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Topic: Random Triangle Problem
Replies: 57   Last Post: Aug 17, 1997 10:51 PM

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Kevin Brown

Posts: 360
Registered: 12/6/04
Re: Random Triangle Problem (LONG summary)
Posted: Aug 16, 1997 3:22 PM
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On Fri, 08 Aug 1997 T.Moore@massey.ac.nz (Terry Moore) wrote:
>Keith Ramsay has drawn our attention to an American
>Mathematical Monthly article which has discussed this
>question and looked at several distributions for the
>vertices.


In addition to that article, the same and similar questions
about "random triangles" have been discussed in print and in
this newsgroup several times before. For example, there's a
little Dover book called "The Surprise Attack in Mathematical
Problems" by L. A. Graham, consisting of a collection of
problems and solutions that appeared in Graham's newspaper
column over the years. One whole chapter is devoted to the
problem "What is the chance that the altitudes of a triangle
may themselves form another triangle?" In reviewing the range
of answers and arguments that were prompted by this question,
Graham dryly notes that "the choice of the optimum answer
introduces matters of philosophy, esthetics, symmetry, and
even psychiatry".

Coincidentally, I recently got an email from two college students
who had determined THE probability that a randomly selected chord
of a regular n-gon (n>3) is shorter than the side of the n-gon,
and they were wondering if perhaps they were due a Fields Medal.
Of course this is a variation of a familiar class of problems,
such as finding the probability that a "random chord" of a circle
is longer than the radius, and as with all such problems it clearly
depends on the assumed distribution. The two guys were quite
disappointed to learn that the definition of a "random chord" they
were using (which they had taken from a Cliff Pickover book) was
not universally regarded as THE unique definition of a random
chord, and that other equally plausible definitions give different
answers.

Anyway, the assumption they were using was that the endpoints of
the chord are uniformly distributed on the perimeter of the polygon.
They chose not to reveal their solution (thinking there might still
be big money in it), but I imagine they reasoned that there's a 1/n
probability of the two ends of the chord falling on the same edge,
and a 2/n probability of falling on adjacent edges, in which case
the probability of the chord being shorter than an edge length is
just the area in the first quadrant inside the ellipse

x^2 + 2xycos(2pi/n) + y^2 = 1

giving an overall probability of

1 / 2pi/n --- ( 1 + ----------- )
n \ sin(2pi/n) /







Date Subject Author
7/16/97
Read Random Triangle Problem
Mike Housky
7/21/97
Read Re: Random Triangle Problem
Bill Taylor
7/22/97
Read Re: Random Triangle Problem
tony richards
7/24/97
Read Re: Random Triangle Problem
Brian M. Scott
7/23/97
Read Re: Random Triangle Problem
tony richards
7/23/97
Read Re: Random Triangle Problem
T. Sheridan
7/24/97
Read Re: Random Triangle Problem
Bill Taylor
7/24/97
Read Re: Random Triangle Problem
Bill Taylor
7/25/97
Read Re: Random Triangle Problem
Ilias Kastanas
7/23/97
Read Re: Random Triangle Problem
Robert Hill
7/23/97
Read Re: Random Triangle Problem
tony richards
7/27/97
Read Re: Random Triangle Problem
Bill Taylor
7/24/97
Read Re: Random Triangle Problem
Robert Hill
7/28/97
Read Re: Random Triangle Problem
tony richards
7/30/97
Read Re: Random Triangle Problem
Bill Taylor
7/30/97
Read Re: Random Triangle Problem
tony richards
8/1/97
Read Re: Random Triangle Problem
Bill Taylor
7/24/97
Read Re: Random Triangle Problem
Robert Hill
7/24/97
Read Re: Random Triangle Problem
Robert Hill
7/24/97
Read Re: Random Triangle Problem
Robert Hill
7/25/97
Read Re: Random Triangle Problem
Robert Hill
7/30/97
Read Re: Random Triangle Problem
Bill Taylor
8/1/97
Read Re: Random Triangle Problem
Charles H. Giffen
8/1/97
Read Re: Random Triangle Problem
John Rickard
8/1/97
Read Re: Random Triangle Problem
Chris Thompson
8/1/97
Read Re: Random Triangle Problem
John Rickard
8/4/97
Read Re: Random Triangle Problem
Bill Taylor
8/5/97
Read Re: Random Triangle Problem
John Rickard
7/25/97
Read Re: Random Triangle Problem
Charles H. Giffen
7/25/97
Read Re: Random Triangle Problem
Charles H. Giffen
7/28/97
Read Re: Random Triangle Problem
Hauke Reddmann
7/28/97
Read Re: Random Triangle Problem
Robert Hill
7/28/97
Read Re: Random Triangle Problem
Robert Hill
7/28/97
Read Re: Random Triangle Problem
Robert Hill
7/29/97
Read Re: Random Triangle Problem
tony richards
7/30/97
Read Re: Random Triangle Problem
Keith Ramsay
7/30/97
Read Re: Random Triangle Problem
tony richards
8/2/97
Read Re: Random Triangle Problem
Keith Ramsay
7/29/97
Read Re: Random Triangle Problem
tony richards
8/4/97
Read Re: Random Triangle Problem
Bill Taylor
8/5/97
Read Re: Random Triangle Problem
Charles H. Giffen
8/6/97
Read Re: Random Triangle Problem
Terry Moore
8/7/97
Read Re: Random Triangle Problem (LONG summary)
Terry Moore
8/16/97
Read Re: Random Triangle Problem (LONG summary)
Kevin Brown
8/17/97
Read Re: Random Triangle Problem (LONG summary)
Kevin Brown
7/30/97
Read Re: Random Triangle Problem
Muad'dib
7/30/97
Read Re: Random Triangle Problem
Robert Hill
7/31/97
Read Re: Random Triangle Problem
tony richards
8/6/97
Read Re: Random Triangle Problem
Terry Moore
7/31/97
Read Re: Random Triangle Problem
John Rickard
7/30/97
Read Re: Random Triangle Problem
Robert Hill
7/31/97
Read Re: Random Triangle Problem
Robert Hill
7/31/97
Read Re: Random Triangle Problem
Robert Hill
8/1/97
Read Re: Random Triangle Problem
R J Morris
8/4/97
Read Re: Random Triangle Problem
Robert Hill
8/4/97
Read Re: Random Triangle Problem
Robert Hill
8/5/97
Read Re: Random Triangle Problem
Charles H. Giffen
8/6/97
Read Re: Random Triangle Problem
Robert Hill

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