From:           israel@math.ubc.ca (Robert Israel)
Newsgroups:     sci.math
Subject:        Re: Geometry problem (points on a plane)
Date:           14 Jan 1996 09:03:51 GMT
Organization:   Mathematics, University of British Columbia, Vancouver, Canada

In article <4d59ie\$lr@frigate.doc.ic.ac.uk>,
Yaniv Shpilberg  <yes@doc.ic.ac.uk> wrote:

>The problem:
>Given: infinitely many points on a plane such that the distance between any
>two points is a whole number.
>To prove or disprove: all those points ore on the same straight line.
>

Proof:
Suppose you have three points A, B and C with integer distances between
them and not all on the same line.  If d(A,Q) (=distance from A to Q)
and d(B,Q) are both integers, note that d(A,Q) - d(B,Q) is one of the
integers from -d(A,B) to d(A,B).  Now for any given k, the points Q
with d(A,Q) - d(B,Q) = k lie on a branch of a hyperbola (or its degenerate
cases, a straight line parallel or perpendicular to AB).  Every point of
your set is an intersection of one of these curves, and one of the
analogous curves for A and C, and one of the curves for B and C.  But
any two of the curves intersect in only a finite number of points.
Therefore there are only a finite number of points with integer distances
from A, B and C.

--
Robert Israel                            israel@math.ubc.ca
Department of Mathematics             (604) 822-3629
University of British Columbia            fax 822-6074
Vancouver, BC, Canada V6T 1Y4

From:           israel@math.ubc.ca (Robert Israel)
Newsgroups:     sci.math
Subject:        Re: Geometry problem (points on a plane)
Date:           14 Jan 1996 08:43:27 GMT
Organization:   Mathematics, University of British Columbia, Vancouver, Canada

In article <4d62ik\$7oo@geraldo.cc.utexas.edu>,
Miguel Lerma <mlerma@arthur.ma.utexas.edu> wrote:

>Given a set S of points in a plane, we define property P(S) = "the distance
>between any two points in S is a whole number, and there are no three
>distinct points of S in the same straight line".

> A related problem is the
>following: a set S of points verifying property P will be called
>"extendable" if there is some point p not in S such that S union {p}
>verifies property P, otherwise it will be called "non extendable".
>By using Zorn's Lemma it is possible to prove that there is some
>(perhaps infinite) non extendable set S in the plane, so it makes
>sense to ask: which is the minimum cardinal of a non extendable set?

That's easy: 3.  Namely, the vertices of an equilateral triangle of
side 1 form a non-extendable set.

Proof: if A and B are two of the vertices, and Q any point with integer
distances from both but not collinear with them, then the absolute
value of the difference between the distances AQ and BQ is less than 1,
so must be 0.  The only point equidistant from all three vertices is
the centroid, but this is not an integer distance from them.

--
Robert Israel                            israel@math.ubc.ca
Department of Mathematics             (604) 822-3629
University of British Columbia            fax 822-6074
Vancouver, BC, Canada V6T 1Y4

Date:           Wed, 03 Apr 1996 13:49:09 +0100
From:           solymosi <solymosi@ss10.nuik.kfki.hu>
To:             eppstein@ics.uci.edu
Subject:        integral distances

In 1945 P.Erdoes published the "nice proof of Robert Israel". Integral
distances on the plane is a well-known research topic of the discrete
geometry.  A famous unsolved promlem: Is there a set of 7 points in general
position (no 3 in line, 4 in circle) and all distances are integer?

Best regards    J.Solymosi