Lattice Point Equidistant Proof

[hermes7]

Hello all,
Could someone help me out with this problem? I tried using circle geometries, perpendicular bisectors, and some more pure algebra. Nothing has been "unifying." Here is the problem:
Is it possible to have a point Q=(r,s), where r and s are rational, where the point Q is not equidistant from ANY two lattice points? where a lattice point is a point of integer x and y coordinates.
Thank you in advance!

 

[Perfection]

This is for nonzero cases only, zero cases one can pretty quickly come up with equidistant points.

I may have screwed up some algebra in here but I'm confident the framework will work:


For any rational r and s for some integers a,b,c,d it is true that: r=a/b and s=c/d

It's pretty obvious Q lies on the line xb/a=yd/c

or:
y=(bc/ad)x

O=(0,0) is another point that lies on that line.

Taking the negative recipricol of the line we find the perpendicular line:

y=-ad/bc(x)

of which these two integer solutions can be found (there are of course, an infinite number of integer solutions):

T=(ad,-bc) and U=(-ad,bc)

TOQ and TUQ both form congruent triangles with the hypotenuse being TQ and UQ, and thus T and U are equidistant from Q