There is a sphere $(O, r)$, a line $(g)$, and two distances $(a, b)$. Construct a line $(e)$ that is parallel to the first image plane and intersects $g$ at $X$, the sphere at $Y, Z$ such that $\overline{X Y}=a$ and $\overline{Y Z}=b$.
If a chord $b=\overline{Y Z}$ of a sphere is of length $b$, then this chord is tangent to the sphere centered at $O$ with radius $\varrho=\sqrt{r^{2}-\frac{b^{2}}{4}}$. The points $X$ on this chord for which $\overline{X Y}=a$ are points on a sphere centered at $O$ with radius $R=\sqrt{\varrho^{2}+\left(\frac{b}{2}+a\right)^{2}}$ (or $R=\sqrt{\varrho^{2}+\left(\frac{b}{2}-a\right)^{2}}$).

![](https://cdn.mathpix.com/cropped/2024_05_02_db16dde41698143aeb1eg-1.jpg?height=444&width=423&top_left_y=431&top_left_x=840)

A plane parallel to the first image plane is constructed from the intersection point $(X)$ of $g$ and the sphere with radius $R$; this plane intersects the sphere with radius $\varrho$ in a circle; drawing tangents from $X$ to this circle yields the desired lines.

There can be four points $X$ on the spheres with radius $R$; from each of these points, two tangents can be drawn to the sphere with radius $\varrho$, so the number of solutions is at most eight.

Béla Szabó (Hunyadi Mátyás g. VIII. o. Kőszeg.)