2022 AMC 12A Problem 25

A circle with integer radius r is centered at (r, r). Distinct line segments of length c_i connect points (0, a_i) to (b_i, 0) for 1 \le i \le 14 and are tangent to the circle, where a_i, b_i, and c_i are all positive integers and c_1 \le c_2 \le \cdots \le c_{14}. What is the ratio \frac{c_{14}}{c_1} for the least possible value of r?

\textbf{(A)} ~\frac{21}{5} \qquad\textbf{(B)} ~\frac{85}{13} \qquad\textbf{(C)} ~7 \qquad\textbf{(D)} ~\frac{39}{5} \qquad\textbf{(E)} ~17

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