Example 3. Five circles are tangent to each other in sequence, and also tangent to lines $a$, $b$. If the diameters of the smallest and largest circles are 18 and 32, respectively, find the diameter of the circle in the middle. (34th American High School Mathematics Competition)
Analyzing the original problem involving the relationship between five circles, it is not easy to grasp. First, consider the case of three circles:

Let three circles $O_{1}, O_{2}, O_{3}$ be tangent to each other in sequence. If their radii are $r_{1}, r_{2}, r_{3}$ respectively, then by the similarity of triangles (as shown in the figure above), we have
$$
\frac{r_{2}+r_{1}}{r_{2}-r_{1}}=\frac{r_{3}+r_{2}}{r_{3}-r_{2}} \text {. }
$$

Using the property of the ratio of the sum and difference, we get $\frac{r_{2}}{r_{1}}=\frac{r_{3}}{r_{2}}$.
By analogy with the above process, it is easy to deduce the case for five circles.
If their radii are $r_{1}, r_{2}, r_{3}, r_{4}, r_{5}$ respectively, then we have
$$
\frac{r_{2}}{r_{1}}=\frac{r_{3}}{r_{2}}=\frac{r_{4}}{r_{3}}=\frac{r_{5}}{r_{4}}=k \text {, }
$$

which means $2 r_{i+1}=2 r_{1} k^{i}(i=1,2,3,4)$.
Therefore, $\left(2 r_{3}\right)^{2}=4 r_{1}^{2} k^{4}=2 r_{1} \cdot 2 r_{5}$.
Thus, $2 r_{3}=24$.