The base of an isosceles trapezoid $AB = a$ and the side parallel to it $DC = b$. Draw a circle that is tangent to the sides of the trapezoid and connect the points of tangency $M$ and $N$ on the sides $AD$ and $BC$. What is the length of $MN$?
Let the intersection point of the diagonals of the trapezoid be $E$. Since $\triangle BNE \sim \triangle BCD$:

$$
CD: \frac{MN}{2} = CB: BN
$$

but

$$
CB = CN + NB = \frac{b}{2} + \frac{a}{2} = \frac{a+b}{2} \text{ and } BN = \frac{a}{2}
$$

thus

$$
b: \frac{MN}{2} = \frac{a+b}{2}: \frac{a}{2}
$$

from which

$$
MN = \frac{2ab}{a+b}
$$

(Mór Kelemen, VII. o. t., Györ.)

The problem was also solved by: Déri Zs., Deutsch I., Haar A., Kertész G., Popoviciu M., Raab R.