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.


Using Pythagoras theorem is not difficult to find that the radius $B$ of red arcs is $8$. It follows that $PQ=6$, $PR=9$ and $QR=5$. Apply now the cosine rule to triangle $PQR$ to find 
$$
\cos\phi={23\over27}.
$$

