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.
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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.
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 Pi and Pj 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.
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.
are you sure Kalkewl8ter? tell me whats's the distance between (0,0) and (1,1)?
ooo i think that he has got you there 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!
:( No thank you for little ol me :'( you hurt my feelings!:mad:
Fine...I'll thank you too Sprite (who knows what for), just to make you happy.:D
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
The "Therefore" is not that which was meant to be proven!Quote:
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.
:confused:
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.
What's a degenerate?
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