9. Two circles are said to be orthogonal if they intersect in two points, and their tangents at either point of intersection are perpendicular. Two circles $\omega_{1}$ and $\omega_{2}$ with radii 10 and 13 , respectively, are externally tangent at point $P$. Another circle $\omega_{3}$ with radius $2 \sqrt{2}$ passes through $P$ and is orthogonal to both $\omega_{1}$ and $\omega_{2}$. A fourth circle $\omega_{4}$, orthogonal to $\omega_{3}$, is externally tangent to $\omega_{1}$ and $\omega_{2}$. Compute the radius of $\omega_{4}$.
power $r_{3}^{2}$ to each of them. Thus $O_{3}$ is the radical center of $\omega_{1}, \omega_{2}, \omega_{4}$, which is equidistant to the three sides of $\triangle O_{1} O_{2} O_{4}$ and therefore its incenter.
For distinct $i, j \in\{1,2,4\}, \omega_{i} \cap \omega_{j}$ lies on the circles with diameters $O_{3} O_{i}$ and $O_{3} O_{j}$, and hence $\omega_{3}$ itself. It follows that $\omega_{3}$ is the incircle of $\triangle O_{1} O_{2} O_{4}$, so $8=r_{3}^{2}=\frac{r_{1} r_{2} r_{4}}{r_{1}+r_{2}+r_{4}}=\frac{130 r_{4}}{23+r_{4}} \Longrightarrow r_{4}=\frac{92}{61}$.
Comment: The condition $P \in \omega_{3}$ is unnecessary.