Re: circle that passes through 3 lattice points on the plane

In article <d7neff$gpg$1@xxxxxxxxxxxxxxxxxxx>,
Keith A. Lewis <klewis@xxxxxxxxxxxxxxx> wrote:
>RainForestMan@xxxxxxxxx writes in article
><1117730062.187698.314620@xxxxxxxxxxxxxxxxxxxxxxxxxxxx> dated 2 Jun 2005
>09:34:22 -0700:
>>Curious problem that a friend asked: draw circle on 2D plane with
>>center on line x=r, where r is real, which passes through any 3 lattice
>>points (x,y are integers).


>Suppose there was an algorithm which would generate y and r for any given x,
>so that the circle of radius r centered at (x,y) would contain 3 lattice
>points.

>Let x=e. Find the lattice points.

>Set up equations for perpendicular bisectors of 2 sides of the lattice
>triangle. They will be equations containing only rational coefficients.

>Solve the simultaneous equations to find the value of x at intersection of
>the perpendicular bisectors, which is also the center of the circle. You
>now have a rational number which is equal to e. That is absurd; therefore
>there can be no such algorithm.

Another way to see this: three non-collinear points determine a circle.
There are only countably many triples of lattice points, therefore only
countably many circles that pass through three lattice points, and only
countably many x-coordinates of centres of such circles.

Robert Israel israel@xxxxxxxxxxx
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada

.

Relevant Pages

  • Re: circle that passes through 3 lattice points on the plane
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  • Re: lattice points in a circle
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  • Re: circle that passes through 3 lattice points on the plane
    ... so that the circle of radius r centered at would contain 3 lattice ... Set up equations for perpendicular bisectors of 2 sides of the lattice ... They will be equations containing only rational coefficients. ... which is also the center of the circle. ...
    (sci.math)
  • Re: lattice points in a circle
    ... lattice points inside any circle centered at the origin ... the circumference of a circle to the center is equidistant, ... By rotational symmetry (or alternatively, ... the quadrants is the same for each quadrant, ...
    (sci.math)