Example 14 Consider two concentric circles on the same plane with radii $R$ and $r (R > r)$. Let $P$ be a fixed point on the circumference of the smaller circle, and $B$ be a moving point on the circumference of the larger circle. The line $BP$ intersects the larger circle at another point $C$. The line $l$ passing through point $P$ and perpendicular to $BP$ intersects the smaller circle at another point $A$ (if $l$ is tangent to the smaller circle at $P$, then $A = P$).
(1) Find the set of values taken by the expression $BC^2 + CA^2 + AB^2$;
(2) Find the locus of the midpoint of segment $AB$.
(1) As shown in Figure $21-12$, construct the rectangle $B C C^{\prime} B^{\prime}$. Clearly, $B^{\prime} C^{\prime}$ passes through point $A$. Let the other intersection point of $B^{\prime} C^{\prime}$ with the smaller circle be $A^{\prime}$. Since the distance from $O$ to $B C$ is equal to the distance from $O$ to $A A^{\prime}$, we have
$$
\begin{array}{l}
z_{B}+z_{r}+z_{A}+z_{A}=0, \text { i.e., } z_{A}+z_{B}+z_{C}=-z_{A^{\prime}} . \text { Therefore, } \\
B C^{2}+C A^{2}+A B^{2} \\
=\left(z_{C}-z_{B}\right)\left(\overline{z_{C}}-\overline{z_{B}}\right)+\left(z_{A}-z_{C}\right)\left(\overline{z_{A}}-\overline{z_{C}}\right)+\left(z_{B}-z_{A}\right)\left(\overline{z_{B}}-\overline{z_{A}}\right) \\
=4 R^{2}+2 r^{2}-\left(z_{B} \cdot \overline{z_{A}}+z_{A} \cdot \overline{z_{B}}+z_{C} \cdot \overline{z_{B}}+z_{B} \cdot \overline{z_{C}}+z_{A} \cdot \overline{z_{C}}+z_{C} \cdot \overline{z_{A}}\right) \\
=6 R^{2}+3 r^{2}-\left(z_{A}+z_{B}+z_{C}\right)\left(\overline{z_{A}}+\overline{z_{B}}+\overline{z_{C}}\right) \\
=6 R^{2}+2 r^{2} \text {. (constant) }
\end{array}
$$

When $A$ coincides with $P$, it is easy to verify that $B C^{2}+C A^{2}+A B^{2}$ also equals $6 R^{2}+2 r^{2}$. Therefore, the set of values for the expression is $\left\{6 R^{2}+2 r^{2}\right\}$.
(2) Clearly, $P B B^{\prime} A$ is also a rectangle, so the midpoint $Q$ of $A B$ is also the midpoint of $P B^{\prime}$, hence $z_{B^{\prime}}+z_{P}=2 z_{Q}$, which implies $z_{H^{\prime}}=2 z_{Q}-z_{P}$.
Since $B^{\prime}$ lies on the larger circle, we have $\left|z_{B^{\prime}}\right|=R$, thus
$\left|2 z_{Q}-z_{P}\right|=R$, or $\left|z_{Q}-\frac{z_{P}}{2}\right|=\frac{R}{2}$.
This indicates that the locus is a circle with the midpoint of segment $O P$ as its center and $\frac{R}{2}$ as its radius.