## Task 32/77

For a right-angled triangle, let the perimeter $U$ and the circumradius $r$ be known. Calculate the area $A$ from these!
We denote the two catheti with $k_{1}$ and $k_{2}$, and the hypotenuse with $h$. According to Thales' theorem, $h=2 r$, and according to the Pythagorean theorem,

$$
k_{1}^{2}+k_{2}^{2}=h^{2}=4 r^{2}
$$

Furthermore, $k_{1}+k_{2}=u-h=u-2 r$. Squaring this equation yields

$$
k_{1}^{2}+2 k_{1} k_{2}+k_{2}^{2}=u^{2}+4 u r+4 r^{2} \rightarrow 2 k_{1} k_{2}=u^{2}-4 u r
$$

Since $A=\frac{1}{2} k_{1} k_{2}$, it follows that $A=\frac{1}{4} u^{2}-u r=u\left(\frac{1}{4} u-r\right)$.