Example 17 As shown in Figure 1.4.19, a semicircle is constructed inside trapezoid $ABCD$, tangent to the upper base and the two non-parallel sides, with its diameter lying on the lower base. If $AB=2, CD=3$, find the length of the lower base $BC$.

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Complete the circle, draw a tangent line $E F$ of circle $O$ parallel to $A D$, intersecting the extensions of $A B, D C$ at $E, F$ respectively. It is easy to prove that $B C$ is the midline of trapezoid $A E F D$, then
$$
A E=2 A B=4, D F=2 D C=6
$$

Since trapezoid $A E F D$ is circumscribed about circle $O$, then
$$
\begin{array}{c}
A D+E F=A E+D F=10 \\
B C=\frac{1}{2}(A D+E F)=5
\end{array}
$$