A ¡§lattice point¡¨ has integer coordinates. Then, A = (m, n) is a lattice point if both
m and n are integers. Let¡¦s call a point P = (x, y) ¡§generic¡¦¡¦ if all the distances from P
to lattice points are different.
With some algebraic work, I checked that the point S = ( , ) is generic.
However, the point T = (0, £k) is not generic because it is equally distant from the lattice
points (1, 0) and (-1, 0).
„½ Is there some generic point with rational coordinates?
That is, if Q = (r, s) for rational numbers r and s, must there exist two lattice points
equidistant from Q ?
As a first step, show that R = ( , ) is not generic. (Find lattice points A, B equidistant from R.)
Can you use those ideas to answer the general question?
But how ???
To determine if there exists a generic point with rational coordinates, we need to find a point Q = (r, s) for rational numbers r and s such that there are no two lattice points equidistant from Q.
Let's consider the point R = (r, s) where r and s are both rational numbers. We want to show that this point is not generic by finding two lattice points A and B that are equidistant from R.
To do this, we need to find two lattice points A = (m1, n1) and B = (m2, n2) such that the distance between A and R is equal to the distance between B and R.
The distance formula between two points (x1, y1) and (x2, y2) is given by:
d = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Let's start by finding the distance between A = (m1, n1) and R = (r, s):
d1 = sqrt((r - m1)^2 + (s - n1)^2)
Now, we need to find the distance between B = (m2, n2) and R = (r, s):
d2 = sqrt((r - m2)^2 + (s - n2)^2)
Since we want these distances to be equal, we can set up the equation:
(r - m1)^2 + (s - n1)^2 = (r - m2)^2 + (s - n2)^2
Expanding both sides of the equation:
r^2 - 2rm1 + m1^2 + s^2 - 2sn1 + n1^2 = r^2 - 2rm2 + m2^2 + s^2 - 2sn2 + n2^2
Simplifying the equation:
m1^2 - m2^2 + n1^2 - n2^2 - 2(r(m1 - m2) + s(n1 - n2)) = 0
We need to find values for m1, n1, m2, n2, r, and s that satisfy this equation. One simple solution is to let A = (0, 0) and B = (1, 0). In this case, m1 = 0, n1 = 0, m2 = 1, n2 = 0, r = r, and s = s.
Substituting these values into the equation:
0^2 - 1^2 + 0^2 - 0^2 - 2(r(0 - 1) + s(0 - 0)) = 0
Simplifying further:
-1 + 2r = 0
This equation implies that r = 1/2. Therefore, the point R = (1/2, s) is not generic as there are two lattice points, A = (0, 0) and B = (1, 0), which are equidistant from R.
Based on this finding, we can conclude that there exist generic points with rational coordinates since we showed that the point R = (1/2, s) is not generic.