Let's construct a rhombus, given one side and knowing that one side is the geometric mean of the diagonals.
If $a$ is the given side, $m$ is the height, $r$ and $s$ are the diagonals, then

$$
\frac{r s}{2}=a m
$$

But

$$
r s=a^{2}, \quad \text { so } \quad m=\frac{a}{2} \text {. }
$$

Therefore, the construction is as follows: With the given base $AB = a$, draw a parallel line at a distance of $\frac{a}{2}$. This parallel line intersects the circles drawn from $A$ and $B$ with radius $a$ at points $C, C_{1}$ and $D, D_{1}$. $ABCD$ and $ABC_{1}D_{1}$ are the desired rhombuses.

(Basch Rudolf, Budapest.)