75. Two regular hexagons. Find, without using radicals, the ratio of the areas of two regular hexagons, one of which is inscribed in a given circle, and the other is circumscribed around it.
75. Consider a regular inscribed hexagon, whose vertices divide the corresponding sides of the circumscribed hexagon in half. Connect the center of the circle with the vertices of the inscribed hexagon, and then connect the centers of the resulting equilateral triangles with the vertices of these triangles. As a result, we obtain 24 equal triangles, which together form the

![](https://cdn.mathpix.com/cropped/2024_05_21_2b40d984baf9da275c68g-099.jpg?height=526&width=466&top_left_y=1099&top_left_x=795)

circumscribed hexagon. Eighteen of them are located inside the inscribed hexagon. Therefore, the ratio of the areas of the inscribed and circumscribed hexagons is $18: 24=3: 4$.

$[M$. M., 35, 70 (March 1962).]