7. Rotate(B_1, 60°, A) → B'_1
  8. Angle(B_1, A, B'_1) → γ
  9. Line(A, B'_1) → e
  10. Intersect(c, e, 2) → D
  11. Polygon(A, D, 3) → poly1, f, g, h, E
  12. Line(A, B') → line_{AB}
  13. Intersect(c, line_{AB}, 2) → B
  14. Segment(A, B) → seg_{AB}
  15. Line(A, O) → diamLine
  16. Intersect(c, diamLine, 2) → C
  17. Segment(A, C) → diam
  18. Circle(A, (m * diam)) → MConic
  19. Intersect(MConic, diam, 1) → M_{PQ}
  20. OrthogonalLine(M_{PQ}, diam) → perpToDiamLine
  21. Intersect(c, perpToDiamLine, 2) → Q
  22. Intersect(c, perpToDiamLine, 1) → P
  23. Segment(P, Q) → seg_{PQ}
  24. Rotate(A, 30°, C) → A'
  25. Angle(A, C, A') → δ
  26. Line(C, A') → i
  27. Intersect(c, i, 2) → F
  28. Polygon(F, C, 3) → poly2, j, k, l, G
  29. Circle(O, (0.25 * diam)) → d
  30. PointIn(c) → M_{ST}
  31. Segment(O, M_{ST}) → seg_{OMST}
  32. OrthogonalLine(M_{ST}, seg_{OMST}) → lineST
  33. Intersect(c, lineST, 1) → S
  34. Intersect(c, lineST, 2) → T
  35. Segment(S, T) → seg_{ST}

AI Drawing Prompt Reference

Draw a circle and select a random point on it to draw a tangent line. Rotate the intersection of the tangent line with the circle to form two triangles. Connect the rotated point with the center of the circle to intersect the circle at two points, forming a line segment. Draw a circle with this line segment as the diameter. Intersect this circle with a line connecting the center to one vertex of the triangle formed earlier. Draw a perpendicular line from the intersection point to the diameter, intersecting the circle at two points, forming a line segment. Rotate the line connecting the center to another vertex of the triangle, intersecting the circle at a point. Connect the center to this point, intersecting the circle at another point, forming a triangle. Draw another circle with the same diameter, select a random point on it, connect the center to this point, and draw a perpendicular line from the point to the connecting line, intersecting the circle at two points, forming a line segment.
The prompt above was compiled from the original drawing commands and is for reference only.