10.093. The perimeter of a right-angled triangle is $2 p$, and the hypotenuse is $c$. Determine the area of the circle inscribed in the triangle.
Solution.

$\triangle A B C$ is a right triangle (Fig. 10.90); $A B=x, B C=y$. It is known that $2 p=x+y+c$. Therefore, $x+y=2 p-c$. The radius of the inscribed circle $r=\frac{x+y-c}{2}=\frac{2 p-c-c}{2}=p-c$. The area of the circle is found using the formula $S=\pi r^{2}=\pi(p-c)^{2}$.

Answer: $\pi(p-c)^{2}$.