Prove that the maximum number of points with rational coordinates on a circle whose center is `(sqrt(3),0)` is two.

Explanation

There cannot be three points on the circle with rational coordinates as for then the center of the circle, being the circumcenter of a triangle whose vertices have rational coordinates, must have rational coordinates ( since the coordinates will be obtained by solving two linear equations in x, y having rational coefficients ). But the point `( sqrt(3) , 0)` does not have rational coordinates.
Also, the equation of the circle is
`(x-sqrt(3))^(2)+y^(2)=r^(2)`
or `x=sqrt(3)+-sqrt(r^(2)-y^(2))`
For suitable r and x, where x is rational, y may have two rational values.
For example, `r=2,x=0,y=1,-1` satisfy `x=sqrt(3)+-sqrt(r^(2)-y^(2))` .
So, we get two points (0,1) and (0,-1) which have rational coordinates.