Consider two spheres $\left(O_{1}, r ; O_{2}, r\right)$, a point $A$, and a distance $d$. The radii of the spheres are equal. The triangle $ABC$ is to be constructed, with one vertex $(A)$ given. On the side $a$ of the triangle, the spheres determine chords of length $d$, whose midpoints are $B$ and $C$. The side $a$ forms an angle $\alpha$ with the central line of the spheres. What is the geometric locus of the lines $a$? What can $d$ and $\alpha$ be?
Let the midpoint of a chord of length $d$ of a sphere with radius $r$ be $F$, then $\overline{O_{1} F}=\sqrt{r^{2}-\frac{d^{2}}{4}}=\varrho$. Thus, the geometric locus of all points $F$ is a sphere with center $O_{1}$ and radius $\varrho$. The side $AB$ of $\triangle ABC$ therefore touches the spheres centered at $O_{1}$ and $O_{2}$ with radius $\varrho$. The points of tangency are $B$ and $C$. This also implies that

$$
0<d<2 r
$$

![](https://cdn.mathpix.com/cropped/2024_05_02_0f6ddfa90c7721e5c720g-1.jpg?height=331&width=333&top_left_y=450&top_left_x=885)

Cut the spheres with a plane $S$ that is parallel to $\left|O_{1} O_{2}\right|$. This plane cuts two equal-radius circles from the sphere. Let the angle between the common internal tangents of these circles be $2\alpha$, then the tangents form an angle $\alpha$ with the central line of the spheres. If we rotate these tangents around the central line as the axis, the tangents remain tangents. The geometric locus of these tangents is called a one-sheet hyperboloid of revolution. (If we rotate a hyperbola around its imaginary axis, we get this surface.) If $\alpha=0$, the hyperboloid becomes the common tangent cylinder of the spheres. If $\alpha$ is maximal, the hyperboloid becomes the common tangent cone of the spheres. During this process, the radius of the hyperboloid's throat circle decreases from $\varrho$ to 0.

If the plane $S$ is not parallel to the central line, the two circles cut out are not equal. Among the common tangents, there can be one that forms an angle $\alpha$ with the central line, but this tangent already belongs to the mentioned hyperboloid.

Pfeifer Sándor (Vörösmarthy Mihály g. VIII. r. o. Budapest.)