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Bertrand Paradox - First Method

Bertrand Paradox - First Method
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2017-01-23 03:22
CircleProbability & StatisticsGeometric Construction
In collections probability
This example illustrates the first solution to Bertrand's paradox. It involves probability and geometric construction. Using GeoGebra commands such as Circle, Point, Tangent, Rotate, etc., a circle and a random point within it are constructed to explore the results under different probability models. Suitable for high school mathematics education and students learning probability and statistics.

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. Segment(M_{ST}, O) → seg_{OMST}
  31. OrthogonalLine(M_{ST}, seg_{OMST}) → line_{ST}
  32. Intersect(c, line_{ST}, 1) → S
  33. Intersect(c, line_{ST}, 2) → T
  34. Segment(S, T) → seg_{ST}
  35. Point(tangentLine) → H
  36. Point(tangentLine) → I
  37. Point(tangentLine) → J
  38. Point(tangentLine) → K
  39. Segment(I, K) → n
  40. Segment(K, A) → p
  41. Segment(A, H) → q
  42. Segment(H, J) → r

AI Drawing Prompt Reference

Draw a circle with radius 3 centered at O. Randomly select a point A inside the circle and draw a tangent line at A. Choose two points B1 and B' on the tangent line. Rotate B' to get B'1 and calculate the angles β and γ between B1, A, and B'1. Draw a line e through A and B'1, intersecting the circle at D to form a triangle. Then, draw a line through A and B, intersecting the circle at B. Draw the diameter AC, intersecting the circle at C. Construct a circle M with center A and diameter AC, intersecting the diameter at M_{PQ}. Draw a perpendicular line to M_{PQ} and intersect the circle at P and Q, then connect P and Q. Rotate A to get A', calculate the angle δ between A, C, and A', and draw a line i through C and A', intersecting the circle at F to form another triangle. Construct a circle d with center O and radius half the diameter, intersecting the diameter at M_{ST}. Draw a perpendicular line to M_{ST} and intersect the circle at S and T, then connect S and T. Finally, select four points I, J, K, and H on the tangent line and connect them.
The prompt above was compiled from the original drawing commands and is for reference only.