In a right triangle with a perimeter of 30, a circle is inscribed. One of the legs is divided by the point of tangency in the ratio $2: 3$, counting from the vertex of the right angle. Find the sides of the triangle.
Let $r$ be the radius of the inscribed circle. Then one of the legs is $5 r / 2$, and the area of the triangle is $15 r$. Therefore, the second leg is $2 \cdot 15: \frac{5}{2}=12$. Hence, the hypotenuse is $12+{ }^{3 r} / 2$, and the perimeter is $24+5 r=30$, so $r=2$.

## Answer

$5,12,13$.