Given one side of a triangle, as well as the radius of the circumscribed circle and the radius of the inscribed circle. What are the lengths of the other two sides? (Numerical example: $a=79, R=65, r=28$ units.) (Try to solve without using trigonometric tables!)
We denote the center of the circumscribed circle of the written, circumscribed circle by $O$, the midpoint of $BC$ by $A_{1}$, the projection of $O$ onto $BC$ by $A_{2}$, the intersection of $AO$ with $BC$ and the circumscribed circle by $A_{3}$ and $D$, the point opposite to $D$ by $E$. It is known that

$$
B D=O D=C D
$$

since, for example, the angle $\angle BOD$ and the angle $\angle OBD$ are each equal to half the sum of the angles at $A$ and $B$ of the triangle $ABC$. We can assume that $ABc$ under the condition; if we omit this condition, the roles of $b$ and $c$ can naturally be swapped. The $d$ in (4) is the root of the equation (1). From this, we get $e$ based on (2), assuming that a positive number stands on the right side of (2). Therefore, the number of possible solutions can be any number between 0 and 4.

In the numerical case mentioned, (1) represents the equation

$$
d^{2}-130 d+39^{2}=0
$$

from which $d_{1}=13, d_{2}=117$. Accordingly, based on (2),

$$
e_{1}^{2}=78^{2}-4 \cdot 28 \cdot 54=36
$$

and a negative value is obtained for $e_{2}^{2}$. Finally, from (4), we get

$$
b_{1}=126, \quad c_{1}=120
$$

Remark. In the publication, the value of $a$ was given as 79 due to a typographical error (the correct text appeared in the Russian and English translations). From this, we get the values $b_{1}=127.4$, and $c_{1}=117.0$ ( $e_{2}^{2}$ is also negative here).