From:Aleksandrs Mihailovs <mihailov@math.upenn.edu>Newsgroups:sci.math.researchSubject:Isosceles trianglesDate:Mon, 06 May 1996 17:40:52 -0700Organization:University of Pennsylvania

Does anybody know how to show that for any set of n points in the plane, the number of triples that determine an isosceles triangle is O(n^(7/3))? I would appreciate any suggestions and references. Thank you. Have a nice day!! Alec

From:David Eppstein <eppstein@ICS.UCI.EDU>Newsgroups:sci.math.researchSubject:Re: Isosceles trianglesDate:6 May 1996 18:24:46 -0700Organization:UC Irvine Department of ICS

Aleksandrs Mihailovs <mihailov@math.upenn.edu> writes: > Does anybody know how to show that for any set of n points in the plane, > the number of triples that determine an isosceles triangle is O(n^(7/3))? Here's one reference: J. Pach and P. K. Agarwal, "Combinatorial Geometry", Wiley, 1995, theorem 12.2, page 184. The proof is omitted but should be obvious from the surrounding context. -- David Eppstein UC Irvine Dept. of Information & Computer Science eppstein@ics.uci.edu http://www.ics.uci.edu/~eppstein/

Date:Tue, 7 May 1996 15:51:19 -0700From:"Daniel A. Asimov" <asimov@nas.nasa.gov>To:eppstein@ICS.UCI.EDUSubject:Re: Isosceles trianglesNewsgroups:sci.math.researchOrganization:NAS - NASA Ames Research Center, Moffett Field, CA

In article <4mm8ou$qd8@wormwood.ics.uci.edu> you write: >Aleksandrs Mihailovs <mihailov@math.upenn.edu> writes: >> Does anybody know how to show that for any set of n points in the plane, >> the number of triples that determine an isosceles triangle is O(n^(7/3))? > >Here's one reference: [...] ------------------------------------------------------------------------- Can you please try to clarify my perplexity over this question? Obviously, most sets of n points in the plane will have no repeated distances. None. So: Whence the isosceles triangles? (I am sure this question must have a very obvious answer, but I don't see it.) Thanks, Dan Asimov

To:asimov@nas.nasa.govSubject:Isosceles trianglesDate:Tue, 07 May 1996 16:48:26 -0700From:David Eppstein <eppstein@ICS.UCI.EDU>

Obviously, most sets of n points in the plane will have no repeated distances. None. So: Whence the isosceles triangles? Obviously, the problem asks about worst case bounds that are true for all arrangements of points, rather than just a measure-1 subset of the arrangements. For instance, if you happen to choose points that form a sqrt(n)*sqrt(n) integer grid, there will be many isosceles triangles. (More than n^2 of them, but fewer than n^(7/3).) -- David Eppstein UC Irvine Dept. of Information & Computer Science eppstein@ics.uci.edu http://www.ics.uci.edu/~eppstein/

Date:Wed, 8 May 1996 09:37:28 -0700From:"Daniel A. Asimov" <asimov@nas.nasa.gov>To:David Eppstein <eppstein@ICS.UCI.EDU>Subject:Re: Isosceles triangles

Thanks -- I guess your "obviously" is not necessarily the same as my "obviously"... --Dan