We have taken a circle, $k_{1}$, in one of two parallel planes, and another circle, $k_{2}$, in the other. Point $M_{1}$ moves on circle $k_{1}$, and point $M_{2}$ moves on circle $k_{2}$. Determine the geometric locus of the midpoint of segment $M_{1} M_{2}$.
Solution. Let the center of the circle $k_{i}$ be denoted by $O_{i}$, and its radius by $r_{i}$ $(i=1,2)$, the midpoint of the segment $O_{1} O_{2}$ by $O$, and the plane passing through $O$ and parallel to the plane of $k_{1}$ by $S$. We can assume that $r_{1} \geqslant r_{2}$.

![](https://cdn.mathpix.com/cropped/2024_05_02_2fc09804127aea386ba4g-1.jpg?height=881&width=1022&top_left_y=264&top_left_x=538)

Figure 1

Let $F_{\alpha}$ be a fixed point on the circle $k_{1}$. Shrink the circle $k_{2}$ from the point $F_{\alpha}$ to half its size. Denote the resulting circle by $c_{\alpha}$, and its center by $K_{\alpha}$ (Figure 1). From the properties of central similarity, it follows that if $M_{2}$ is any point on the circle $k_{2}$, then the midpoint of the segment $F_{\alpha} M_{2}$ lies on $c_{\alpha}$, and conversely, if $P_{\alpha}$ is any point on the circle $c_{\alpha}$, then there is a point $M_{2}$ on $k_{2}$ such that $P_{\alpha}$ is the midpoint of the segment $F_{\alpha} M_{2}$. Furthermore, $K_{\alpha}$ is the midpoint of the segment $F_{\alpha} O_{2}$, so in the triangle $F_{\alpha} O_{1} O_{2}$, $K_{\alpha} O$ is the midline, hence $K_{\alpha} O = F_{\alpha} O_{1} / 2 = r_{1} / 2$, and the radius of the circle $c_{\alpha}$ is $r_{2} / 2$. If the point $F_{\alpha}$ traces the circle $k_{1}$, then the corresponding points $K_{\alpha}$, due to $K_{\alpha} O = r_{1} / 2$, trace a circle centered at $O$ with radius $r_{1} / 2$, and the circles $c_{\alpha}$ trace a circular ring centered at $O$ with outer radius $O K_{\alpha} + r_{2} / 2 = (r_{1} + r_{2}) / 2$ and inner radius $O K_{\alpha} - r_{2} / 2 = (r_{1} - r_{2}) / 2$ (Figure 2).

![](https://cdn.mathpix.com/cropped/2024_05_02_2fc09804127aea386ba4g-1.jpg?height=537&width=489&top_left_y=1621&top_left_x=799)

Thus, if $M_{1} M_{2}$ is any segment as described in the problem, then $M_{1}$ coincides with some $F_{\alpha}$, and consequently, the midpoint of $M_{1} M_{2}$ lies on the circle $c_{\alpha}$ corresponding to $F_{\alpha}$, and is therefore an element of the circular ring.

Conversely, if $T$ is any point in the circular ring, then there is a circle $c_{\alpha}$ (exactly 2 if $T$ is an interior point and exactly 1 if $T$ is a boundary point) that passes through $T$, so if we double the point $T$ from the point $F_{\alpha}$ corresponding to $c_{\alpha}$, the resulting point $T_{2}$ is a point on the circle $k_{2}$, and the midpoint of the segment $F_{\alpha} T_{2}$ is $T$.

Thus, the desired geometric locus is the circular ring in the plane $S$ centered at $O$ with outer radius $(r_{1} + r_{2}) / 2$ and inner radius $(r_{1} - r_{2}) / 2$.