Let's calculate the radius of the circle that touches the sides of the rhombus, if the diagonals of the rhombus are $2a$ and $2b$.
The diagonals of a rhombus are perpendicular to each other and bisect each other, so one side of the rhombus is

$$
c=\sqrt{a^{2}+b^{2}}
$$

If $r$ is the radius of the inscribed circle, then twice the area of the triangle determined by one side of the rhombus and half of each diagonal is:

$$
r c=a b
$$

and thus

$$
r=\frac{a b}{c}=\frac{a b}{\sqrt{a^{2}+b^{2}}}
$$

(Hirschfeld Gyula, Pécs.)