On the sides of the right triangle, outside are constructed regular nonagons, which are constructed on one of the catheti and on the hypotenuse, with areas equal to $1602$  $cm^2$ and $2019$  $cm^2$, respectively. What is the area of the nonagon that is constructed on the other cathetus of this triangle?

(Vladislav Kirilyuk)
1. Let the right triangle have sides \(a\) (hypotenuse), \(b\) (one cathetus), and \(c\) (the other cathetus). According to the Pythagorean theorem, we have:
   \[
   a^2 = b^2 + c^2
   \]

2. Denote \(A_1\), \(A_2\), and \(A_3\) as the areas of the regular nonagons constructed on the sides \(a\), \(b\), and \(c\) respectively. Given:
   \[
   A_1 = 2019 \, \text{cm}^2 \quad \text{and} \quad A_2 = 1602 \, \text{cm}^2
   \]

3. Since all regular nonagons are similar to each other, the areas of the nonagons are proportional to the squares of the sides of the triangle. Therefore, we have:
   \[
   \frac{A_1}{a^2} = \frac{A_2}{b^2} = \frac{A_3}{c^2}
   \]

4. From the above proportionality, we can write:
   \[
   \frac{A_1}{a^2} = \frac{A_2}{b^2} = \frac{A_3}{c^2} = k \quad \text{(some constant \(k\))}
   \]

5. Using the Pythagorean theorem \(a^2 = b^2 + c^2\), we can express \(k\) as:
   \[
   k = \frac{A_1}{a^2} = \frac{A_2 + A_3}{b^2 + c^2} = \frac{A_2 + A_3}{a^2}
   \]

6. Since \(k = \frac{A_1}{a^2}\), we have:
   \[
   A_1 = A_2 + A_3
   \]

7. Substituting the given values:
   \[
   2019 = 1602 + A_3
   \]

8. Solving for \(A_3\):
   \[
   A_3 = 2019 - 1602 = 417 \, \text{cm}^2
   \]

The final answer is \(\boxed{417 \, \text{cm}^2}\)