## Task 19/84

Under what condition can a triangle be inscribed in two concentric circles $K_{1}$ and $K_{2}$ such that the larger circle is the circumcircle and the smaller circle is the incircle of the triangle?
Exactly when the circumcircle and incircle of a triangle are concentric, their centers, i.e., the intersection of the perpendicular bisectors and the intersection of the angle bisectors, coincide. From this, it follows that the triangle is equilateral.

Thus, the intersection of the medians also coincides with the center of the concentric circles. Since the medians bisect each other in a ratio of 1:2, the ratio of the radii of the incircle and circumcircle, $\rho$ and $r$ respectively, is $\rho: r = 0.5$ and $r = 2 \rho$.

Thus, the desired condition is found: the radius of the larger circle must be twice the radius of the smaller circle. This condition is both necessary and sufficient.