At some point, the circle will pass through a grid point for the first time. No other grid point may lie on the same circle. From that moment on and until the circle passes through a grid point for the second time, the circle will contain exactly 1 grid point. Afterwards and until it passes through another grid point for the third time, it will contain only 2 grid points, and so on. Which implies rationality of 2.
Field is a set closed under arithmetic operations.
But this is not true in general. Z 2 that contains no other elements but 0 and 1, does not, in particular, contain 2. All extension fields of Q naturally contain 2. Now, if two numbers a and b are equidistant from 2, then they lie on the opposite sides from 2 with the latter serving as their average. Assume, e.g., that a 1 is not 0.