305. On the plane, there are three points. Three lines are drawn through these points, forming an equilateral triangle. Find the geometric locus of the centers of these triangles.
305. Let three given points form a triangle $ABC$. There are two families of equilateral triangles circumscribed about $\triangle ABC$. The first family is constructed as follows. Construct circles on the sides of the triangle such that the arcs of these circles, located outside the triangle, measure an angle of $4\pi / 3$. Take an arbitrary point $A_{1}$ on the circle constructed on $BC$, the line $A_{1}B$ intersects the circle constructed on $BA$ again at point $C_{1}$, and the line $A_{1}C$ intersects the circle constructed on $CA$ at point $B_{1}$. The triangle $A_{1}B_{1}C_{1}$ is one of the triangles of the first family. Denote by $E, F$, and $G$ the points of intersection of the angle bisectors of $\triangle A_{1}B_{1}C_{1}$ with the circles constructed on the sides of the given triangle. The points $E, F$, and $G$ are fixed ($E$ is the midpoint of the arc of the circle constructed on $BC$ and located on the same side of $BC$ as $\triangle ABC$). The points $E, F$, and $G$ are the centers of the equilateral triangles constructed on the sides of $\triangle ABC$ inside it. The triangle $EFG$ is equilateral (see problem II.304), its center coincides with the centroid of triangle $ABC$. The center of $\triangle A_{1}B_{1}C_{1}$ lies on the circumcircle of $EFG$, the square of the radius of this circle is (see the solution to problem II.304) $\frac{1}{9}\left(\frac{a^{2}+b^{2}+c^{2}}{2}-2 S \sqrt{3}\right)$, where $a, b, c$ are the sides, and $S$ is the area of $\triangle ABC$.

The second family of equilateral triangles circumscribed about $\triangle ABC$ is obtained by constructing circles on the sides of $\triangle ABC$ such that the arcs located outside $\triangle ABC$ are equal to $2\pi / 3$.

The desired geometric locus of points consists of two concentric circles, the centers of which coincide with the centroid of $\triangle ABC$, and the radii are equal to $\frac{1}{3} \sqrt{\frac{1}{2}\left(a^{2}+b^{2}+c^{2}\right) \pm 2 S \sqrt{3}}$.