From:           Aleksandrs Mihailovs <mihailov@math.upenn.edu>
Newsgroups:     sci.math.research
Subject:        Isosceles triangles
Date:           Mon, 06 May 1996 17:40:52 -0700
Organization:   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.research
Subject:        Re: Isosceles triangles
Date:           6 May 1996 18:24:46 -0700
Organization:   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 -0700
From:           "Daniel A. Asimov" <asimov@nas.nasa.gov>
To:             eppstein@ICS.UCI.EDU
Subject:        Re: Isosceles triangles
Newsgroups:     sci.math.research
Organization:   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.gov
Subject:        Isosceles triangles
Date:           Tue, 07 May 1996 16:48:26 -0700
From:           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 -0700
From:           "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