If A is rational then A can be expressed as the ratio of integers (p,q), q<>0.
Plug that value in eq 1 to find B = q/(2q-p), p<>2q.
Then equating the two expressions for C after substituting in eq 2 and 3 gives p*(quadratic in p and q) = 514. Possible values for p are factors of 514, namely 1,2,257,514.