3-2. Find the geometric locus of the midpoints of segments with endpoints on two different non-intersecting circles, one lying outside the other.
Solution 2. Let \(S_{1}\) and \(S_{2}\) be the given circles, \(O_{1}\) and \(O_{2}\) their centers. Consider the circle \(S_{2}^{\prime}\), which is obtained from the circle \(S_{2}\) by translation by the vector \(\overrightarrow{O_{2} \overrightarrow{O_{1}}}\); the center of this circle coincides with the center of the circle \(S_{1}\). Let \(A_{1}\) be a point on the circle \(S_{1}\), \(A_{2}\) and \(A_{2}^{\prime}\) be points on the circles \(S_{2}\) and \(S_{2}^{\prime}\), corresponding to each other. If \(M\) is the midpoint of the segment \(A_{1} A_{2}\), and \(M^{\prime}\) is the midpoint of the segment \(A_{1} A_{2}^{\prime}\), then \(\overrightarrow{M M^{\prime}}=\frac{1}{2} \overrightarrow{O_{1} O_{2}}\). Therefore, we can consider the case when two concentric circles are given: the obtained geometric locus needs to be simply translated by the vector \(\frac{1}{2} \overrightarrow{O_{1} O_{2}}\).

Let \(O\) be the common center of two circles with radii \(R\) and \(r\), where \(R \geqslant r\). Fix a point \(A\) on the circle with radius \(r\) and consider the midpoints of all segments \(A B\), where the point \(B\) moves along the circle with radius \(R\). They form a circle, and its closest point to \(O\) is at a distance \(\frac{R-r}{2}\), while the farthest point is at a distance \(\frac{R+r}{2}\). If the point \(A\) moves along the entire circle, we obtain
a ring with an inner radius of \(\frac{R-r}{2}\) and an outer radius of \(\frac{R+r}{2}\) (if \(R=r\), then we get a circle, not a ring).