Let's construct an isosceles triangle, given the radius of the circumscribed circle $r$ and the radius of the inscribed circle $\varrho$.
We denote the center of the circumscribed circle by $K$ and the center of the inscribed circle by $O$. It is evident that if we construct the distance $K O=d$, then constructing the triangle becomes straightforward.

![](https://cdn.mathpix.com/cropped/2024_05_02_07cdf70ebca22f6e2d05g-1.jpg?height=648&width=596&top_left_y=267&top_left_x=754)

To construct the distance $d$, we can use Euler's well-known theorem (see the 2nd note attached to problem 1897/2 in part I of Mathematical Competition Theorems, page 41), which states that

$$
d^{2}=r^{2}-2 r \varrho=r(r-2 \varrho) .
$$

Thus, $d$ can be constructed as the geometric mean between $r$ and $r-2 \varrho$. If $r>2 \varrho$, there is always one and only one solution for $d$. For the constructed $d$, there are always 2 triangles (see the figure) depending on whether we take the endpoint $A_{1}$ or $A^{2}$ of the diameter $O K$ as the vertex.

Tamás Endrödy (Bp., III., Árpád g. I. o. t.)