2025 AMC 10A Problem 22

A circle of radius r is surrounded by three circles, whose radii are 1, 2, and 3, all externally tangent to the inner circle and externally tangent to each other, as shown in the diagram below.

What is r?

\textbf{(A)} \dfrac{1}{4} \qquad \textbf{(B)} \dfrac{6}{23} \qquad \textbf{(C)} \dfrac{3}{11} \qquad \textbf{(D)} \dfrac{5}{17} \qquad \textbf{(E)} \dfrac{3}{10}

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.