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Four Point Polygons (Posted on 2011-12-06) Difficulty: 3 of 5
Four points have been chosen randomly from the vertices of a n-sided regular polygon.

Determine the probability (in terms of n) that they form (a) a cyclic quadrilateral; (b) a rhombus.

No Solution Yet Submitted by K Sengupta    
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Solution solution Comment 2 of 2 |

(a) The vertices of a regular polygon all lie on one circle, and so the probability is 1 that the vertices form a cyclic quadrilateral.

(b)To be a rhombus, all the sides must have the same length. There are two cases:

n is not divisible by 4: probability zero.

n is divisible by 4: After the first point is chosen, only one of the C(n-1,3) combinations of the remaining three points satisfies the condition (which is that of a square), so the probability is 1/C(n-1,3).  The first few of these are:

  n    prob of rhombus (square)
  4     1/1
  8     1/35
 12     1/165
 16     1/455
 20     1/969
 24     1/1771

  Posted by Charlie on 2011-12-06 12:36:00
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