From:israel@math.ubc.ca (Robert Israel)Newsgroups:sci.mathSubject:Re: Geometry problem (points on a plane)Date:14 Jan 1996 09:03:51 GMTOrganization: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.mathSubject:Re: Geometry problem (points on a plane)Date:14 Jan 1996 08:43:27 GMTOrganization: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 +0100From:solymosi <solymosi@ss10.nuik.kfki.hu>To:eppstein@ics.uci.eduSubject: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