Three 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) an acute triangle; (b) a right triangle?

(In reply to computer findings for triangles through 50-gons by Charlie)

It looks like the probability of a right triangle is zero for odd n and 3/(n-1) for even n.

In reduced proper fractions, all the numbers given before are:

 4  0  1
 5  2//3  0
 6  1//5  3//5
 7  3//5  0
 8  2//7  3//7
 9  4//7  0
 10  1//3  1//3
 11  5//9  0
 12  4//11  3//11
 13  6//11  0
 14  5//13  3//13
 15  7//13  0
 16  2//5  1//5
 17  8//15  0
 18  7//17  3//17
 19  9//17  0
 20  8//19  3//19
 21  10//19  0
 22  3//7  1//7
 23  11//21  0
 24  10//23  3//23
 25  12//23  0
 26  11//25  3//25
 27  13//25  0
 28  4//9  1//9
 29  14//27  0
 30  13//29  3//29
 31  15//29  0
 32  14//31  3//31
 33  16//31  0
 34  5//11  1//11
 35  17//33  0
 36  16//35  3//35
 37  18//35  0
 38  17//37  3//37
 39  19//37  0
 40  6//13  1//13
 41  20//39  0
 42  19//41  3//41
 43  21//41  0
 44  20//43  3//43
 45  22//43  0
 46  7//15  1//15
 47  23//45  0
 48  22//47  3//47
 49  24//47  0
 50  23//49  3//49

The double slashes are an artifact of UBASIC's processing.

Edited on August 14, 2011, 5:10 pm

Posted by Charlie on 2011-08-14 17:08:52