10.253. For a triangle with sides 26, 28, and 30 cm, find the product of the radii of the circumscribed and inscribed circles.
## Solution.

Let the sides of the triangle be: $a=26 \text{ cm}, b=28$ cm, $c=30$ cm. The semi-perimeter of the triangle $p=\frac{26+28+30}{2}=42$ cm. Let $S$ be the area of the triangle. The radius of the circumscribed circle $R=\frac{a b c}{4 S}$. The radius of the inscribed circle $r=\frac{S}{p}$. Then $R \cdot r=\frac{a b c}{4 S} \cdot \frac{S}{p}=\frac{a b c}{4 p}=130 \text{ cm}^2$.

Answer: $130 \text{ cm}^2$.

![](https://cdn.mathpix.com/cropped/2024_05_21_3edb7be591bff54ce350g-0750.jpg?height=524&width=497&top_left_y=709&top_left_x=430)

Fig. 10.59