Loc(K)i and trigonometry

The equation of a circle that passes through $(0,1)$ and $(2,3)$ and has its centre on the equation of locus of point which is equidistant from $(-1,1)$ and $(4,-2)$ is $x^2+y^2 + 2gx + 2fy + c = 0$ .

If $\tan^{-1} (2g) + \tan^{-1} (2f) + \tan^{-1}(c) = \tan^{-1}(A)$ , where $A$ can be written in the form of $\dfrac ab$ , where $a$ and $b$ are coprime positive integers, find $a+b$ .