89. A circle of radius $r$ touches internally a circle of radius $R$. $A$ is the point of tangency. A line perpendicular to the line of centers intersects one circle at point $B$, and the other at point $C$. Find the radius of the circle circumscribed about triangle $A B C$.
89. Let $M$ be the point of intersection of the line $CB$ with the line of centers of the given circles. Denote $|AM|=x, \widehat{ACB}=\varphi$, $|AB|^2=2rx, \quad|AC|^2=2Rx, \quad \sin \varphi=\frac{x}{|AC|}$. If $\rho$ is the radius of the circumcircle of $\triangle ABC$, then $\rho=\frac{|AB|}{2 \sin \varphi}=\frac{|AB| \cdot|AC|}{2 x}=$ $=\frac{\sqrt{2rx} \cdot \sqrt{2Rx}}{2 x}=\sqrt{Rr}$.

Answer: $\sqrt{\overline{Rr}}$.