A point $(a,b)$ is called a rational point when $a$ and $b$ are both rationals.
Find the maximum possible rational points on a circle centred at $(\pi,2)$.
Suppose there exist $3$ (or more) such rational points. Consider a triangle formed by three points.
It is fairly easy to prove that such a "rational triangle" has a rational point as the circumcentre.
But this contradicts the fact that the actual circumcentre is, in fact, not a rational point. Hence, there can exist at most two rational points.
The existence of two rational points:
Let the radius of the circle be $r$. Then, the equation of the circle happens to be $$(x-\pi)^2+(y-2)^2=r^2=r_r+r_i$$ Where $r_r$ denotes the rational part of $r^2$ and the $r_i$ denotes the irrational part of $r^2$.
Let $(a,b)$ be a rational point on this circle.
It is fairly easy to arrive at $$r_r=a^2+(b-2)^2$$ And $$r_i=\pi^2-2\pi a$$
Now, for certain values of $r$, each of them generates a single value each of $r_r$ and $r_i$. And, each value of $r_i$ generates a single value of $a$, which on substituting in the second equation, gives two values of $b$.
Hence, there may exist two such points.
I somehow feel this proof of mine isn't exactly correct. Please correct me if I'm wrong. Other solutions are welcome as well.