The target board shown in the figure consists of three concentric circles and two perpendicular lines that pass through the centers of the circles. The 12 resulting sections have equal area. Determine the ratio of the radii of the three circles.
Let the radii of the circles be denoted by \( r_{1} < r_{2} < r_{3} \). According to the condition, the areas of the regions marked with thick lines on the diagram are equal, that is,

\[
t_{1} = \frac{r_{1}^{2} \pi}{4} = t_{2} = \frac{\left(r_{2}^{2} - r_{1}^{2}\right) \pi}{4} = t_{3} = \frac{\left(r_{3}^{2} - r_{2}^{2}\right) \pi}{4}
\]

From the first two equalities: \( r_{1}^{2} = r_{2}^{2} - r_{1}^{2} \), hence \( r_{2} = \sqrt{2} r_{1} \).

From the second and third equalities: \( r_{2}^{2} - r_{1}^{2} = r_{3}^{2} - r_{2}^{2} \).

Substituting \( r_{2}^{2} \) with the previously obtained \( 2 r_{1}^{2} \): \( r_{1}^{2} = r_{3}^{2} - 2 r_{1}^{2} \), hence \( r_{3} = \sqrt{3} r_{1} \). The ratio of the radii is:

\[
r_{1} : r_{2} : r_{3} = 1 : \sqrt{2} : \sqrt{3}
\]

![](https://cdn.mathpix.com/cropped/2024_05_02_f66254d0c43aa3a72c27g-1.jpg?height=1108&width=854&top_left_y=579&top_left_x=641)