## Task $9 / 85$

Given are two similar right-angled triangles, one inscribed and the other circumscribed to the same circle. Determine the minimal similarity ratio $k>1$!
1. Since $k>1$, $k$ is the ratio of the segment lengths of the circumscribed triangle to the corresponding segment lengths of the inscribed triangle.
2. If one of the cathetus lengths of the inscribed triangle approaches zero, the corresponding cathetus length of the circumscribed triangle approaches $r$ (where $r$ is the radius of the circle). Therefore, in this case, $k \rightarrow \infty$.
3. By symmetry, an extremum of $k$ occurs in the case of the isosceles triangle; because
4. this is the sought minimum.
5. For the height $h$ on the hypotenuse, in the case of the isosceles triangle,

a) for the inscribed triangle $h_{e}=4$

b) for the circumscribed triangle $h_{u}=r(1+\sqrt{2})$

(as one can easily see from a sketch using Thales' and Pythagoras' theorems). Thus, we obtain

$$
k=h_{u}: h_{e}=1+\sqrt{2} \approx 2.41 \ldots
$$