420. Using only a compass, it is required to construct a fourth proportional segment to three given segments $a, b, c$.
420. We describe two concentric circles from the center $O$ and make $A A_{1}=B B_{1}$ (Fig. 92). Then $\triangle A O A_{1}=\triangle B O B_{1}$, therefore $\triangle A O B \backsim \triangle A_{1} O B_{1}$, from which: $\triangle O: O A_{1}=A B: A_{1} B_{1}$

Thus, if we describe concentric circles with radii $a$ and $b$, make the segment $A B$ equal to $c$ and the segments 
$A A_{1}, B B_{1}$ equal to the same arbitrary segment, then the segment $A_{1} B_{1}$ will be the desired one.

Note. If the segment $c$ is so large that it cannot serve as a chord of the circle ($O A$), then we take the segment $\frac{c}{n}$, where $n$ is an integer. In this case, not $x$ but $\frac{x}{n}$ is constructed.