Let's examine the variations in the area of an isosceles trapezoid circumscribed around a given circle, and determine the case of the eminent value of the area.

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Let's examine the variations in the area of an isosceles trapezoid circumscribed around a given circle, and determine the case of the eminent value of the area.
I. solution. Let the isosceles trapezoid $ABCD$ be circumscribed around a circle of radius $r$. Let $AD = BC = x$, and $AB = a$ and $CD = d$; then the area $t$ of the trapezoid is:

$$
t = \frac{(a+d)}{2} \cdot 2r = (a+d)r
$$

However, since the trapezoid is also a tangential quadrilateral, we have

$$
(a+d) = 2x
$$

thus

$$
t = 2x \cdot r
$$

Since $x \geq 2r$, it follows that the area has a minimum value when $x = 2r$, and thus the minimum area is $4r^2$. Since $t = 2xr$, the change in $t$ is proportional to $x$.

(Pilczer Pál, Kaposvár.)

II. solution. For a circle of radius $r$, the height of any isosceles trapezoid circumscribed around it is always $2r$. Therefore, the area of the trapezoids depends only on the midline, and the one with the smallest area will have the smallest midline. The minimum midline is $2r$, and in this case, the trapezoid is a square.

(Bayer Béla, Losoncz.)

The problem was also solved by: Aczél F., Baranyó E., Bartók I., Blau A., Bogdán G., Dessauer A., Enyedi B., Haar A., Hirschfeld Gy., Kertész F., Lázár L., Papp F., Pintér M., Pivnyik I., Póka Gy., Raab R., Riesz K., Sasvári J., Schlesinger A., Szmodics H., Tóbiás J. L., Weisz P.