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

 Messages: [ Previous | Next ]
 Kevin Brown Posts: 360 Registered: 12/6/04
Re: Random Triangle Problem (LONG summary)
Posted: Aug 16, 1997 3:22 PM

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

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 Mike Housky
7/21/97 Bill Taylor
7/22/97 tony richards
7/24/97 Brian M. Scott
7/23/97 tony richards
7/23/97 T. Sheridan
7/24/97 Bill Taylor
7/24/97 Bill Taylor
7/25/97 Ilias Kastanas
7/23/97 Robert Hill
7/23/97 tony richards
7/27/97 Bill Taylor
7/24/97 Robert Hill
7/28/97 tony richards
7/30/97 Bill Taylor
7/30/97 tony richards
8/1/97 Bill Taylor
7/24/97 Robert Hill
7/24/97 Robert Hill
7/24/97 Robert Hill
7/25/97 Robert Hill
7/30/97 Bill Taylor
8/1/97 Charles H. Giffen
8/1/97 John Rickard
8/1/97 Chris Thompson
8/1/97 John Rickard
8/4/97 Bill Taylor
8/5/97 John Rickard
7/25/97 Charles H. Giffen
7/25/97 Charles H. Giffen
7/28/97 Hauke Reddmann
7/28/97 Robert Hill
7/28/97 Robert Hill
7/28/97 Robert Hill
7/29/97 tony richards
7/30/97 Keith Ramsay
7/30/97 tony richards
8/2/97 Keith Ramsay
7/29/97 tony richards
8/4/97 Bill Taylor
8/5/97 Charles H. Giffen
8/6/97 Terry Moore
8/7/97 Terry Moore
8/16/97 Kevin Brown
8/17/97 Kevin Brown
7/30/97 Robert Hill
7/31/97 tony richards
8/6/97 Terry Moore
7/31/97 John Rickard
7/30/97 Robert Hill
7/31/97 Robert Hill
7/31/97 Robert Hill
8/1/97 R J Morris
8/4/97 Robert Hill
8/4/97 Robert Hill
8/5/97 Charles H. Giffen
8/6/97 Robert Hill