In article <Pine.SUN.3.91.970730115937.13347Bfirstname.lastname@example.org>, Muad'dib <email@example.com> writes: > Just wondering: > > Suppose you are given a "random" triangle. > It does have a longest side. Call it n, and its enpoints a and b. > Call the third point of the triangle c. > c > / \ > / \ > /________ > a n b > > What is wrong with the assumption that the distances from a to c and the > distances from b to c are independent, and that the distance from a to c > (and likewise the distance from b to c) is uniformly distributed over (0,n]? > (ie the ratios of the two other sides to the longest side should be > uniformly distributed over (0,1])
Well, if you assume this, then there is a finite probability that diatance ac + distance bc < n, and then no triangle is possible.
You could modify your assumption to say that, if x,y are the ratios of the two shorter sides to the longest side, then the point (x,y) is uniformly distributed over the region
0 <= x <= 1, 0 <= y <= 1, x + y >= 1
of the x,y plane, and would then get an answer different from 3/4.
> These seem like viable, intuitive assumptions about a random triangle. > If not, then in what way should the lengths of the sides be dependent? > > The reason I ask is because these simple assumptions alone (not > assumptions involving random points in R^2 and the like) allow the > construction of a measure space of triangles that does not give 3/4 as > the probability of obtuseness. If 3/4 is the accepted answer, then this > implies that a "random" triangle follows nonuniform distributions in the > ratios of its sides. This doesn't make sense to me.
There are perhaps many ways of defining what is meant by a random triangle, and most of them are incompatible with most of the others. Each of them gives rise to a separate version of the question "what is the probability of obtuseness", which may have a different answer from the other versions. Many of the versions of this question might be interesting but they are separate from one another. I am not arguing that one of them is best.
I have forgotten the exact wording used by the person who started this thread, but I believe he fairly clearly implied that he was interested in the assumption that the vertices are uniformly and indepndently distributed over the whole plane. This brings in the extra problem that it is not a proper distribution. This is what makes all the disagreement possible. In what way, if any, can one preserve the spirit of this idea while translating it into a well-defined problem?
-- Robert Hill
University Computing Service, Leeds University, England
"Though all my wares be trash, the heart is true." - John Dowland, Fine Knacks for Ladies (1600)