Erdos-Anning Theorem

[Kalkewl8ter]

Can anyone explain this theorem??? It doesn't seem to make sense:

If an infinite number of points in the plane are all separated by integer distances, then all the points lie on a straight line.

[sql_lall]

It means that if there are heaps and heaps of points P₁, P₂, P₃ etc... all in 2 dimensions (like on a cartesian plane) and the distance between any two points P_i and P_j is always an integer value (like 1,2,3 etc) then the only way this can happen is if all the poinst P₁, P₂, P₃ etc... all line in a straight line.

[Kalkewl8ter]

But if you consider all the lattice points (points with integer coordinates), they are all separated by integer distances, and they certainly don't all lie on one line.

[bugzpodder]

are you sure Kalkewl8ter? tell me whats's the distance between (0,0) and (1,1)?

[SilverSprite]

ooo i think that he has got you there kalkewl8ter.

[Kalkewl8ter]

You stay out of this, Sprite! Ok, bugz, you're right...sqrt(2) usually isn't considered an integer...that makes more sense now. Thx bugz and sql_lall!

[SilverSprite]

No thank you for little ol me :'( you hurt my feelings!

[Kalkewl8ter]

Fine...I'll thank you too Sprite (who knows what for), just to make you happy.

[bugzpodder]

In article <[email protected]>,
Yaniv Shpilberg <[email protected]> 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 [email protected]
Department of Mathematics (604) 822-3629
University of British Columbia fax 822-6074
Vancouver, BC, Canada V6T 1Y4

[NotLKH]

Originally posted by bugzpodder
>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 are 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................................................................................................ ...............
Therefore there are only a finite number of points with integer distances
from A, B and C.

The "Therefore" is not that which was meant to be proven!

[bugzpodder]

he assumed that some of the points are NOT on the straight line, then then proved that the number of points are finite. although i agree that he didn't exactly prove that if they are on a straight line, there could be infinitely many points.

[Kalkewl8ter]

What's a degenerate?

[bugzpodder]

a degenerated polygon is a polygon with no area (or maybe at least one degree less). if it has no area then its vertices are probably on a line