18.2 (IMO 29) Consider two concentric circles 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 $C$, and $BP \perp PA$ intersects the smaller circle at another point $A$. (If $PA$ is tangent to the smaller circle at $P$, then consider $A=P$). (1) Find the set of values taken by the expression $BC^2 + CA^2 + AB^2$; (2) Find the locus of the midpoint $Q$ of the segment $AB$.
18. 2 Using the complex number method. (1) Construct a rectangle $B C C^{\prime} B^{\prime}$ inscribed in the large circle, obviously $B^{\prime} C^{\prime}$ passes through point $A$. Let $B^{\prime} C^{\prime}$ intersect the small circle at another point $A^{\prime}$. Since the distance from $O$ to $B C$ = the distance from $O$ to $A A^{\prime}$, $\therefore \frac{Z_{B}+Z_{C}}{2}=-\frac{Z_{A}+Z_{A}^{\prime}}{2}, Z_{C} \cdot \overline{Z_{C}}=R^{2}, Z_{A} \cdot \overline{Z_{A}}=r^{2}$. 
Thus, we calculate $B C^{2}+C A^{2}+A B^{2}=\left(Z_{C}-Z_{B}\right)\left(\overline{Z_{C}}-\overline{Z_{B}}\right)+\cdots=\cdots=6 R^{2}+2 r^{2}$.
(2) The midpoint of $A B$ is also the midpoint of $P B^{\prime}$, $Z_{B^{\prime}}+Z_{P}=2 Z_{Q}$, thus $\left|Z_{Q}-\frac{Z_{P}}{2}\right|=\left|\frac{Z_{B^{\prime}}}{2}\right|=\frac{R}{2}$, 
the locus is a circle with the midpoint of segment $O P$ as its center and $\frac{R}{2}$ as its radius.