52. $A B C D$ is a circumscribed quadrilateral, the lengths of the segments from $A$ to the points of tangency are $a$, the lengths of the segments from $C$ to the points of tangency are $b$. In what ratio does the diagonal $A C$ divide the diagonal $B D$?
52. Let $K, L, M$ and $N$ be the points of tangency of the sides $A B, B C, C D$ and $D A$ with the circle. Denote by $P$ the intersection point of $A C$ and $K M$. If $\widehat{A K M}=\varphi$, then $\widehat{K M C}=180^{\circ}-\varphi$. Therefore,

$\frac{|A P|}{|P C|}=\frac{S_{A K M}}{S_{K M C}}=\frac{\frac{1}{2}|A K| \cdot|K M| \sin \varphi}{\frac{1}{2}|K M| \cdot|M C| \sin \left(180^{\circ}-\varphi\right)}=\frac{|A K|}{|M C|}=\frac{a}{b}$.

But the line $N L$ will also divide $A C$ in the same ratio. Thus, the lines $A C, K M$ and $N L$ intersect at one point. Applying the same reasoning to the diagonal $B D$, we get that $B D$ also passes through the point $P$. The desired ratio is $a / b$.