1.23. The sides of a triangle are 13, 14, and 15 cm. Find the ratio of the areas of the circumscribed and inscribed circles of the triangle.
1.23. The radii \( r \) and \( R \) of the inscribed and circumscribed circles are related to the area \( S \) of the triangle by formulas (1.4) and (1.5): \( r = \frac{S}{p}, R = \frac{a b c}{4 S} \). The area of the triangle can be found using Heron's formula:

\[
S = \sqrt{p(p-a)(p-b)(p-c)} = \sqrt{21 \cdot 8 \cdot 7 \cdot 6} = 84 \, \text{cm}^2
\]

Thus, \( r = 4 \, \text{cm}, R = \frac{65}{8} \, \text{cm} \), from which we can find the desired ratio of the areas: \(\frac{\pi R^2}{\pi r^2} = \left(\frac{65}{32}\right)^2\).

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

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