Mekkora a kör köré rajzolható egyenlőoldalú háromszög és a körbe rajzolható szabályos hatszög területeinek aránya?

What is the ratio of the areas of an equilateral triangle circumscribed around a circle and a regular hexagon inscribed in the same circle?
From the diagram, it can be seen that the area of $\triangle AOK$ is twice the area of $\triangle AOB$.

![](https://cdn.mathpix.com/cropped/2024_05_02_cff24be492005e084eedg-1.jpg?height=518&width=596&top_left_y=191&top_left_x=754)

However, the area $t_6$ of the inscribed regular hexagon is six times the area of $\triangle AOB$, and the area $T_3$ of the circumscribed regular triangle is six times the area of $\triangle AOK$, thus

$$
T_3 : t_6 = 2 : 1
$$

(Kirchknopf Ervin, Budapest.)

The problem was also solved by: Chambré M., Erdős V., Füstös P., Kiss J., Koffler B., Kovács Gy., Miklóssy K., Neumann L., Paunz A., Rajz E., Petrik S., Rosenthal M., Sárközy P., Szilárd V., Szőke D., Tóth B., Wáhl V.