10.163. The sides of a triangle are 13, 14, and 15 cm. Find the ratio of the areas of the circumscribed and inscribed circles of this triangle.
Solution.

Let $r$ and $R$ be the radii of the inscribed and circumscribed circles. Then $r=\frac{S}{p}, R=\frac{a b c}{4 S} \cdot$ We find

\[
\begin{aligned}
& S=\sqrt{p(p-a)(p-b)(p-c)}= \\
& =\sqrt{21 \cdot 8 \cdot 7 \cdot 6}=84\left(\mathrm{~cm}^{2}\right)
\end{aligned}
\]

Therefore, $r=4 \mathrm{~cm}, R=\frac{65}{8} \mathrm{~cm}$, from which we obtain the desired ratio of areas: $\frac{\pi R^{2}}{\pi r^{2}}=\left(\frac{65}{32}\right)^{2}$.

Answer: $\left(\frac{65}{32}\right)^{2}$

![](https://cdn.mathpix.com/cropped/2024_05_22_fa9db84b44b98ec0a5c7g-632.jpg?height=470&width=500&top_left_y=81&top_left_x=702)

Fig. 10.137