Angles between vectors of center of two incircles I have two two incircle between rectangle and two 
quadrilateral circlein. It's possible to determine exact value of $\phi,$ angles between vectors of center of two circles.


There are constraints that $A$ and $B$ must fullfill, i.e., the following system of equations coming from Pythagoras theorem applied to certain right triangles :
$$\begin{cases}(B-2)^2+2^2=(A/2)^2\\(B-1)^2+(A/2)^2=(B+1)^2\end{cases}$$
giving $A=8 \sqrt{2}$ and $B=8$.
If now we take equations as in the partial solution you gave (a good idea) :
$$\tan(\theta) = \dfrac{2}{A/2} = \dfrac{\sqrt{2}}{4}\tag{1}$$
and 
$$\tan(\phi + \theta) = \dfrac{B-1}{A/2} = \dfrac{7}{4 \sqrt{2}}\tag{2}$$
Equation (2) can also be written :
$$\dfrac{\tan(\phi) + \tan(\theta)}{1-\tan(\phi)\tan(\theta)} = \dfrac{7\sqrt{2}}{8}\tag{3}$$
Let $T:=\tan(\phi)$. (3) is equivalent to :
$$\dfrac{T + \sqrt{2}/4}{1-T\sqrt{2}/4} = \dfrac{7\sqrt{2}}{8}\tag{4}$$
giving $T=\dfrac{10 \sqrt{2}}{23}$. Therefore 


$\phi=\arctan(\dfrac{10 \sqrt{2}}{23})\approx 31.5863 \ \text{degrees}.$


