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 Four Point Polygons (Posted on 2011-12-06)
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 No Rating

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(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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